Nonlinear nonnested 2-D spline approximation
aa r X i v : . [ m a t h . C A ] J un NONLINEAR NONNESTED 2-D SPLINE APPROXIMATION
M. LIND AND P. PETRUSHEV
Abstract.
Nonlinear approximation from regular piecewise polynomials (splines)supported on rings in R is studied. By definition a ring is a set in R obtainedby subtracting a compact convex set with polygonal boundary from anothersuch a set, but without creating uncontrollably narrow elongated subregions.Nested structure of the rings is not assumed, however, uniform boundedness ofthe eccentricities of the underlying convex sets is required. It is also assumedthat the splines have maximum smoothness. Bernstein type inequalities forthis sort of splines are proved which allow to establish sharp inverse estimatesin terms of Besov spaces. Introduction
Nonlinear approximation from piecewise polynomials (splines) in dimensions d > L p . While thistheory is simple and well understood in the univariate case, it is underdevelopedand challenging in dimensions d > L p (Ω), 0 < p < ∞ , fromregular piecewise polynomials in R or on compact subsets of R with polygonalboudaries. Our goal is to obtain complete characterization of the rates of approxi-mation (the associated approximation spaces). To describe our results we begin byintroducing in more detail our Setting and approximation tool.
We are interested in approximation in L p ,0 < p < ∞ , from the class of regular piecewise polynomials S ( n, k ) of degree k − k ≥ n rings. More specifically, with Ω beinga compact polygonal domain in R or Ω = R , we denote by S ( n, k ) the set of allpiecewise polynomials S of the form(1.1) S = n X j =1 P j R j , S ∈ W k − (Ω) , P j ∈ Π k , where R , . . . , R n are rings with disjoint interiors. Here Π k denotes the set of allalgebraic polynomials of degree k − R and S ∈ W k − (Ω) means that all partialderivatives ∂ α S ∈ C (Ω), | α | ≤ k −
2. In the case where k = 1, these are simplypiecewise constants.A set R ⊂ R is called a ring if R is a compact convex set with polygonal bound-ary or the difference of two such sets. All convex sets we consider are with uniformly Date : June 16, 2015.2010
Mathematics Subject Classification.
Key words and phrases.
Spline approximation, Nonlinear approximation, Besov spaces. bounded eccentricity and we do not allow uncontrollably narrow elongated subre-gions. For the precise definitions, see § § Motivation.
Our setting would simplify considerably if the rings R j in (1.1) arereplaced by regular convex sets with polygonal boundaries or simply triangles. How-ever, this would restrict considerably the approximation power of our approximationtool. As will be seen the piecewise polynomials as defined above with rings allow tocapture well point singularities of functions, which is not quite possible with piece-wise polynomials over convex polygonal sets. The idea of using rings has alreadybeen utilized in [1].It is important to point out that our tool for approximation although regularis highly nonlinear. In particular, we do not assume any nested structure of therings involved in the definition of different splines S in (1.1). The case of approxi-mation from splines over nested (anisotropic) rings induced by hierarchical nestedtriangulations is developed in [2, 4].Denote by S kn ( f ) p the best L p -approximation of a function f ∈ L p (Ω) from S ( n, k ). Our goal is to completely characterize the approximation spaces A αq , α > < q ≤ ∞ , defined by the (quasi)norm k f k A αq := k f k L p + (cid:16) ∞ X n =1 (cid:0) n α S kn ( f ) p (cid:1) q n (cid:17) /q with the ℓ q -norm replaced by the sup-norm if q = ∞ . To this end we utilizethe standard machinery of Jackson and Bernstein estimates. The Besov spaces B s,kτ := B s,kττ with 1 /τ = s/ /p naturally appear in our regular setting. TheJackson estimate takes the form: For any f ∈ B s,kτ (1.2) S kn ( f ) p ≤ cn − s/ | f | B s,kτ . For k = 1 , k >
2. Estimate (1.2) implies the direct estimate(1.3) S kn ( f ) p ≤ cK ( f, n − s/ ) , where K ( f, t ) = K ( f, t ; L p , B s,kτ ) is the K -functional induced by L p and B s,kτ .It is a major problem to establish a companion inverse estimate. The followingBernstein estimate would imply such an estimate:(1.4) | S − S | B s,kτ ≤ cn s/ k S − S k L p , S , S ∈ S ( n, k ) . However, as is easy to show this estimate is not valid. The problem is that S − S may have one or more uncontrollably elongated parts such as [0 ,ε ] × [0 , with small ε , which create problems for the Besov norm, see Example 3.2 below.The main idea of this article is to replace (1.4) by the Bernstein type estimate:(1.5) | S | λB s,kτ ≤ | S | λB s,kτ + cn λs/ k S − S k λL p , λ := min { τ, } , where 0 < s/ < k − /p . This estimate leads to the needed inverse estimate:(1.6) K ( f, n − s/ ) ≤ cn − s/ (cid:16) n X ν =1 ν (cid:2) ν s/ S ν ( f ) p (cid:3) λ + k f k λp (cid:17) /λ . ONLINEAR SPLINE APPROXIMATION 3
In turn, this estimate and (1.3) yield a characterization of the associated approxi-mation spaces A αq in terms of real interpolation spaces:(1.7) A αq = ( L p , B s,kτ ) αs ,q , < α < s, < q ≤ ∞ . See e.g. [3, 6].A natural restriction on the Bernstein estimate (1.5) is the requirement that thesplines S , S ∈ S ( n, k ) have maximum smoothness. For instance, if we considerapproximation from piecewise linear functions S ( k = 2), it is assumed that S iscontinuous. As will be shown in Example 4.4 estimate (1.5) is no longer valid fordiscontinuous piecewise linear functions.The proof of estimate (1.5) is quite involved. To make it more understandablewe first prove it in § § d = 2.However, there is a great deal of geometric arguments involved in our proofs andto avoid more complicated considerations we consider only spline approximation indimension d = 2 here. Useful notation.
Throughout this article we shall use | G | to denote the Lebesguemeasure a set G ⊂ R , G ◦ , G , and ∂G will denote the interior, closure, and bound-ary of G , d ( G ) will stand for the diameter of G , and G will denote the characteristicfunction of G . If G is finite, then G will stand for the number of elements of G .If γ is e polygon in R , then ℓ ( γ ) will denote its length. Positive constants willbe denoted by c , c , c ′ , . . . and they may vary at every occurrence. Some impor-tant constants will be denoted by c , N , β, . . . and they will remain unchangedthroughout. The notation a ∼ b will stand for c ≤ a/b ≤ c .2. Background
Besov spaces.
Besov spaces naturally appear in spline approximation.The Besov space B s,kτ = B s,kττ , s > k ≥
1, 1 /τ := s/ /p is defined as theset of all functions f ∈ L τ (Ω) such that(2.1) | f | B s,kτ := (cid:16) Z ∞ (cid:2) t − s ω k ( f, t ) τ (cid:3) τ dtt (cid:17) /τ < ∞ , with the usual modification when q = ∞ . Here ω k ( f, t ) τ := sup | h |≤ t k ∆ kh f ( · ) k L τ (Ω) with ∆ kh f ( x ) := P kν =0 ( − k + ν (cid:0) kν (cid:1) f ( x + νh ) if the segment [ x, x + kh ] ⊂ Ω and∆ kh f ( x ) := 0 otherwise.Observe that for the standard Besov spaces B spq with s > ≤ p, q ≤ ∞ the norm is independent of the index k > s . However, in the Besov spaces above ingeneral τ <
1, which changes the nature of the Besov space and k should no longerbe directly connected to s . For more details, see the discussion in [4], pp. 202-203.2.2. Nonlinear spline approximation in dimension d = 1. For comparison,here we provide a brief account of nonlinear spline approximation in the univariatecase. Denote by S kn ( f ) p the best L p -approximation of f ∈ L p ( R ) from the set S ( n, k ) of all picewise polynomials S of degree < k with n + 1 free knots. Thus, S ∈ S ( n, k ) if S = P nj =1 P j I j , where P j ∈ Π k and I j , j = 1 , . . . , n , are arbitrarycompact intervals with disjoin interiors and ∪ j I j is an interval. No smoothness of S is required. M. LIND AND P. PETRUSHEV
Let s >
0, 0 < p < ∞ , and 1 /τ = s + 1 /p . The following Jackson and Bernsteinestimates hold (see [5]): If f ∈ L p ( R ) and n ≥
1, then(2.2) S kn ( f ) p ≤ cn − s | f | B s,kτ and(2.3) | S | B s,kτ ≤ cn s k S k L p , S ∈ S ( n, k ) , where c > s and p . These estimates implydirect and inverse estimates which allow to characterise completely the respectiveapproximation spaces. For more details, see [5] or [3, 6].Several remarks are in order. (1) Above no smoothness is imposed on the piece-wise polynomials from S ( n, k ). The point is that the rates of approximation fromsmooth splines are the same as for nonsmooth splines. A key observation is thatin dimension d = 1 the discontinuous piecewise polynomials are infinitely smoothwith respect to the Besov spaces B s,kτ . This is not the case in dimensions d > s >
0. (3) If S , S ∈ S ( n, k ), then S − S ∈ S (2 n, k ), and hence (2.3)is sufficient for establishing the respective inverse estimate. This is not true in themultivariate case and one needs estimates like (1.4) (if valid) or (1.5) (in our case).(4) There is a great deal of geometry involved in multivariate spline approximation,while in dimension d = 1 there is none.2.3. Nonlinear nested spline approximation in dimension d = 2. The ratesof approximation in L p , 0 < p < ∞ , from splines generated by multilevel anisotropicnested triangulations in R are studied in [2, 4]. The respective approximationspaces are completely characterized in terms of Besov type spaces (B-spaces) de-fined by local piecewise polynomial approximation. The setting in [2, 4] allows todeal with piecewise polynomials over triangulations with arbitrarily sharp angles.However, the nested structure of the underlying triangulations is quite restrictive.In this article we consider nonlinear approximation from nonnested splines, but ina regular setting. It is a setting that frequently appears in applications.3. Nonlinear approximation from piecewise constants
Setting.
Here we describe all components of our setting, including the re-gion Ω where the approximation will take place and the tool for approximation weconsider.
The region Ω.
We shall consider two scenarios for Ω: (a) Ω = R or (b) Ω is acompact polygonal domain in R . More explicitly, in the second case we assumethat Ω can be represented as the union of finitely many triangles with disjointinteriors obeying the minimum angle condition. Therefore, the boundary ∂ Ω of Ωis the union of finitely many polygons consisting of finitely many segments (edges).
The approximation tool.
To describe our tool for approximation we first intro-duce rings in R . Definition 3.1.
We say that R ⊂ R is a ring if R can be represented in the form R = Q \ Q , where Q , Q satisfy the following conditions:(a) Q ⊂ Q or Q = ∅ ;(b) Each of Q and Q is a compact regular convex set in R whose boundary isa polygon consisting of no more than N ( N fixed) line segments. Here a compact ONLINEAR SPLINE APPROXIMATION 5 convex set Q ⊂ R is deemed regular if Q has a bounded eccentricity , that is, thereexists balls B , B , B j = B ( x j , r j ), such that B ⊂ Q ⊂ B and r ≤ c r , where c > R contains no uncontrollably narrow and elongated subregions, which isspecified as follows: Each edge (segment) E of the boundary of R can be subdividedinto the union of at most two segments E , E ( E = E ∪ E ) with disjoint (onedimensional) interiors such that there exist triangles △ with a side E and adjacentto E angles of magnitude β , and △ with a side E and adjacent to E angles ofmagnitude β such that △ j ⊂ R , j = 1 ,
2, where 0 < β ≤ π/ Figure 1.
Left: a ring R = Q \ Q . Right: R with the trianglesassociated to the segments of ∂R . Remark.
Observe that from the above definition it readily follows that for anyring R in R (3.1) | R | ∼ d ( R ) with constants of equivalence depending only on the parameters N , c , and β .In the case when Ω is a compact polygonal domain in R , we assume that thereexists a constant n ≥ n rings R j with disjoint interiors: Ω = ∪ n j =1 R j . If Ω = R , then we set n := 1.We now can introduce the class of regular piecewise constants. Case 1:
Ω is a compact polygonal domain in R . We denote by S ( n,
1) ( n ≥ n )the set of all piecewise constants S of the form(3.2) S = n X j =1 c j R j , c j ∈ R , where R , . . . , R n are rings with disjoint interiors such that Ω = ∪ nj =1 R j . Case 2:
Ω = R . In this case we denote by S ( n,
1) the set of all piecewise constantfunctions S of the form (3.2), where R , . . . , R n are rings with disjoint interiors suchthat the support R := ∪ nj =1 R j of S is a ring in the sense of Definition 3.1. Example.
A simple case of the above setting is when Ω = [0 , and the rings R are of the form R = Q \ Q , where Q , Q are dyadic squares in R . These kindof dyadic rings have been used in [1]. M. LIND AND P. PETRUSHEV
A bit more general is the setting when Ω is a regular rectangle in R withsides parallel to the coordinate axes or Ω = R and the rings R are of the form R = Q \ Q , where Q , Q are regular rectangles with sides parallel to the coor-dinate axes, and no narrow and elongated subregions are allowed in the sense ofDefinition 3.1 (c).Clearly the set S ( n,
1) in nonlinear since the rings { R j } and the constants { c j } in (3.2) may vary with S .We denote by S n ( f ) p the best approximation of f ∈ L p (Ω) from S ( n,
1) in L p (Ω),0 < p < ∞ , i.e.(3.3) S n ( f ) p := inf S ∈S ( n, k f − S k L p . Besov spaces.
When approximating in L p , 0 < p < ∞ , from piecewise constantsthe Besov spaces B s, τ with 1 /τ = s/ /p naturally appear. In this section, weshall use the abbreviated notation B sτ for these spaces.3.2. Direct and inverse estimates.
The following
Jackson estimate is quite easyto establish (see [4]): If f ∈ B sτ , s >
0, 1 /τ := s/ /p , 0 < p < ∞ , then f ∈ L p (Ω)and(3.4) S n ( f ) p ≤ cn − s/ | f | B sτ for n ≥ n , where c > s, p and the structural constants N , c , and β of the setting.This estimate leads immediately to the following direct estimate: If f ∈ L p (Ω),then(3.5) S n ( f ) p ≤ cK ( f, n − s/ ) , n ≥ , where K ( f, t ) is the K -functional induced by L p and B sτ , namely,(3.6) K ( f, t ) = K ( f, t ; L p , B sτ ) := inf g ∈ B sτ {k f − g k p + t | g | B sτ } , t > . The main problem here is to prove a matching inverse estimate. Observe thatthe following Bernstein estimate holds: If S ∈ S ( n, n ≥ n , and 0 < p < ∞ ,0 < s < /p , 1 /τ = s/ /p , then(3.7) | S | B sτ ≤ cn s/ k S k L p , where the constant c > s, p , and the structural constants of thesetting (see the proof of Theorem 4.5). The point is that this estimate does notimply a companion to (3.5) inverse estimate. The following estimate would implysuch an estimate:(3.8) | S − S | B sτ ≤ cn s/ k S − S k L p , S , S ∈ S ( n, . However, as the following example shows this estimate is not valid.
Example 3.2.
Consider the function f := [0 ,ε ] × [0 , , where ε > ω ( f, t ) ττ ∼ (cid:26) t if t ≤ εε if t > ε and hence for 0 < s < /p and 1 /τ = s/ /p we have | f | B sτ ∼ ε /τ − s ∼ ε /p − s/ ∼ ε − s/ k f k L p , implying | f | B sτ c k f k L p , ONLINEAR SPLINE APPROXIMATION 7 since ε can be arbitrarily small. It is easy to see that one comes to the sameconclusion if f is the characteristic function of any convex elongated set in R . Thepoint is that if S , S ∈ S ( n, S − S can be a constant multiple of thecharacteristic function of one or more elongated convex sets in R and, therefore,estimate (3.8) is in general not possible.We overcome the problem with estimate (3.8) by establishing the following mainresult: Theorem 3.3.
Let < p < ∞ , < s < /p , and /τ = s/ /p . Then for any S , S ∈ S ( n, , n ≥ n , we have | S | B sτ ≤ | S | B sτ + cn s/ k S − S k L p , if τ ≥ , and (3.9) | S | τB sτ ≤ | S | τB sτ + cn τs/ k S − S k τL p , if τ < , (3.10) where the constant c > depends only on s, p , and the structural constants N , c ,and β . In the limiting case we have this result:
Theorem 3.4. If S , S ∈ S ( n, , n ≥ n , then (3.11) | S | BV ≤ | S | BV + cn / k S − S k L , where the constant c > depends only on the structural constants N , c , and β . We next show that estimates (3.9)-(3.10) and (3.11) imply the desired inverseestimate.
Theorem 3.5.
Let p , s , and τ be as in Theorem 3.3 and set λ := min { τ, } . Thenfor any f ∈ L p (Ω) we have (3.12) K ( f, n − s/ ) ≤ cn − s/ (cid:16) n X ℓ = n ℓ (cid:2) ℓ s/ S ℓ ( f ) p (cid:3) λ + k f k λp (cid:17) /λ , n ≥ n . Here K ( f, t ) = K ( f, t ; L p , B sτ ) is the K -functional defined in (3 . and c > is aconstant depending only on s, p , and the structural constants of the setting.Furthermore, in the case when p = 2 and s = 1 estimated (3 . holds with B sτ replaced by BV and λ = 1 .Proof. Let τ ≤ f ∈ L p (Ω). We may assume that for any n ≥ n thereexists S n ∈ S ( n,
1) such that k f − S n k p = S n ( f ) p . Clearly, for any m ≥ m with m := ⌈ log n ⌉ we have(3.13) K ( f, − ms/ ) ≤ k f − S m k p + 2 − ms/ | S m | B sτ . We now estimate | S m | τB sτ using iteratively estimate (3.10). For ν ≥ m + 1 we get | S ν | τB sτ ≤ | S ν − | τB sτ + c τνs/ k S ν − S ν − k τp ≤ | S ν − | τB sτ + c τνs/ (cid:0) k f − S ν k τp + k f − S ν − k τp (cid:1) ≤ | S ν − | τB sτ + c ′ τνs/ S ν − ( f ) τp . From (3.7) we also have | S m | B sτ ≤ c k S m k p ≤ c k f − S m k p + c k f k p = cS m ( f ) p + c k f k p . M. LIND AND P. PETRUSHEV
Summing up these estimates we arrive at | S m | τB sτ ≤ c m − X ν = m τνs/ S ν ( f ) τp + c k f k τp . Clearly, this estimate and (3.13) imply (3.12). The proof in the cases λ > p = 2, s = 1, and B sτ replaced by BV is the same. (cid:3) Observe that the direct and inverse estimates (3.5) and (3.9)-(3.11) imply imme-diately a characterization of the approximation spaces A αq associated with piecewiseconstant approximation from above just like in (1.7).3.3. Proof of Theorems 3.3.
We shall only consider the case when Ω ⊂ R is acompact polygonal domain. The proof in the case Ω = R is similar.Assume S , S ∈ S ( n, n ≥ n . Then S , S can be represented in the form S j = P R ∈R j c R R , where R j is a set of at most n rings in the sense of Definition 3.1with disjoint interiors and such that Ω = ∪ R ∈R j R , j = 1 , U the set of all maximal compact connected subsets U of Ω obtainby intersecting all rings from R and R with the property U ◦ = U (the closure ofthe interior of U is U ). Here U being maximal means that it is not contained inanother such set.Observe first that each U ∈ U is obtained from the intersection of exactly tworings R ′ ∈ R and R ′′ ∈ R , and is a subset of Ω with polygonal boundary ∂U consisting of ≤ N line segments (edges). Secondly, the sets in U have disjointinteriors and Ω = ∪ U ∈U U .It is easy to see that there exists a constant c > U ≤ cn.
Indeed, each U ∈ U is obtain by intersecting two rings, say, R ′ ∈ R and R ′′ ∈ R .If | R ′ | ≤ | R ′′ | , we associate R ′ to U , and if | R ′ | > | R ′′ | we associate R ′′ to U .However, because of condition (b) in Definition 3.1 every ring R from R or R canbe intersected by only finitely many, say, N ⋆ rings from R or R , respectively, ofarea ≥ | R | . Here N ⋆ depends only on the structural constants N and c . Also,the intersection of any two rings may have only finitely many, say N ⋆⋆ , connectedcomponents. Therefore, every ring R ∈ R ∪ R can be associated to only N ⋆ N ⋆⋆ sets U ∈ U , which implies (3.14) with c = 2 N ⋆ N ⋆⋆ .Example 3.2 clearly indicates that our main problem will be in dealing with sets U ∈ U or parts of them with diam much larger than their area. To overcome theproblem with these sets we shall subdivide each of them using the following Construction of good triangles.
According to Definition 3.1, each segment E from the boundary of every ring R ∈ R j can be subdivided into the union of atmost two segments E , E ( E = E ∪ E ) with disjoint interiors such that thereexist triangles △ with a side E and adjacent to E angles of size β > △ with a side E and adjacent to E angles β such that △ ℓ ⊂ R , ℓ = 1 ,
2. We nowassociate with △ the triangle ˜ △ ⊂ △ with one side E and adjacent to E anglesof size β/
2; just in the same way we construct the triangle ˜ △ ⊂ △ with a side E .We proceed in the same way for each edge E from ∂R , R ∈ R j , j = 1 ,
2. We denoteby T R the set of all triangles ˜ △ , ˜ △ associated in the above manner with all edges E from ∂R . We shall call the triangles from T R the good triangles associated with R . Observe that due to △ , △ ⊂ R for the triangles from above it readily follows ONLINEAR SPLINE APPROXIMATION 9 that the good triangles associated with R ( R ∈ R j , j = 1 ,
2) have disjoint interiors;this was the purpose of the above construction.
Figure 2.
The ring from Figure 1 with good triangles (angles = β/ E from ∂R that has been subdivided into E and E as above we shall consider E and E as segments from ∂R in place of E .We denote by E R the set of all (new) segments from ∂R . We now associate witheach E ∈ E R the good triangle which has E as a side and denote it by △ E .To summarize, we have subdivided the boundary ∂R of each ring R ∈ R j , j = 1 ,
2, into a set E R of segments with disjoint interiors ( ∂R = ∪ E ∈E R E ) andassociated with each E ∈ E R a good triangle △ E ⊂ R such that E is a side of △ E and the triangles {△ E } E ∈E R have disjoint interiors. In addition, if E ′ ⊂ E isa subsegment of E , then we associate with E ′ the triangle △ E ′ ⊂ △ E with oneside E ′ and the other two sides parallel to the respective sides of △ E ; hence △ E ′ is similar to △ E . We shall call △ E ′ a good triangle as well. Subdivision of the sets from U . We next subdivide each set U ∈ U by using thegood triangles constructed above. Suppose U ∈ U is obtained from the intersectionof rings R ′ ∈ R and R ′′ ∈ R . Then the boundary ∂U of U consists of two sets ofsegments E ′ U and E ′′ U , where each E ∈ E ′ U is a segment or subsegment of a segmentfrom E R ′ and each E ∈ E ′′ U is a segment or subsegment of a segment from E R ′′ .Clearly, ∂U = ∪ E ∈E ′ U ∪E ′′ U E and the segments from E ′ U ∪ E ′′ U have disjoint interiors.For each E ∈ E ′ U ∪ E ′′ U we denote by △ E the good triangle with a side E , definedabove.Consider the collection of all sets of the form △ E ∩ △ E with the properties:(a) E ∈ E U ′ , E ∈ E U ′′ .(b) There exists an isosceles trapezoid or an isosceles triangle T ⊂ △ E ∩ △ E such that its two legs (of equal length) are contained in E and E , respectively,and its height is not smaller than its larger base. We assume that T is a maximalisosceles trapezoid (or triangle) with these properties. Observe that it may happenthat there are no trapezoids like this.We denote by T U the set of all trapezoids as above. We also denote by A U the set of all maximal compact connected subsets A of U \ ∪ T ∈T U T ◦ . Clearly, U = ∪ T ∈T U T ∪ A ∈A U A and the sets in T U ∪ A U have disjoint interiors.The following lemma will be instrumental for the rest of this proof. Lemma 3.6.
There exist constants c ⋆ > and β ⋆ > depending only on N , c ,and β , such that if A ∈ A U for some U ∈ U , then d ( A ) ≤ c ⋆ | A | , and there existsa triangle △ ⊂ A whose minimum angle is ≥ β ⋆ such that | A | ≤ c ⋆ |△| .Proof. There are several cases to be considered, depending on the shapes of U and A . Since in each case the argument will be geometric we shall illustrate thegeometry involved in a number of figures. Case 1.
Let U be the closure of a connected subset of Q \ Q , where Q , Q are convex polygonal sets just as in Definition 3.1. This may happen if rings R , R of the form R = ˜ Q \ Q and R = Q \ ˜ Q intersect as illustrated in Figure 3. Q Q R ˜ Q ˜ Q R Figure 3.
One configuration for U = Q \ Q .Denote γ := ∂U ∩ ∂Q and γ := ∂U ∩ ∂Q . Thus γ is the “inner” part ofthe boundary ∂U of U , which is a subset of ∂Q , and γ is the “outer” part of ∂U , which is a subset of ∂Q . The polygons γ and γ may have two points ofintersection, one point of intersection or none. With no loss of generality we shallassume that γ and γ have two points of intersection just as in Figure 3.Each γ and γ is a polygon consisting of no more than N segments. For any suchsegment E we denote by △ E the good triangle with a side E whose construction isdescribed above. The set U with its good triangles is displayed in Figure 4. Figure 4.
The set U with the good triangles associated to it. ONLINEAR SPLINE APPROXIMATION 11
Let E ⊂ γ and E ⊂ γ be two edges of γ such that △ E U , △ E U , andeither E and E have a common end point, say v or E and E are connected bya chain of segments I , . . . , I m , I j ⊂ γ , such that △ I j ⊂ U , j = 1 , . . . , m . Denoteby v the common end point of E and I , and by v the common end point of E and I m . See Figure 5 below p p p p v I j v q q q q E ˜ E E ˜ E γ γ T T Figure 5.
The case m ≥ E ⊂ γ and ˜ E ⊂ γ be edges of γ such that △ ˜ E U , △ ˜ E U , and △ ˜ E ∩ △ E = ∅ , △ ˜ E ∩ △ E = ∅ . Assume that there exist isosceles trapezoids T ⊂ △ E ∩ △ ˜ E , T ⊂ △ E ∩ △ ˜ E , and T , T are maximal. Let p , p , p , p bethe vertices of T and q , q , q , q be the vertices of T as shown in Figure 5.Let η be the part of γ enclosed by the points p and q , and let η be the partof γ between the points p and q .Consider now the polygonal set A ⊂ U bounded by η , η and the segments[ p , p ] and [ q , q ]. We next show that(3.15) d ( A ) ≤ c | A | for some constant c > Q , Q are convex sets with uniformly bounded eccentricities it iseasy to see that ℓ ( η ) ≤ cℓ ( η ). Consider the case when E and E are connectedby segments I , . . . , I m . Denote I := [ p , v ] and △ I := [ p , v , p ] the trianglewith vertices p , v , p . Also, denote I m +1 := [ v , q ] and set △ I m +1 := [ v , q , q ].Now, let j max := arg max ≤ j ≤ m +1 |△ I j | . Then d ( A ) ≤ c max { ℓ ( η ) , ℓ ( η ) } ≤ cℓ ( η ) ≤ c | I j max | ≤ c |△ j max | ≤ c | A | as claimed. Here △ j max is the triangle whose existence is claimed in Lemma 3.6.Just as above we establish estimate (3.15) for a set A as above where the rolesof γ and γ are interchanged. Case 2.
Let U be the closure of the Q ∩ Q , where Q , Q are convex polygonalsets as in Figure 6. ˜ Q Q Q ˜ Q Figure 6.
One configuration for U = Q ∩ Q .Then ∂U consists of two polygonal curves γ and γ with two points of intersec-tion, each having no more than N segments. The argument is now simpler thanthe one in Case 1. Case 3.
It may also happen that we have a situation just as in Case 1, wherein addition the set ˜ Q intersects Q \ Q (see Figure 3) or the situation is as inCase 2, where ˜ Q or ˜ Q or both ˜ Q and ˜ Q intersect Q ∩ Q (see Figure 6). Weonly consider in detail the first scenario, the second one is similar.With the notation from Case 1, let Q \ Q = ∅ and assume that ˜ Q intersects Q \ Q . Let U be the closure of a connected subset of Q \ ( Q ∪ ˜ Q ). Then U issubdivided by applying the procedure described above.Several subcases are to be considered here. Case 3 (a).
If ˜ Q and the good triangles attached to ˜ Q are contained in someset A ∈ A U from Case 1, then apparently | A | ≤ c | A \ ˜ Q | and hence d ( A ) ≤ c | A | ≤ c | A \ ˜ Q | .T ′ T T ˜ Q γ Figure 7.
The case when ˜ Q ⊂ U and ˜ Q is close to γ . Case 3 (b).
The most dangerous situation is when ˜ Q is contained in U andan edge of ˜ Q is located close to the inner part γ of ∂U as shown in Figure 7.However, in this situation a good triangle attached to ˜ Q would intersect γ (seeFigure 7) and would create a trapezoid in T U . ONLINEAR SPLINE APPROXIMATION 13
The set ˜ Q may intersect Q \ Q in various other ways. The point is thatafter subtracting from U the trapezoids T ∈ T U constructed above the remainingconnected components A ∈ A U cannot be uncontrollably elongated. We omit thefurther details.Also, an important point is that by construction ˜ Q cannot intersect any trape-zoid from T U . (cid:3) In what follows we shall need the following obvious property of the trapezoidsfrom T . Property 3.7.
There exists a constant 0 < ˆ c < L = [ v , v ] is oneof the legs of a trapezoid T ∈ T and T ⊂ △ E ∩ △ E (see the construction oftrapezoids), then for any x ∈ L with | x − v j | ≥ ρ , j = 1 ,
2, for some ρ > B ( x, ˆ cρ ) ⊂ △ E ∪ △ E . Moreover, if D = [ v , v ] is one of the bases of thetrapezoid T , then for any x ∈ D with | x − v j | ≥ ρ , j = 1 ,
2, for some ρ > B ( x, ˆ cρ ) ⊂ △ E ∩ △ E .Let A := ∪ U ∈U A U and T := ∪ U ∈U T U . We have Ω = ∪ A ∈A A ∪ T ∈T T and,clearly, the sets in A ∪ T have disjoint interiors. From these we obtain the followingrepresentation of S ( x ) − S ( x ) for x ∈ Ω which is not on any of the edges:(3.16) S ( x ) − S ( x ) = X A ∈A c A A ( x ) + X T ∈T c T T ( x ) , where c A and c T are constants.For future reference, we note that(3.17) A ≤ cn and T ≤ cn.
These estimates follow readily by (3.14) and the fact that the number of edges ofeach U ∈ U is ≤ N .Let 0 < s/ < /p and assume τ ≤
1. Fix t > h ∈ R with norm | h | ≤ t . Write ν := | h | − h and assume ν =: (cos θ, sin θ ), − π < θ ≤ π .We shall frequently use the following obvious identities: If S is a constant ona measurable set G ⊂ R and H ⊂ G ( H measurable), then(3.18) k S k L τ ( G ) = | G | /τ − /p k S k L p ( G ) = | G | s/ k S k L p ( G ) and(3.19) k S k L τ ( H ) = ( | H | / | G | ) /τ k S k L τ ( G ) . We next estimate k ∆ h S k τL τ ( G ) − k ∆ h S k τL τ ( G ) for different subsets G of Ω. Case 1.
Let T ∈ T be such that d ( T ) > t/ ˆ c with ˆ c the constant from Property 3.7.Denote T h := { x ∈ Ω : [ x, x + h ] ⊂ Ω and [ x, x + h ] ∩ T = ∅} . We now estimate k ∆ h S k τL τ ( T h ) − k ∆ h S k τL τ ( T h ) .We may assume that T is an isosceles trapezoid contained in △ E ∩ △ E , where △ E j ( j = 1 ,
2) is a good triangle for a ring R j ∈ R j , and T is positioned so thatits vertices are the points: v := ( − δ / , , v := ( δ / , , v := ( δ / , H ) , v := ( − δ / , H ) , where 0 ≤ δ ≤ δ and H > δ . Let L := [ v , v ] and L := [ v , v ] be the twoequal (long) legs of T . We assume that L ⊂ E and L ⊂ E . We denote by D := [ v , v ] and D := [ v , v ] the two bases of T . Set V T := { v , v , v , v } . SeeFigure 8 below. I tT v v v v B v B v L L △ E △ E Figure 8.
A trapezoid T .Furthermore, let γ ≤ π/ D and L and assume that ν =: (cos θ, sin θ ) with θ ∈ [ γ, π ]. The case θ ∈ [ − γ,
0] is just the same. The casewhen θ ∈ [0 , γ ] ∪ [ − π, − γ ] is considered similarly.Denote B v := B ( v, t/ ˆ c ), v ∈ V T , A tT := (cid:8) A ∈ A : d ( A ) > t and A ∩ ( T + B (0 , t )) = ∅ (cid:9) , A tT := (cid:8) A ∈ A : d ( A ) ≤ t and A ∩ ( T + B (0 , t )) = ∅ (cid:9) and T tT := (cid:8) T ′ ∈ T : d ( T ′ ) > t/ ˆ c and T ′ ∩ ( T + B (0 , t )) = ∅ (cid:9) , T tT := (cid:8) T ′ ∈ T : d ( T ′ ) ≤ t/ ˆ c and T ′ ∩ ( T + B (0 , t )) = ∅ (cid:9) . Case 1 (a).
If [ x, x + h ] ∈ △ ◦ E , then ∆ h S ( x ) = 0 because S is a constant on △ E . Hence no estimate is needed. Case 1 (b).
If [ x, x + h ] ⊂ ∪ v ∈V T B v , we estimate | ∆ h S ( x ) | using the obviousinequality(3.20) | ∆ h S ( x ) | ≤ | ∆ h S ( x ) | + | S ( x ) − S ( x ) | + | S ( x + h ) − S ( x + h ) | . ONLINEAR SPLINE APPROXIMATION 15
Clearly, the contribution of this case to estimating k ∆ h S k τL τ ( T h ) −k ∆ h S k τL τ ( T h ) is ≤ c X v ∈V T X A ∈A tT k S − S k τL τ ( B v ∩ A ) + c X v ∈V T X T ′ ∈T tT k S − S k τL τ ( B v ∩ T ′ ) + c X v ∈V T X A ∈ A tT k S − S k τL τ ( B v ∩ A ) + c X v ∈V T X T ′ ∈ T tT k S − S k τL τ ( B v ∩ T ′ ) ≤ X A ∈A tT ct d ( A ) τs − k S − S k τL p ( A ) + X T ′ ∈T tT ct τs/ d ( T ′ ) τs/ − k S − S k τL p ( T ′ ) + X A ∈ A tT cd ( A ) τs k S − S k τL p ( A ) + X T ′ ∈ T tT cd ( T ′ ) τs k S − S k τL p ( T ′ ) . Here we used these estimates, obtained using Lemma 3.6 and (3.18) or/and (3.19):(1) If A ∈ A tT and v ∈ V T , then k S − S k τL τ ( B v ∩ A ) = ( | B v | / | A | ) k S − S k τL τ ( A ) ≤ ct d ( A ) − k S − S k τL τ ( A ) ≤ ct d ( A ) τs − k S − S k τL p ( A ) . (2) If T ′ ∈ T tT and δ ( T ′ ) > t/ ˆ c with δ ( T ′ ) being the maximal base of T ′ , thenfor any v ∈ V T we have k S − S k τL τ ( B v ∩ T ′ ) = ( | B v | / | T ′ | ) k S − S k τL τ ( T ′ ) ≤ ct | T ′ | τs/ − k S − S k τL p ( T ′ ) ≤ ct δ ( T ′ ) τs/ − d ( T ′ ) τs/ − k S − S k τL p ( T ′ ) ≤ ct τs/ d ( T ′ ) τs/ − k S − S k τL p ( T ′ ) , where we used that τ s/ <
1, which is equivalent to s < s + 2 /p .(3) If T ′ ∈ T tT and δ ( T ′ ) ≤ t/ ˆ c , then for any v ∈ V T k S − S k τL τ ( B v ∩ T ′ ) = ( | B v ∩ T ′ | / | T ′ | ) k S − S k τL τ ( T ′ ) = | B v ∩ T ′ || T ′ | τs/ − k S − S k τL p ( T ′ ) ≤ ctδ ( T ′ )[ δ ( T ′ ) d ( T ′ )] τs/ − k S − S k τL p ( T ′ ) = ctδ ( T ′ ) τs/ d ( T ′ ) τs/ − k S − S k τL p ( T ′ ) ≤ ct τs/ d ( T ′ ) τs/ − k S − S k τL p ( T ′ ) . (4) If A ∈ A tT , then k S − S k τL τ ( B v ∩ A ) ≤ k S − S k τL τ ( A ) ≤ c | A | τs/ k S − S k τL p ( A ) ≤ cd ( A ) τs k S − S k τL p ( A ) . (5) If T ′ ∈ T tT , then k S − S k τL τ ( B v ∩ T ′ ) ≤ k S − S k τL τ ( T ′ ) ≤ c | T ′ | τs/ k S − S k τL p ( T ′ ) (3.21) ≤ cd ( T ′ ) τs k S − S k τL p ( T ′ ) . Case 1 (c).
If [ x, x + h ]
6⊂ ∪ v ∈V T B v and [ x, x + h ] intersects D or D , then δ > t/ ˆ c > t or δ > t and hence [ x, x + h ] ⊂ △ E ∩ △ E , which implies∆ h S ( x ) = 0. No estimate is needed. Case 1 (d).
Let I tT be the set defined by(3.22) I tT := { x ∈ T : x is between L and L + εe } \ (cid:0) B ( v , t/ ˆ c ) ∪ B ( v , t/ ˆ c ) (cid:1) , where ε := ( δ − δ ) M − t , e := h , i , and M := | L | = | L | . Set J hT := I tT + [0 , h ].See Figure 8.In this case we use again (3.20) to estimate | ∆ h S ( x ) | . We obtain k ∆ h S k τL τ ( I tT ) ≤ k ∆ h S k τL τ ( I tT ) + k S − S k τL τ ( I tT ) + X A ∈A tT k S − S k τL τ ( J hT ∩ A ) + X A ∈ A tT k S − S k τL τ ( J hT ∩ A ) . Clearly, | I tT | ≤ ctδ ( T ) and | T | ∼ δ ( T ) d ( T ). Then using (3.18)-(3.19) we infer k S − S k τL τ ( I tT ) = ( | I tT | / | T | ) k S − S k τL τ ( T ) ≤ ctd ( T ) − k S − S k τL τ ( T ) = ctd ( T ) − | T | τs/ k S − S k τL p ( T ) ≤ ctd ( T ) τs − k S − S k τL p ( T ) . Similarly, for A ∈ A tT we use that | J hT ∩ A | ≤ ctd ( A ) and | A | ∼ d ( A ) to obtain k S − S k τL τ ( J hT ∩ A ) ≤ ctd ( A ) k S − S k τL ∞ ( A ) = ctd ( A ) | A | − τ/p k S − S k τL p ( A ) ≤ ctd ( A ) − τ/p k S − S k τL p ( A ) ≤ ctd ( A ) τs − k S − S k τL p ( A ) . For A ∈ A tT , we have k S − S k τL τ ( J hT ∩ A ) ≤ k S − S k τL τ ( A ) = | A | τs/ k S − S k τL p ( A ) ≤ cd ( A ) τs k S − S k τL p ( A ) . Putting the above estimates together we get k ∆ h S k τL τ ( I tT ) ≤ k ∆ h S k τL τ ( I tT ) + ctd ( T ) τs − k S − S k τL p ( T ) + X A ∈A tT ctd ( A ) τs − k S − S k τL ∞ ( A ) + X A ∈ A tT cd ( A ) τs k S − S k τL p ( A ) . Case 1 (e) (Main).
Let T ⋆h ⊂ T h be defined by(3.23) T ⋆h := { x ∈ T h : [ x, x + h ] ∩ L = ∅ , x I tT , [ x, x + h ] [ v ∈V T B v } . We next estimate k ∆ kh S k τL τ ( T ⋆h ) .Recall that by assumption h = | h | ν with ν =: (cos θ, sin θ ) and θ ∈ [ γ, π ], where γ ≤ π/ D and L .Let x ∈ T ⋆h . With the notation x = ( x , x ) we let ( − a, x ) ∈ L and ( a, x ) ∈ L , a >
0, be the points of intersection of the horizontal line through x with L and L . Set b := 2 a − ε with ε := ( δ − δ ) M − t , see (3.22).We associate the points x + be and x + be + h to x and x + h . A simple geometricargument shows that x + be ∈ △ E \ T , while x + be + h ∈ T ◦ .Now, using that S = constant on △ ◦ E we have S ( x ) = S ( x + be ) and since S = constant on △ ◦ E we have S ( x + h ) = S ( x + be + h ). We use these twoidentities to obtain S ( x + h ) − S ( x ) = S ( x + be + h ) − S ( x + be )+ [ S ( x + h ) − S ( x + h )] − [ S ( x + be ) − S ( x + be )]and, therefore, | ∆ h S ( x ) | ≤ | ∆ h S ( x + be ) | (3.24) + | S ( x + h ) − S ( x + h ) | + | S ( x + be ) − S ( x + be ) | . ONLINEAR SPLINE APPROXIMATION 17
Some words of explanation are in order here. The purpose of the set I tT is that thereis one-to-one correspondence between pairs of points x ∈ T ◦ \ I tT , x + h ∈ △ E \ T and x + be ∈ △ E \ T , x + be + h ∈ T ◦ . Due to δ < δ , this would not be true if I tT was not removed from T ◦ . Thus there is one-to-one correspondence between thedifferences | ∆ h S ( x ) | and | ∆ h S ( x + be ) | in the case under consideration. Also, itis important that ∆ h S ( x + be ) = 0 and hence | ∆ h S ( x + be ) | need not be usedto estimate | ∆ h S ( x + be ) | .Another important point here is that x + h T ◦ and x + be T ◦ . Therefore, noquantities | S ( x ) − S ( x ) | with x ∈ T ◦ \ I tT are involved in (3.24), which is critical.Observe that for x ∈ T ⋆h we have[ x, x + h ] [ v ∈V T B v and hence [ x + be , x + be + h ] [ v ∈V T B v . Therefore, by Property 3.7 it follows that [ x, x + h ] and [ x + be , x + be + h ] do notintersect any trapezoid T ′ ∈ T , T ′ = T .Let T ⋆⋆h := { x + be : x ∈ T ⋆h } . For any A ∈ A and t > A t := { x ∈ A : dist( x, ∂A ) ≤ t } . From all of the above we get k ∆ h S k τL τ ( T ⋆h ) ≤ k ∆ h S k τL τ ( T ⋆⋆h ) + X A ∈A tT k S − S k τL τ ( A t ) + X A ∈ A tT k S − S k τL τ ( A ) . Now, using that | A t | ≤ ctd ( A ) and | A | ∼ d ( A ) for A ∈ A tT we obtain k S − S k τL τ ( A t ) = ( | A t | / | A | ) | A | τs/ k S − S k τL p ( A ) (3.26) ≤ ctd ( A ) τs − k S − S k τL p ( A ) . For A ∈ A tT we use that | A | ∼ d ( A ) and obtain k S − S k τL τ ( A ) = | A | τs/ k S − S k τL p ( A ) ≤ cd ( A ) τs k S − S k τL p ( A ) . (3.27)Inserting these estimates above we get k ∆ h S k τL τ ( T ⋆h ) ≤ k ∆ h S k τL τ ( T ⋆⋆h ) + X A ∈A tT ctd ( A ) τs − k S − S k τL p ( A ) (3.28) + X A ∈ A tT cd ( A ) τs k S − S k τL p ( A ) . Case 2.
Let Ω ⋆h be the set of all x ∈ Ω such that [ x, x + h ] ⊂ Ω and [ x, x + h ] ∩ T = ∅ for all T ∈ T with d ( T ) ≥ t/ ˆ c . To estimate | ∆ h S ( x ) | we use again (3.20). Withthe notation from (3.25) we get k ∆ h S k τL τ (Ω ⋆h ) ≤ k ∆ h S k τL τ (Ω ⋆h ) + X T ∈T : d ( T ) ≤ t/ ˆ c k S − S k τL τ ( T ) + X A ∈A : d ( A ) >t k S − S k τL τ ( A t ) + X A ∈A : d ( A ) ≤ t k S − S k τL τ ( A ) For the first sum above we have just as in (3.21) X T ∈T : d ( T ) ≤ t/ ˆ c k S − S k τL τ ( T ) ≤ X T ∈T : d ( T ) ≤ t/ ˆ c cd ( T ) τs k S − S k τL p ( T ) . We estimate the other two sums as in (3.26) and (3.27). We obtain k ∆ h S k τL τ (Ω ⋆h ) ≤ k ∆ h S k τL τ (Ω ⋆h ) + X T ∈T : d ( T ) ≤ t/ ˆ c cd ( T ) τs k S − S k τL p ( T ) + X A ∈A : d ( A ) >t ctd ( A ) τs − k S − S k τL p ( A ) + X A ∈A : d ( A ) ≤ t cd ( A ) τs k S − S k τL p ( A ) . It is an important observation that each trapezoid T ∈ T with d ( T ) > t/ ˆ c may share trapezoids T ′ ∈ T tT and sets A ∈ A tT with only finitely many trapezoidswith the same properties. Also, for every such trapezoid T we have T tT ≤ c and A tT ≤ c with c > ω ( S , t ) ττ ≤ ω ( S , t ) ττ + Y + Y , where Y = X A ∈A : d ( A ) >t ctd ( A ) τs − k S − S k τL p ( A ) + X A ∈A : d ( A ) >t ct d ( A ) τs − k S − S k τL p ( A ) + X A ∈A : d ( A ) ≤ t cd ( A ) τs k S − S k τL p ( A ) and Y = X T ∈T : d ( T ) > t/ ˆ c ctd ( T ) τs − k S − S k τL p ( T ) + X T ∈T : d ( T ) > t/ ˆ c ct τs/ d ( T ) τs/ − k S − S k τL p ( T ) + X T ∈T : d ( T ) ≤ t/ ˆ c cd ( T ) τs k S − S k τL p ( T ) . We now turn to the estimation of | S | B sτ . Using the above and interchanging theorder of integration and summation we get | S | τB sτ = Z ∞ t − sτ − ω ( S , t ) ττ dt ≤ | S | τB sτ + Z + Z , where Z = X A ∈A cd ( A ) τs − k S − S k τL p ( A ) Z d ( A )0 t − τs dt + X A ∈A cd ( A ) τs − k S − S k τL p ( A ) Z d ( A )0 t − τs +1 dt + X A ∈A cd ( A ) τs k S − S k τL p ( A ) Z ∞ d ( A ) t − τs − dt ONLINEAR SPLINE APPROXIMATION 19 and Z = X T ∈T cd ( T ) τs − k S − S k τL p ( T ) Z ˆ cd ( T ) / t − τs dt + X T ∈T cd ( T ) τs/ − k S − S k τL p ( T ) Z ˆ cd ( T ) / t − τs/ dt + X T ∈T cd ( T ) τs k S − S k τL p ( T ) Z ∞ ˆ cd ( T ) / t − τs − dt. Observe that − τ s > − s/ < /p , which is one of the assumptions,and − τ s/ > − s < s + 2 /p , which is obvious. Therefore, all ofthe above integrals are convergent, and we obtain | S | τB sτ ≤ | S | τB sτ + X A ∈A c k S − S k τL p ( A ) + X T ∈T c k S − S k τL p ( T ) . Finally, applying H¨older’s inequality and using (3.17) we arrive at | S | τB sτ ≤ | S | τB sτ + c (cid:0) A (cid:1) τ (1 /τ − /p ) (cid:16) X A ∈A k S − S k pL p ( A ) (cid:17) τ/p + c (cid:0) T (cid:1) τ (1 /τ − /p ) (cid:16) X T ∈T k S − S k pL p ( T ) (cid:17) τ/p ≤ cn τ (1 /τ − /p ) k S − S k τL p (Ω) = cn τs/ k S − S k τL p (Ω) . This confirms estimate (3.10). The proof in the case when τ > (cid:3)
The proof of Theorem 3.4 is easier than the above proof. We omit it.4.
Nonlinear approximation from smooth splines
In this section we focus on Bernstein estimates in nonlinear approximation in L p , 0 < p < ∞ , from regular nonnested smooth piecewise polynomial functionsin R .4.1. Setting and approximation tool.
We first introduce the class of regularpiecewise polynomials S ( n, k ) of degree k − k ≥ n rings of maximumsmoothness. As in § R . Case 1:
Ω is a compact polygonal domain in R . We denote by S ( n, k ) ( n ≥ n )the set of all piecewise polynomials S of the form(4.1) S = n X j =1 P j R j , S ∈ W k − (Ω) , P j ∈ Π k , where R , . . . , R n are rings in the sense of Definition 3.1 with disjoint interiors suchthat Ω = ∪ nj =1 R j . Here Π k stands for the set of all polynomials of degree < k intwo variables and S ∈ W k − (Ω) means that all partial derivatives ∂ α S ∈ C (Ω), | α | ≤ k − Case 2:
Ω = R . In this case we denote by S ( n, k ) the set of all piecewisepolynomials S of degree k − R of the form (3.2), where R , . . . , R n are ringswith disjoint interiors such that the support Λ = ∪ nj =1 R j of S is a ring in the senseof Definition 3.1. We denote by S kn ( f ) p the best approximation of f ∈ L p (Ω) from S ( n, k ) in L p (Ω), 0 < p < ∞ , i.e.(4.2) S kn ( f ) p := inf S ∈S ( n,k ) k f − S k L p . Remark.
Observe that in our setting the splines are of maximum smoothness andthis is critical for our development. As will be shown in Example 4.4 below inthe nonnested case our Bernstein type inequality is not valid in the case when thesmoothness of the splines is not maximal.We next consider several scenarios for constructing of regular piecewise polyno-mials of maximum smoothness:
Example 1.
Suppose that T is an initial subdivision of Ω into triangles whichobey the minimum angle condition and is with no hanging vertices in the interiorof Ω. In the case of Ω = R we assume for simplicity that the triangles △ ∈ T are of similar areas, i.e. c ≤ |△ | / |△ | ≤ c for any △ , △ ∈ T . Next wesubdivide each triangle △ ∈ T into 4 triangles by introducing the mid points onthe sides of △ . The result is a triangulation T of Ω. In the same way we define thetriangulations T , T , etc. Each triangulation T j supports Courant hat functions(linear finite elements) ϕ θ , each of them supported on the union θ of all trianglesfrom T j which have a common vertex, say, v . Thus ϕ θ ( v ) = 1, ϕ θ takes values zeroat all other vertices of triangles from T j , and ϕ θ is continuous and piecewise linearover the triangles from T j . Clearly, each piecewise liner function over the trianglesfrom T j can be represented as a linear combination of Courant hat functions likethese.Denote by Θ j the set of all supports θ of Courant elements supported by T j andset Θ := ∪ j ≥ Θ j . Consider the nonlinear set S n of all piecewise linear functions S of the form S = X θ ⊂M n c θ ϕ θ , where M n ⊂ Θ and M n ≤ n ; the elements θ ∈ M n may come from differentlevels and locations. It is not hard to see that S n ⊂ S ( cn, Example 2.
More generally, one can consider piecewise linear functions S of theform S = X θ ⊂M n c θ ϕ θ , where { ϕ θ } are Courant hat functions as above, M n ≤ n , and M n consists ofcells θ as above that are not necessarily induced by a hierarchical collection oftriangulations of Ω, however, there exists a underlying subdivision of Ω into ringsobeying the conditions from § Example 3.
The C quadratic box-splines on the four-directional mesh (theso called “Powell-Zwart finite elements”) and the piecewise cubics in R or on arectangular domain, endowed with the Powell–Sabin triangulation generated bya uniform 6-direction mesh provide examples of quadratic and cubic splines ofmaximum smoothness.Other examples are to be identified or developed. ONLINEAR SPLINE APPROXIMATION 21
Splines with defect.
To make the difference between approximation from nonnestedand nested splines more transparent and for future references we now introduce thesplines with arbitrary smoothness. Given a set Ω ⊂ R with polygonal boundaryor Ω := R , k ≥
2, and 0 ≤ r ≤ k −
2, we denote by S ( n, k, r ) ( n ≥ n ) the set ofall piecewise polynomials S of the form(4.3) S = n X j =1 P j R j , S ∈ W r (Ω) , P j ∈ Π k , where R , . . . , R n are rings with disjoint interiors such that Ω = ∪ nj =1 R j . We set(4.4) S k,rn ( f ) p := inf S ∈S ( n,k,r ) k f − S k L p . Jackson estimate.
Jackson estimates in spline approximation are relativelyeasy to prove. Such estimates (also in anisotropic settings) are established in [2,4]. For example the Jackson estimate we need in the case of approximation frompiecewise linear functions ( k = 2) follows by [4, Theorem 3.6] and takes the form: Theorem 4.1.
Let < p < ∞ , s > , and /τ = s/ /p . Assume Ω = R or Ω ⊂ R is a compact set with polygonal boundary and initial triangulation consistingof n triangles with no hanging interior vertices and obeying the minimum anglecondition. Then for any f ∈ B s, τ we have f ∈ L p (Ω) and (4.5) S n ( f ) p ≤ cn − s/ | f | B s, τ , n ≥ n . Consequently, for any f ∈ L p (Ω)(4.6) S n ( f ) p ≤ cK ( f, n − s/ ) , n ≥ n . Here K ( f, t ) = K ( f, t ; L p , B sτ ) is the K -functional defined in (3 . and c > is aconstant depending only on s, p , and the structural constants of the setting. Similar Jackson and direct estimates for nonlinear approximation from splines ofdegrees 2 and higher and of maximum smoothness do not follow automatically fromthe results in [2]. The reason being the fact that the basis functions for splines ofdegree 2 and 3 that we are familiar with are not stable. The stability is required in[2]. The problem for establishing Jackson estimates for approximation from splinesof degree 2 and higher of maximum smoothness remains open.4.3.
Bernstein estimate in the nonnested case.
We come now to one of themain result of this article. Here we operate in the setting described above in § Theorem 4.2.
Let < p < ∞ , k ≥ , < s/ < k − /p , and /τ = s/ /p .Then for any S , S ∈ S ( n, k ) , n ≥ n , we have | S | B s,kτ ≤ | S | B s,kτ + cn s/ k S − S k L p , if τ ≥ , and (4.7) | S | τB s,kτ ≤ | S | τB s, τ + cn τs/ k S − S k τL p , if τ < . (4.8) where the constant c > depends only on s, p, k , and the structural constants of thesetting. An immediate consequence of this theorem is the inverse estimate given in
Corollary 4.3.
Let < p < ∞ , k ≥ , < s/ < k − /p , and /τ = s/ /p .Set λ := min { τ, } . Then for any f ∈ L p (Ω) we have (4.9) K ( f, n − s/ ) ≤ cn − s/ (cid:16) n X ℓ = n ℓ (cid:2) ℓ s/ S kℓ ( f ) p (cid:3) λ + k f k λp (cid:17) /λ , n ≥ n . Here K ( f, t ) = K ( f, t ; L p , B sτ ) is the K -functional defined just as in (3 . and c > is a constant depending only on s, p, k , and the structural constants of the setting. The proof of this corollary is just a repetition of the proof of Theorem 3.5. Weomit it.In turn, estimates (4.6) and (4.9) imply a characterization of the approxima-tion spaces associated with nonlinear nonnested piecewise linear approximation,see (1.7).The proof of Theorem 4.2 relies on the idea we used in the proof of Theorem 3.3.However, there is an important complication to overcome. The fact that manyrings with relatively small supports can be located next to a large ring is a majorobstacle in implementing this idea in the case of smooth splines. An additionalconstruction is needed. To make the proof more accessible, we shall proceed in twosteps. We first develop the needed additional construction and implement it in § § Example 4.4.
We now show that estimates (4.7)-(4.8) fail without the assumptionthat S , S ∈ W k − (Ω) (i.e., both splines have maximum smoothness). We shallonly consider the case when k = 2 and τ ≤
1. Let Ω = [ − , × [0 ,
1] and0 < ε < /
4. Set S ( x ) := x [0 , ( x ) , S ( x ) := x [ ε, × [0 , ( x ) , x = ( x , x ) . Clearly, S is continuous on Ω, while S is discontinuous along x = ε . A straight-forward calculation shows that(4.10) ω ( S , t ) ττ = 2 t τ +1 τ + 1 and ω ( S , t ) ττ = Z t − t | w + ε | τ dw for 0 ≤ t ≤ / . Further,(4.11) Z t − t | w + ε | τ dw = 1 τ + 1 (cid:2) ( t + ε ) τ +1 + sign( t − ε ) | t − ε | τ +1 (cid:3) . On the other hand, obviously ω ( S − S , t ) ττ ≤ k S − S k τL τ ≤ ε τ +1 yielding(4.12) ω ( S , t ) ττ ≥ ω ( S , t ) ττ − ε τ +1 . We shall use this estimate for t > /
4. From (2.1) and (4.10)-(4.12) we obtain | S | τB s, τ − | S | τB s, τ ≥ τ + 1 h Z ε t − sτ − [( t + ε ) τ +1 − ( ε − t ) τ +1 − t τ +1 ] dt + Z / ε t − sτ − [( ε + t ) τ +1 + ( t − ε ) τ +1 − t τ +1 ] dt i − ε τ +1 Z ∞ / t − sτ − dt =: I + I − (4 sτ +1 /sτ ) ε τ +1 . ONLINEAR SPLINE APPROXIMATION 23
Substituting t = εu in I and I , we get I + I = ε τ − sτ +1 τ + 1 h Z u − sτ − φ ( u ) du + Z / ε u − sτ − φ ( u ) du i , where φ ( u ) = (1 + u ) τ +1 − (1 − u ) τ +1 − u τ +1 and φ ( u ) = (1 + u ) τ +1 + ( u − τ +1 − u τ +1 . We clearly have φ ≥ ,
1] and φ ≥ , ∞ ). Therefore, | S | τB s, τ − | S | τB s, τ ≥ c ε τ − sτ +1 − c ε τ +1 = ε τ − sτ +1 ( c − c ε sτ ) , where c := 1 τ + 1 Z t − sτ − φ ( u ) du > c := 4 sτ +1 /sτ. By taking ε sufficiently small, we get(4.13) | S | τB s, τ − | S | τB s, τ ≥ ( c / ε τ − sτ +1 . Evidently,(4.14) k S − S k L p ≤ ε /p . By (4.13) and (4.14), | S | τB s, τ − | S | τB s, τ k S − S k τL p ≥ ( c / ε − sτ − τ/p = ( c / ε − sτ/ . Since ε − sτ/ → ∞ as ε →
0, estimate (4.8) cannot hold.4.4.
Bernstein estimate in the nested case.
We next prove a Bernstein esti-mate which yields an inverse estimate in the case of nested spline approximation.
Theorem 4.5.
Let < p < ∞ , k ≥ , ≤ r ≤ k − , < s/ < r + 1 /p , and /τ = s/ /p . Then for any S ∈ S ( n, k, r ) , n ≥ n , we have (4.15) | S | B s,kτ ≤ cn s/ k S k L p , where the constant c > depends only on s, p, k, r , and the structural constant ofour setting. Additional subdivision of Ω.
Situations where there are many small rings lo-cated next to a large ring create problems. To be able to deal with such cases weshall additionally subdivide Ω in two steps.
Subdivision of all rings R ∈ R n into nested hierarchies of rings. Lemma 4.6.
There exists a subdivision of every ring R ∈ R n into a nested multi-level collection of rings K R = ∪ ∞ m = m R K Rm with the following properties, where we use the abbreviated notation K m := K Rm : ( a ) Every level K m defines a partition of R into rings with disjoint interiors suchthat R = ∪ K ∈K m K . ( b ) The levels {K m } m ≥ m R are nested, i.e. K m +1 is a refinement of K m , andeach K ∈ K m has at least 4 and at most M children in K m +1 , where M ≥ is aconstant. ( c ) | R | ≤ c | K | for all K ∈ K m R . ( d ) We have (4.16) c − − m ≤ | K | ≤ c − m , ∀ K ∈ K m , ∀ m ≥ m R . As a consequence we have c − − m R ≤ | R | ≤ c − m R and (4.17) c − − m ≤ d ( K ) ≤ c − m , ∀ K ∈ K m , ∀ m ≥ m R . ( e ) All rings K ∈ K R are rings without a hole, except for finitely many of themin the case when R = Q \ Q and Q is small relative to Q . Then the rings witha hole form a chain R ⊃ K ⊃ K ⊃ · · · ⊃ K ℓ ⊃ Q . All sets K ∈ K R are ringsin the sense of Definition 3.1 with structural constants (parameters) N ∗ , c ⋆ , and β ⋆ . These and the constants M and c , c , c , c > from above depend only on theinitial structural constants N , c , and β .Proof. Observe first that if we are in a setting as the one described in Scenario 1from § R = Q \ Q be a ring in the sense of Definition 3.1, andassume that Q = ∅ . We subdivide the polygonal convex set Q into subrings byconnecting the center of eccentricity of Q with, say, 6 points from the boundary ∂R of R , preferably end points of segments on the boundary, so that the minimum anglecondition is obeyed. After that we subdivide the resulting rings using mid pointsand connecting them with segments. Necessary adjustments are made around Q depending on the size and location of Q . (cid:3) Subdivision of all rings from R n into subrings with disjoint interiors. We first pickup all rings from each K R , R ∈ R n , see Lemma 4.6, that are needed to handlesituations where many small rings are located next to a large ring.We shall only need the rings in K R that intersect the boundary ∂R of R . Denotethe set all such rings by Γ R and set Γ Rm := Γ R ∩ K Rm . We shall make use of thetree structure in Γ R . More precisely, we shall use the parent-child relation in Γ R induced by the inclusion relation: Each ring K ∈ Γ Rm has (contains) at least 1 andat most M children in Γ Rm +1 and has a single parent in Γ Rm − or no parent.We now construct a set Λ R of rings from Γ R which will help prevent situationswhere a ring may have many small neighbors.Given R ∈ R n , we denote by R Rn the set of all rings ˜ R ∈ R n , ˜ R = R , such that˜ R ∩ R = ∅ and d ( ˜ R ) ≤ d ( R ). These are all rings from R n that are small relative to R and intersect R (are neighbors of R ).It will be convenient to introduce the following somewhat geometric terminology: We say that a ring K ∈ Γ R can see ˜ R ∈ R Rn or that ˜ R is in the range of K if d ( K ) ≥ d ( ˜ R ) and K ∩ ˜ R = ∅ . We now construct Λ R by applying the following Rule : We place K ∈ Γ R in Λ R if K can see some (at least one) rings from R Rn but neither of the children of K in Γ R can see all of them. We now extend Λ R to ˜Λ R by adding to Λ R all same level neighbors of all K ∈ Λ R ,i.e. if K ∈ Λ R and K ∈ Γ Rm , then we add to Λ R each K ′ ∈ Γ Rm such that K ′ ∩ K = ∅ .The next step is to construct a subdivision of each R ∈ R n into rings by using˜Λ R . We fix R ∈ R n and shall suppress the superscript R for the new sets that willbe introduced next and depend on R .Let ˜Γ ⊂ Γ R be the minimal subtree of Γ R that contains ˜Λ R , i.e. ˜Γ is the setof all K ∈ Γ R such that K ⊃ K ′ for some K ′ ∈ ˜Λ R . We denote by ˜Γ b the set of ONLINEAR SPLINE APPROXIMATION 25 all branching rings in ˜Γ (rings with more than one child in ˜Γ) and by ˜Γ ′ b the setof all children in ˜Γ of branching rings (each of them may or may not belong to ˜Γ).Furthermore, we let ˜Γ ℓ denote the set of all leaves in ˜Γ (rings in ˜Γ containing noother rings from ˜Γ).Evidently, ˜Γ ℓ ⊂ ˜Λ R . However, rings from ˜Γ b and ˜Γ ′ b may or may not belong to˜Λ R . We extend ˜Λ R to ˜˜Λ R := ˜Λ R ∪ ˜Γ b ∪ ˜Γ ′ b . In addition, we add to ˜˜Λ R all ringsfrom K Rm R , if they are not there yet.It is readily seen that each ring ˜ R ∈ R Rn can be in the range of only finitely many K ∈ ˜Γ ℓ and each ring ˜ R ∈ R n may have only finitely many neighbors R ∈ R n suchthat d ( R ) ≥ d ( ˜ R ). Therefore, X R ∈R n Rℓ ≤ cn. Obviously b ≤ ℓ , ′ b ≤ M b ≤ M ℓ , implying R ≤ ℓ + b ≤ c ℓ , and hence R ≤ c ′ ℓ . Putting these estimates together implies(4.18) X R ∈R n R ≤ cn. Observe that, with the exception of all branching rings in ˜Λ R , by constructionevery other ring K ∈ ˜Λ R is either a leaf, and hence contains no other rings from˜˜Λ R , or contains only one ring K ′ ∈ ˜˜Λ R of minimum level, i.e. K has one descendent K ′ in ˜˜Λ R .We now make the final step in our construction: We denote by F R the set of allrings from ˜Γ Rℓ along with all new rings of the form K \ K ′ , where K ∈ ˜Γ ′ b , K ′ ∈ ˜˜Λ R , K ′ ⊂ K and K ′ is of minimum level with these properties. Set F := ∪ R ∈R n F R .The purpose of the above construction becomes clear from the the following Lemma 4.7.
The set F consists of rings in the sense of Definition 3.1 with pa-rameters depending only on the structural constants N , c and β . Also, for any R ∈ R n the rings in F R have disjoint interiors, R = ∪ K ∈F R K , and F R ≤ c R .Hence, (4.19) Ω = [ R ∈R n [ K ∈F R K and X R ∈R n F R ≤ cn. Most importantly, each ring K ∈ F has only finitely many neighbors in F , that is,there exists a constant N such that for any K ∈ F there are at most N rings in F intersecting K .Proof. All properties of the newly constructed rings but the last one given in thislemma follow readily from their construction.To show that each ring K ∈ F has only finitely many neighbors in F we shallneed the following technical Lemma 4.8.
Suppose K ⊃ K ⊃ K , K ∈ Γ R , K , K ∈ ˜Λ R , and both K and K share parts of an edge E of K located in the interior of R . Then there exists K ⋆ ∈ ˜Λ R such that K ⋆ ∩ K ◦ = ∅ , K ⋆ ∩ E = ∅ , and K ⋆ is either a neighbor of K or K , or K ⋆ is a neighbor of the parent of K in Γ R . Proof. If K ∈ Λ R , then by construction all same level neighbors of K belong ˜Λ R and hence the one that shares the edge of K contained in E will be in ˜Λ R . Wedenote this ring by K ⋆ and apparently it has the claimed properties. By the sametoken, if K ∈ Λ R , then one of his neighbors will do the job.Suppose K , K ∈ ˜Λ R \ Λ R . Then K has a neighbor, say, ˆ K that belongs toΛ R and ˆ K is at the level of K . If ˆ K has an edge contained in E , then K ⋆ := ˆ K has the claimed property. Similarly, K has a neighbor ˆ K ∈ Λ R at the level of K .If ˆ K has an edge contained in E , then K ⋆ := ˆ K will do the job.Assume that neither of the above is true. Then since K , ˆ K ∈ Γ R they musthave the same parent in Γ R that has an edge contained in E . Denote this commonparent by K ♯ . For the same reason, K , ˆ K ∈ Γ R have a common parent, say, K ♯♯ in Γ R . Clearly, K ♯ and K ♯♯ have some edges contained in E . Also, ˆ K ⊂ K ♯ ,ˆ K ⊂ K ♯ , and ˆ K ◦ ∩ ˆ K ◦ = ∅ .We claim that K ♯ belongs to Λ R . Indeed, the rings from R n that are in therange of ˆ K are also in the range of K ♯ . Also, the rings from R n that are in therange of ˆ K are also in the range of K ♯ . However, obviously neither of the childrenof K ♯ can have the range of K ♯ . Therefore, K ♯ belongs to Λ R . Now, just as abovewe conclude that one of the neighbors of K ♯ has the claimed property. (cid:3) We are now prepared to show that each ring K ∈ F has only finitely manyneighbors in F . By the construction any K ∈ F R , R ∈ R n , has only finitely manyneighbors that do not belong to F R . Thus, it remains to show that it cannot happenthat there exist rings K ⊂ K ⊂ · · · ⊂ K J , K j ∈ ˜Λ R , with J uncontrollably largethat have edges contained in an edge of a single ring K ∈ ˜Λ R whose interior doesnot intersect K j , j = 1 , . . . , J . But this assertion readily follows by Lemma 4.8. (cid:3) The following lemma will be instrumental in the proof of this theorem.
Lemma 4.9.
Assume < p, q ≤ ∞ , k ≥ , r ≥ , and ν ∈ R with | ν | = 1 .Let the sets G, H ⊂ R be measurable, G ⊂ H , and such that there exist balls B , B , B , B , B j = B ( x j , r j ) , with the properties: B ⊂ G ⊂ B , r ≤ c ♭ r , and B ⊂ H ⊂ B , r ≤ c ♭ r , where c ♭ ≥ is a constant. Then for any P ∈ Π k (4.20) k P k L p ( G ) ≤ c | G | /p − /q k P k L q ( G ) , (4.21) k D rν P k L p ( G ) ≤ cd ( G ) − r k P k L p ( G ) , and (4.22) k P k L p ( G ) ≤ c ( | G | / | H | ) /p k P k L p ( H ) , where c > is a constant depending on p, q, k, r, c ♭ , and the parameters N , c ,and β from Definition 3.1. Here D rν S is the r th directional derivative of S in thedirection of ν .Furthermore, inequality (4 . holds with Q and H replaced by their images L ( G ) and L ( H ) , where L is a nonsingular linear transformation of R .Proof. Inequality (4.20) holds whenever B = B (0 ,
1) and B = B (0 , c ⋄ ) with c ⋄ = constant by the fact that any two (quasi)norms on Π k are equivalent. Thisimplies that (4.20) is valid in the case when B = B (0 ,
1) and B ⊂ B , where B = B ( x , c ⋄ / p = ∞ . In general, it follows from the case p = ∞ and application of(4.20) to G with p and q = ∞ and to H with p = ∞ , q = p . Inequality (4.21) is an ONLINEAR SPLINE APPROXIMATION 27 easy consequence of the Markov inequality for univariate polynomials whenever G is a square. Then in general it follows by inscribing B in a smallest possible cubeand then applying it for the cube and using (4.22). The last claim in the lemma isobvious. (cid:3) Proof of Theorem 4.5.
We shall only consider the case when Ω ⊂ R is a compactpolygonal domain. Let S ∈ S ( n, k, r ) and suppose S is represented as in (4.1), thatis,(4.23) S = X R ∈R n P R R , S ∈ W r (Ω) , P R ∈ Π k , where R n is a collection of ≤ n rings with disjoint interiors such that Ω = ∪ R ∈R n R .We are now prepared to complete the proof of Theorem 4.5. From (4.23) andbecause F is a refinement of R n it follows that S can be represented in the form(4.24) S = X K ∈F P K K , S ∈ W r (Ω) , P K ∈ Π k . Here F is the collection of at most cn rings from above with disjoint interiors suchthat Ω = ∪ K ∈F K .We next introduce some convenient notation. For any ring K ∈ F we denoteby N K the set of all rings K ′ ∈ F such that K ∩ K ′ = ∅ , E K will denote the setof all segments (edges) from the boundary ∂K of K , and V K will be the set of allvertices of the polygon ∂K (end points of edges from E K ).The fact that F consists of rings in the sense of Definition 3.1 implies the fol-lowing Property 4.10.
There exists a constant 0 < ˇ c < E = [ v , v ] is anedge shared by two rings K, K ′ ∈ F then for any x ∈ E with | x − v j | ≥ ρ , j = 1 , ρ > B ( x, ˇ cρ ) ⊂ K ∪ K ′ .Fix t >
0. For each ring K ∈ F we define K t := { x ∈ K : dist( x, ∂K ) ≤ kt } . Write Ω t := ∪ K ∈F K t .Let h ∈ R with norm | h | ≤ t and set ν := | h | − h . For S is a polynomial ofdegree ≤ k − K ∈ F we have ∆ kh S ( x ) = 0 for x ∈ ∪ K ∈F K \ K t . Therefore,(4.25) k ∆ kh S k L τ (Ω) = k ∆ kh S k L τ (Ω t ) . Let K ∈ F and assume d ( K ) > kt/ ˇ c with 0 < ˇ c < N tK := { K ′ ∈ N K : d ( K ) > kt/ ˇ c } , B v := B ( v, kt/ ˇ c ), v ∈ V K , and N tK := { K ′ ∈ F : d ( K ′ ) > kt/ ˇ c and K ′ ∩ ( K + B (0 , kt/ ˇ c )) = ∅} . Observe that because d ( K ) > kt/ ˇ c the number of rings in N tK is uniformlybounded.Let x ∈ Ω t be such that [ x, x + kh ] ∩ K = ∅ . Two cases are to be consideredhere.(a) Let [ x, x + kh ]
6⊂ ∪ v ∈V K B v . Then [ x, x + kh ] intersects some edge E ∈ E K such that ℓ ( E ) ≥ kt/ ˇ c , and [ x, x + kh ] cannot intersect another edge E ′ ∈ E K withthis property or an edge E ′ ∈ E K with ℓ ( E ′ ) < kt/ ˇ c . Suppose that the edge E =: [ v , v ] is shared with K ′ ∈ F and y := E ∩ [ x, x + kh ].Evidently, | y − v j | > kt/ ˇ c , j = 1 ,
2, and in light of Property 4.10 we have [ x, x + kh ] ⊂ B ( y, kt ) ⊂ K ∪ K ′ . Clearly,(4.26) | ∆ kh S ( x ) | ≤ ct r k D rν S k L ∞ ([ x,x + kh ]) ≤ ct r k D rν S k L ∞ ( K ) + ct r k D rν S k L ∞ ( K ′ ) . (b) Let [ x, x + kh ] ⊂ ∪ v ∈V K B v . Then we estimate | ∆ kh S ( x ) | trivially:(4.27) | ∆ kh S ( x ) | ≤ k k X ℓ =0 | S ( x + ℓh ) | . Using (4.26) - (4.27) we obtain k ∆ kh S k τL τ ( K t ) ≤ c X K ′ ∈N tK td ( K ′ ) t rτ k D rν S k τL ∞ ( K ′ ) + c X K ′ ∈ N tK X v ∈V K k S k τL τ ( B v ∩ K ′ ) + c X K ′′ ∈F : d ( K ′′ ) ≤ kt/ ˇ c k S k τL τ ( K ′′ ∩ ( K +[0 ,kh ])) . (4.28)Note that the number of rings K ′ ∈ N tK such that K ′ ∩ B v = ∅ for some v ∈ V K isuniformly bounded.By Lemma 4.9 it follows that k D rν S k L ∞ ( K ′ ) ≤ cd ( K ′ ) − r − /p k S k L p ( K ′ ) and if thering K ′ ∈ N tK and v ∈ V K , then k S k τL τ ( B v ∩ K ′ ) ≤ c ( | B v | / | K ′ | ) k S k τL τ ( K ′ ) ≤ ct | K ′ | − k S k τL τ ( K ′ ) ≤ ct | K ′ | − τ (1 /τ − /p ) k S k τL p ( K ′ ) ≤ ct d ( K ′ ) τs − k S k τL p ( K ′ ) . We use the above estimates in (4.28) to obtain k ∆ kh S k τL τ ( K t ) ≤ c X K ′ ∈N tK t rτ d ( K ′ ) − rτ − τ/p k S k τL p ( K ′ ) + c X K ′ ∈ N tK t d ( K ′ ) τs − k S k τL p ( K ′ ) + c X K ′′ ∈F : d ( K ′′ ) ≤ kt/ ˇ c k S k τL τ ( K ′′ ∩ ( K +[0 ,kh ])) . (4.29)Denote by Ω ⋆t the set of all x ∈ Ω t such that [ x, x + kh ] ⊂ Ω and[ x, x + kh ] ⊂ ∪{ K ∈ F : d ( K ) ≤ kt/ ˇ c } . In this case we shall use the obvious estimate k ∆ kh S k τL τ (Ω ⋆t ) ≤ c X K ∈F : d ( K ) ≤ kt/ ˇ c k S k τL τ ( K ) . This estimate along with (4.29) yields ω k ( S, t ) ττ ≤ c X K ∈F : d ( K ) ≥ kt/ ˇ c t rτ d ( K ) − rτ − τ/p k S k τL p ( K ) + c X K ∈F : d ( K ) ≥ kt/ ˇ c t d ( K ) sτ − k S k τL p ( K ) + c X K ∈F : d ( K ) ≤ kt/ ˇ c k S k τL τ ( K ) . Here we used the fact that only finitely many (uniformly bounded number) of therings involved in the above estimates may overlap at a time due to Lemma 4.7. For
ONLINEAR SPLINE APPROXIMATION 29 the norms involved in the last sum we use the estimate k S k τL τ ( K ) ≤ cd ( K ) sτ k S k τL p ( K ) , which follows by Lemma 4.9, to obtain ω k ( S, t ) ττ ≤ c X K ∈F : d ( K ) ≥ kt/ ˇ c t rτ d ( K ′ ) − rτ − τ/p k S k τL p ( K ′ ) + c X K ∈F : d ( K ) ≥ kt/ ˇ c t d ( K ) sτ − k S k τL p ( K ) + c X K ∈F : d ( K ) ≤ kt/ ˇ c d ( K ) sτ k S k τL p ( K ) . We insert this estimate in (2.1) and interchange the order of integration and sum-mation to obtain | S | τB s,kτ = Z ∞ t − sτ − ω k ( S, t ) ττ dt ≤ c X K ∈F d ( K ) − rτ − τ/p k S k τL p ( K ) Z ˇ cd ( K ) / k t − sτ + rτ dt + c X K ∈F d ( K ) sτ − k S k τL p ( K ) Z ˇ cd ( K ) / k t − sτ +1 dt + c X K ∈F d ( K ) sτ k S k τL p ( K ) Z ∞ ˇ cd ( K ) / k t − sτ − dt. Observe that − sτ + rτ > − s/ < r + 1 /p and − sτ + 1 > − s < /τ = s + 2 /p . Therefore, the above integrals are convergent andtaking into account that 2 − τ /p − sτ = 2 τ (1 /τ − /p − s/
2) = 0 we obtain | S | τB s,kτ ≤ c X K ∈F k S k τL p ( K ) ≤ cn τ (1 /τ − /p ) (cid:16) X K ∈F k S k τL p ( K ) (cid:17) τ/p = cn τs/ k S k τL p (Ω) , where we used H¨older’s inequality. This completes the proof. (cid:3) Proof of the Bernstein estimate (Theorem 4.2) in the nonnested case.
For the proof of Theorem 4.2 we combine ideas from the proofs of Theorem 3.3 andTheorem 4.5. We shall adhere to a large extent to the notation introduced in theproof of Theorem 3.3 in § D k − ν S of any S ∈ S ( n, k )are piecewise constants along the respective straight lines rather than S being apiecewise constant.We consider the case when Ω ⊂ R is a compact polygonal domain. Assume S , S ∈ S ( n, k ), n ≥ n . Then each S j ( j = 1 ,
2) can be represented in the form S j = P R ∈R j P R R , where R j is a set of at most n rings in the sense of Definition 3.1with disjoint interiors and such that Ω = ∪ R ∈R j R , P R ∈ Π k , and S j ∈ W k − (Ω).Just as in the proof of Theorem 4.5 there exist subdivisions F , F of the ringsfrom R , R with the following properties, for j = 1 , F j consists of rings in the sense of Definition 3.1 with parameters N ⋆ , c ⋆ ,and β ⋆ depending only on the structural constants N , c , and β .(b) ∪ R ∈F j R = Ω and F j ≤ cn .(c) There exists a constant N such that for any R ∈ F j there are at most N rings in F j intersecting R ( R has ≤ N neighbors in F j ).(d) S j can be represented in the form S j = P R ∈F j P R R with P R ∈ Π k .Now, just as in the proof of Theorem 3.3 we denote by U the collection of allmaximal connected sets obtained by intersecting rings from F and F . By (3.14)there exists a constant c > U ≤ cn.
We claim that there exists a constant N such that for any U ∈ U there are nomore than N sets U ′ ∈ U which intersect U , i.e. U has at most N neighbors in U . Indeed, let U ∈ U be a maximal connected component of R ∩ R with R ∈ F and R ∈ F . Then using the fact that the ring R has finitely many neighbors in F and R has finitely many neighbors in F we conclude that U has finitely manyneighbors in U .Further, we introduce the sets A and T just as in the proof of Theorem 3.3. Trapezoids.
Our main concern will be in dealing with the trapezoids T ∈ T . Wenext use the fact that any ring from F j , j = 1 ,
2, has at most N neighbors in F j to additionally subdivide the trapezoids from T into trapezoids whose long sidesare sides of good triangles for rings in F or F .Consider an arbitrary trapezoid T ∈ T . Just as in § T is a maximal isosceles trapezoid contained in △ E ∩ △ E , where △ E j ( j = 1 ,
2) isa good triangle for a ring R j ∈ F j , and T is positioned so that its vertices are thepoints: v := ( − δ / , , v := ( δ / , , v := ( δ / , H ) , v := ( − δ / , H ) , where 0 ≤ δ ≤ δ and H > δ . Let L := [ v , v ] and L := [ v , v ] be the twoequal (long) legs of T . We assume that L ⊂ E and L ⊂ E . See Figure 8.By Lemma 4.7 it follows that there exist less than N rings K ′ ℓ ∈ F , ℓ = 1 , . . . , ν ′ ,each of them with an edge or part of an edge contained in L . By Definition 3.1,each of them can be subdivided into at most two segments so that each of these isa side of a good triangle. Denote by I ′ ℓ , ℓ = 1 , . . . , m ′ , these segments, and by △ I ′ ℓ , ℓ = 1 , . . . , m ′ , the respective good triangles attached to them. More precisely, I ′ ℓ isa side of △ I ′ ℓ ⊂ K ′ ℓ and △ I ′ ℓ is a good triangle for K ′ ℓ . Thus we have L = ∪ m ′ ℓ =1 I ′ ℓ ,where the segments I ′ ℓ , ℓ = 1 , . . . , m ′ , are with disjoint interiors.Similarly, there exist segments I ′′ ℓ , ℓ = 1 , . . . , m ′′ , and attached to them goodtriangles △ I ′′ ℓ , ℓ = 1 , . . . , m ′′ , for rings from F , so that L = ∪ m ′′ ℓ =1 I ′′ ℓ .Denote by v ′ ℓ , ℓ = 1 , . . . , m ′ + 1, the vertices of the triangles △ I ′ ℓ , ℓ = 1 , . . . , m ′ ,on L ′ so that I ′ ℓ = [ v ′ ℓ , v ′ ℓ +1 ] and assume that their orthogonal projections onto the x -axis p ′ ℓ , ℓ = 1 , . . . , m ′ + 1, are ordered so that 0 = p ′ < p ′ < · · · < p ′ m ′ +1 = H .Exactly in the same way we define the vertices v ′′ ℓ , ℓ = 1 , . . . , m ′′ +1, of the triangles △ I ′′ ℓ and their projections onto the x -axis 0 = p ′′ < p ′′ < · · · < p ′′ m ′′ +1 = H .For any q ∈ [0 , H ] we let δ ( q ) be the distance between the points where the linewith equation x = q intersects L and L . Thus δ (0) = δ and δ ( H ) = δ , and δ ( q ) is linear.Inductively, starting from q = 0 one can easily subdivide the interval [0 , H ] bymeans of points0 = q < q < · · · < q m +1 = H, m ≤ m ′ + m ′′ ≤ N with the following properties, for k = 1 , . . . , m , either(a) δ ( q k ) ≤ q k +1 − q k < δ ( q k )or (b) q k +1 − q k > δ ( q k ) and ( q k , q k +1 ) contains no points p ′ ℓ or p ′′ ℓ .We use the above points to subdivide the trapezoid T . Let T k , k = 1 , . . . , m ,be the trapezoid bounded by L , L , and the lines with equations x = q k and x = q k +1 . ONLINEAR SPLINE APPROXIMATION 31
We now separate the “bad” from the “good” trapezoids T k . Namely, if property(a) from above is valid then T k is a ring and we place T k in A ; if property (b) isvalid, then T k is a “bad” trapezoid and we place T k in T . We apply the aboveprocedure to all trapezoids. Properties of New Trapezoids.
We now consider an arbitrary trapezoid T fromthe above defined T (the set of bad trapezoids). We next summarise the propertiesof T . It will be convenient to us to use the same notation as above as well as inthe proof of Theorem 3.3. We assume that T is an isosceles trapezoid contained in △ E ∩ △ E , where △ E j , j = 1 ,
2, is a good triangle for a ring R j ∈ F j , and T ispositioned so that its vertices are the points: v := ( − δ / , , v := ( δ / , , v := ( δ / , H ) , v := ( − δ / , H ) , where 0 ≤ δ ≤ δ and H > δ . Let L := [ v , v ] and L := [ v , v ] be the twoequal (long) sides of T . We assume that L ⊂ E and L ⊂ E . See Figure 8.As a result of the above subdivision procedure, there exists a triangle △ L with aside L such that △ L is a good triangle for some ring ˜ R ∈ F and △ ◦ L ∩ △ ◦ E = ∅ .For the same reason, there exists a triangle △ L with a side L such that △ L is agood triangle for some ring ˜ R ∈ F and △ ◦ L ∩ △ ◦ E = ∅ .Observe that △ E and △ E are good triangles and hence the angles of △ E j adjacent to E j are of size β ⋆ / j = 1 ,
2. Likewise, △ L and △ L are good trianglesand hence the angles of △ L j adjacent to L j are of size β ⋆ / j = 1 ,
2. Therefore, wemay assume that △ L ⊂ △ E and △ L ⊂ △ E . Consequently, S is a polynomialof degree < k on △ L and another polynomial of degree < k on △ L . By the sametoken, S is a polynomial of degree < k on △ L and another polynomial of degree < k on △ L . We shall assume that △ L ⊂ A and △ L ⊂ A , where A , A ∈ A .Further, denote by D and D the bottom and top sides of T . We shall denoteby V T = { v , v , v , v } the vertices of T , where v is the point of intersection of L and D and the other vertices are indexed counter clockwise.We shall use the notation δ ( T ) := δ and δ ( T ) := δ . We always assume that δ ( T ) ≥ δ ( T ). Clearly, d ( T ) ∼ H ; more precisely H < d ( T ) < H + δ + δ .Observe that by the construction of the sets T , A , and (3.14) it follows that A ∪ T consists of polygonal sets with disjoint interiors, ∪ A ∈A A ∪ T ∈T T = Ω, thereexists a constant c > A ≤ cn,
T ≤ cn, and there exists a constant N such that each set from A ∪ T has at most N neighbors in A ∪ T .We summarize the most important properties of the sets from T and A in thefollowing Lemma 4.11.
The following properties hold for some constant < ˜ c < dependingonly on the structural constants N , c and β of the setting: ( a ) Let T ∈ T and assume the notation related to T from above. If x ∈ L with | x − v j | ≥ ρ , j = 1 , , then B ( x, ˜ cρ ) ⊂ △ L ∪ △ L . Also, if x ∈ L with | x − v j | ≥ ρ , j = 2 , , then B ( x, ˜ cρ ) ⊂ △ L ∪ △ L . Furthermore, if x ∈ D with | x − v j | ≥ ρ , j = 1 , , then B ( x, ˜ cρ ) ⊂ △ E ∩ △ E , and similarly for x ∈ D . ( b ) Assume that E = [ w , w ] is an edge shared by two sets A, A ′ ∈ A . Let V A be the set of all vertices on ∂A (end points of edges) and let V A ′ be the set of all vertices on ∂A ′ . If x ∈ E with | x − w j | ≥ ρ , j = 1 , , for some ρ > , then (4.31) B ( x, ˜ cρ ) ⊂ A ∪ A ′ ∪ v ∈V A ∪V A ′ B ( v, ρ ) . Proof.
Part (a) of this lemma follows readily from the properties of the trapezoids.Part (b) needs clarification. Suppose that for some x ∈ E with | x − w j | ≥ ρ , j = 1 , ρ >
0, the inclusion (4.31) is not valid. Then exists a point y from an edge˜ E = [ u , u ] of, say, ∂A such that | y − x | < ρ and | y − u j | ≥ ρ , j = 1 ,
2. A simplegeometric argument shows that if the constant ˜ c is sufficiently small (dependingonly on the parameter β of the setting), then there exists an isosceles trapezoidˇ T ⊂ △ E ∩ △ ˜ E with two legs contained in E and ˜ E such that each leg is longerthan its larger base. But then the subdivision of the sets from U (see the proof ofTheorem 3.3) would have created a trapezoid in T that contains part of A . This isa contradiction which shows that Part (b) holds true. (cid:3) We have the representation(4.32) S ( x ) − S ( x ) = X A ∈A P A A ( x ) + X T ∈T P T T ( x ) , where P A , P T ∈ Π k . Note that S − S ∈ W k − (Ω).Let 0 < s/ < k − /p and τ ≤
1. Fix t > h ∈ R with norm | h | ≤ t . Write ν := | h | − h and assume ν =: (cos θ, sin θ ), − π < θ ≤ π .Since S , S ∈ W k − (Ω) we have the following representation of ∆ k − h S j ( x ):∆ k − h S j ( x ) = | h | k − Z R D k − ν S j (cid:0) x + uν (cid:1) M k − ( u ) du, where M k − ( u ) is the B-spline with knots u , u , . . . , u k − , u ℓ := ℓ | h | . In fact, M k − ( u ) = ( k − u , . . . , u k − ]( · − u ) k − is the divided difference. As is wellknown, 0 ≤ M k − ≤ c | h | − , supp M k − ⊂ [0 , ( k − | h | ], and R R M k − ( u ) du = 1.Therefore, by ∆ kh S j ( x ) = ∆ k − h S j ( x + h ) − ∆ k − h S j ( x ), whenever [ x, x + kh ] ⊂ Ω,we arrive at the representation(4.33) ∆ kh S j ( x ) = | h | k − Z k | h | D k − ν S j (cid:0) x + uv (cid:1) M ∗ k ( u ) du, where M ∗ k ( u ) := M k − ( u − | h | ) − M k − ( u ).In what follows we estimate k ∆ kh S k τL τ ( G ) − k ∆ kh S k τL τ ( G ) for different subsets G of Ω. Case 1.
Let T ∈ T be such that d ( T ) > kt/ ˜ c with ˜ c the constant from Lemma 4.11.Denote T h := { x ∈ Ω : [ x, x + kh ] ⊂ Ω and [ x, x + kh ] ∩ T = ∅} . We next estimate k ∆ kh S k τL τ ( T h ) − k ∆ kh S k τL τ ( T h ) .Assume that T ∈ T is a trapezoid positioned as described above in Propertiesof New Trapezoids. We adhere to the notation introduced there.In addition, let v − v =: | v − v | (cos γ, sin γ ) with γ ≤ π/
2, i.e. γ is theangle between D and L . Assume that ν =: (cos θ, sin θ ) with θ ∈ [ γ, π ]. Thecase θ ∈ [ − γ,
0] is just the same. The case when θ ∈ [0 , γ ] ∪ [ − π, − γ ] is consideredsimilarly. ONLINEAR SPLINE APPROXIMATION 33
We set B v := B ( v, kt/ ˜ c ), v ∈ V T . Also, denote A tT := { A ∈ A : d ( A ) > kt/ ˜ c and A ∩ ( T + B (0 , kt )) = ∅} , A tT := { A ∈ A : d ( A ) ≤ kt/ ˜ c and A ∩ ( T + B (0 , kt )) = ∅} and T tT := { T ′ ∈ T : d ( T ′ ) > kt/ ˜ c and T ′ ∩ ( T + B (0 , kt )) = ∅} , T tT := { T ′ ∈ T : d ( T ′ ) ≤ kt/ ˜ c and T ′ ∩ ( T + B (0 , kt )) = ∅} . Clearly, A tT ≤ c and T tT ≤ c for some constant c > Case 1 (a).
If [ x, x + kh ] ⊂ △ E , then ∆ kh S ( x ) = 0 because S is a polynomialof degree < k on △ E . Hence no estimate is needed. Case 1 (b).
If [ x, x + kh ] ⊂ ∪ v ∈V T B v , we estimate | ∆ kh S ( x ) | trivially:(4.34) | ∆ kh S ( x ) | ≤ | ∆ kh S ( x ) | + 2 k k X ℓ =0 | S ( x + ℓh ) − S ( x + ℓh ) | . Clearly, the contribution of this case to estimating k ∆ kh S k τL τ ( T h ) − k ∆ kh S k τL τ ( T h ) is ≤ c X v ∈V T X A ∈A tT k S − S k τL τ ( B v ∩ A ) + c X v ∈V T X T ′ ∈T tT k S − S k τL τ ( B v ∩ T ′ ) + c X v ∈V T X A ∈ A tT k S − S k τL τ ( B v ∩ A ) + c X v ∈V T X T ′ ∈ T tT k S − S k τL τ ( B v ∩ T ′ ) ≤ X A ∈A tT ct d ( A ) τs − k S − S k τL p ( A ) + X T ′ ∈T tT ct τs/ d ( T ′ ) τs/ − k S − S k τL p ( T ′ ) + X A ∈ A tT cd ( A ) τs k S − S k τL p ( A ) + X T ′ ∈ T tT cd ( T ′ ) τs k S − S k τL p ( T ′ ) . Here we used the following estimates, which are a consequence of Lemma 4.9:(1) If A ∈ A tT , then k S − S k τL τ ( B v ∩ A ) ≤ c ( | B v | / | A | ) k S − S k τL τ ( A ) ≤ ct d ( A ) − k S − S k τL τ ( A ) ≤ ct d ( A ) τs − k S − S k τL p ( A ) . (2) If T ′ ∈ T tT and δ ( T ′ ) > kt/ ˜ c , then for any v ∈ V T we have k S − S k τL τ ( B v ∩ T ′ ) ≤ c ( | B v | / | T ′ | ) k S − S k τL τ ( T ′ ) ≤ ct | T ′ | τs/ − k S − S k τL p ( T ′ ) ≤ ct δ ( T ′ ) τs/ − d ( T ′ ) τs/ − k S − S k τL p ( T ′ ) ≤ ct τs/ d ( T ′ ) τs/ − k S − S k τL p ( T ′ ) , where we used that τ s/ <
1, which is equivalent to s < s + 2 /p .(3) If T ′ ∈ T tT and δ ( T ′ ) ≤ kt/ ˜ c , then for any v ∈ V T k S − S k τL τ ( B v ∩ T ′ ) ≤ c ( | B v ∩ T ′ | / | T ′ | ) k S − S k τL τ ( T ′ ) ≤ ctδ ( T ′ )[ δ ( T ′ ) d ( T ′ )] − k S − S k τL τ ( T ′ ) ≤ ctd ( T ′ ) − [ δ ( T ′ ) d ( T ′ )] τs/ k S − S k τL p ( T ′ ) ≤ ct τs/ d ( T ′ ) τs/ − k S − S k τL p ( T ′ ) . (4) If A ∈ A tT , then k S − S k τL τ ( B v ∩ A ) ≤ k S − S k τL τ ( A ) ≤ c | A | τs/ k S − S k τL p ( A ) ≤ cd ( A ) τs k S − S k τL p ( A ) . (5) If T ′ ∈ T tT , then k S − S k τL τ ( B v ∩ T ′ ) ≤ k S − S k τL τ ( T ′ ) ≤ c | T ′ | τs/ k S − S k τL p ( T ′ ) ≤ cd ( T ′ ) τs k S − S k τL p ( T ′ ) . Case 1 (c).
If [ x, x + kh ]
6⊂ ∪ v ∈V T B v and [ x, x + kh ] intersects D or D , then δ > kt/ ˜ c > kt or δ > kt and hence [ x, x + kh ] ⊂ △ E ∩ △ E , which implies∆ kh S ( x ) = 0. No estimate is needed. Case 1 (d).
Let I hT ⊂ T be the quadrilateral bounded by the segments L , L − kh , D and the line with equation x = v + uh , u ∈ R , where v is thepoint of intersection of L with D , whenever this straight line intersects L . Ifthe line x = v + uh , u ∈ R , does not intersect L , then we replace it with the line x = v + uh , u ∈ R . Furthermore, we subtract B v and B v from I hT .Set J hT := I hT + [0 , kh ].A simple geometric argument shows that | J hT | ≤ δ kt .In estimating k ∆ kh S k τL τ ( I hT ) there are two subcases to be considered.If δ ( T ) ≤ kt/ ˜ c , we use (4.34) to obtain k ∆ kh S k τL τ ( I hT ) ≤ k ∆ kh S k τL τ ( I hT ) + k S − S k τL τ ( I hT ) + k S − S k τL τ ( J hT ∩ A ) . We estimate the above norms quite like in
Case 1 ( b ) , using Lemma 4.9. We have k S − S k τL τ ( I hT ) ≤ c ( | I hT | / | T | ) k S − S k τL τ ( T ) ≤ ctδ ( T )[ δ ( T ) d ( T )] − k S − S k τL τ ( T ) ≤ ctd ( T ) − | T | τs/ k S − S k τL p ( T ) ≤ ctd ( T ) − ( δ ( T ) d ( T )) τs/ k S − S k τL p ( T ) ≤ ct τs/ d ( T ) τs/ − k S − S k τL p ( T ) . For the second norm we get k S − S k τL τ ( J hT ∩ A ) ≤ c | J hT |k S − S k τL ∞ ( A ) ≤ ct | A | − τ/p k S − S k τL p ( A ) ≤ ct d ( A ) − τ/p k S − S k τL p ( A ) = ct d ( A ) τs − k S − S k τL p ( A ) , where as before we used the fact that 2 τ /p = 2 − τ s .From the above estimates we infer k ∆ kh S k τL τ ( I hT ) ≤ k ∆ kh S k τL τ ( I hT ) + ct τs/ d ( T ) τs/ − k S − S k τL p ( T ) + ct d ( A ) τs − k S − S k τL p ( A ) . Let δ ( T ) > kt/ ˜ c . We use (4.33) to obtain | ∆ kh S ( x ) | ≤ | ∆ kh S ( x ) | + | ∆ kh ( S − S )( x ) |≤ | ∆ kh S ( x ) | + ct k − k D k − ν ( S − S ) k L ∞ ([ x,x + kh ]) , implying k ∆ kh S k τL τ ( I hT ) ≤ k ∆ kh S k τL τ ( I hT ) + c | I hT | t τ ( k − k D k − ν ( S − S ) k τL ∞ ( I hT ∩ T ) + c | I hT | t τ ( k − k D k − ν ( S − S ) k τL ∞ ( A ) . ONLINEAR SPLINE APPROXIMATION 35
Clearly, k D k − ν ( S − S ) k L ∞ ( I hT ∩ T ) ≤ cδ ( T ) − ( k − k S − S k L ∞ ( T ) ≤ cδ ( T ) − ( k − | T | − /p k S − S k L p ( T ) ≤ cδ ( T ) − ( k − − /p k S − S k L p ( T ) , and k D k − ν ( S − S ) k L ∞ ( A ) ≤ cd ( A ) − ( k − k S − S k L ∞ ( A ) (4.35) ≤ cd ( A ) − ( k − − /p k S − S k L p ( A ) . Therefore, k ∆ kh S k τL τ ( I hT ) ≤ k ∆ kh S k τL τ ( I hT ) + ct τ ( k − δ ( T ) − τ ( k − − τ/p k S − S k τL p ( T ) + ct τ ( k − d ( A ) − τ ( k − − τ/p k S − S k τL p ( A ) . Case 1 (e) (Main).
Let T ⋆h ⊂ T h be the set defined by(4.36) T ⋆h := n x ∈ T h : [ x, x + kh ] ∩ L = ∅ and [ x, x + kh ] I hT [ v ∈V T B v o . We next estimate k ∆ kh S k τL τ ( T ⋆h ) .Let x ∈ T ⋆h . Denote by b and b the points where the line through x and x + kh intersects L and L . Set b = b ( x ) := b − b . We associate the segment[ x + b, x + b + kh ] to [ x, x + kh ] and ∆ kh S ( x + b ) to ∆ kh S ( x ).Since S ∈ Π k on △ E we have D k − ν S ( y ) = constant on [ b , x + b ] and hence(4.37) D k − ν S ( b − uν ) = D k − ν S ( b − uν ) for 0 ≤ u ≤ | x − b | . Similarly, since S ∈ Π k on △ E we have D k − ν S ( y ) = constant on [ x + kh, b ] andhence(4.38) D k − ν S ( b + uν ) = D k − ν S ( b + uν ) for 0 ≤ u ≤ | x + kh − b | . We use (4.33) and (4.37) - (4.38) to obtain∆ kh S ( x ) = | h | k − Z k | h || b − x | D k − ν S ( x + uν ) M ∗ k ( u ) du + | h | k − Z | b − x | D k − ν S ( x + uν ) M ∗ k ( u ) du = | h | k − Z k | h || b − x | D k − ν S ( x + uν ) M ∗ k ( u ) du + | h | k − Z | b − x | D k − ν S ( x + b + uν ) M ∗ k ( u ) du and ∆ kh S ( x + b ) = | h | k − Z k | h || b − x | D k − ν S ( x + b + uν ) M ∗ k ( u ) du + | h | k − Z | b − x | D k − ν S ( x + b + uν ) M ∗ k ( u ) du = | h | k − Z k | h || b − x | D k − ν S ( x + uν ) M ∗ k ( u ) du + | h | k − Z | b − x | D k − ν S ( x + b + uν ) M ∗ k ( u ) du. Therefore,∆ kh S ( x ) = ∆ kh S ( x + b ) + ∆ kh ( S − S )( x )= ∆ kh S ( x + b ) + | h | k − Z k | h || b − x | D k − ν [ S − S ] (cid:0) x + uν (cid:1) M ∗ k ( u ) du + | h | k − Z | b − x | D k − ν [ S − S ] (cid:0) x + b + uν (cid:1) M ∗ k ( u ) du and hence | ∆ kh S ( x ) | ≤ | ∆ kh S ( x + b ) | + ct k − k D k − ν ( S − S ) k L ∞ ([ b ,x + kh ]) (4.39) + ct k − k D k − ν ( S − S ) k L ∞ ([ x + b,b ]) The key here is that ([ b , x + kh ] ∪ [ x + b, b ]) ∩ T ◦ = ∅ .Let T ⋆⋆h := { x + b ( x ) : x ∈ T ⋆h } , where b ( x ) is defined above. By (4.39) we get k ∆ kh S k τL τ ( T ⋆h ) ≤ k ∆ kh S k τL τ ( T ⋆⋆h ) + ctd ( A ) t τ ( k − k D k − ν ( S − S ) k τL ∞ ( A ) + ctd ( A ) t τ ( k − k D k − ν ( S − S ) k τL ∞ ( A ) . Just as (4.35) we have k D k − ν ( S − S ) k L ∞ ( A ) ≤ cd ( A ) − ( k − k S − S k L ∞ ( A ) ≤ cd ( A ) − ( k − − /p k S − S k L p ( A ) , and similar estimates hold with A replaced by A . We use these above to obtain k ∆ kh S k τL τ ( T ⋆h ) ≤ k ∆ kh S k τL τ ( T ⋆⋆h ) + ct τ ( k − d ( A ) − τ ( k − − τ/p k S − S k τL p ( A ) + ct τ ( k − d ( A ) − τ ( k − − τ/p k S − S k τL p ( A ) . It is an important observation that no part of k ∆ kh S k τL τ ( T ⋆⋆h ) has been used forestimation of quantities k ∆ kh S k τL τ ( · ) from previous cases.Putting all of the above estimates together we arrive at k ∆ kh S k τL τ ( T h ) ≤ k ∆ kh S k τL τ ( T h ) + Y + Y + Y + Y , (4.40)where Y := X A ∈A tT ct d ( A ) τs − k S − S k τL p ( A ) + X A ∈ A tT cd ( A ) τs k S − S k τL p ( A ) , ONLINEAR SPLINE APPROXIMATION 37 Y := ct τ ( k − d ( A ) − τ ( k − − τ/p k S − S k τL p ( A ) + ct τ ( k − d ( A ) − τ ( k − − τ/p k S − S k τL p ( A ) ,Y := X T ′ ∈T tT ct τs/ d ( T ′ ) τs/ − k S − S k τL p ( T ′ ) + X T ′ ∈ T tT cd ( T ′ ) τs k S − S k τL p ( T ′ ) + ct τs/ d ( T ) τs/ − k S − S k τL p ( T ) , and Y := ct τ ( k − δ ( T ) − τ ( k − − τ/p k S − S k τL p ( T ) , if δ ( T ) > kt/ ˜ c, otherwise Y := 0. Remark 4.12.
In all cases we considered above but
Case 1 (e) we used the simpleinequality | ∆ kh S ( x ) | ≤ | ∆ kh S ( x ) | + | ∆ kh ( S − S )( x ) | to estimate k ∆ kh S k τL τ ( G ) forvarious sets G and this works because these sets are of relatively small measure. AsExample 3.2 shows this approach in principle cannot be used in Case 1 (e) and thisis the main difficulty in this proof. The gist of our approach in going around is toestimate | ∆ kh S ( x ) | by using | ∆ kh S ( x + b ) | with some shift b , where | ∆ kh S ( x + b ) | isnot used to estimate other terms | ∆ kh S ( x ′ ) | (there is a one-to-one correspondencebetween these quantities). Case 2.
Let Ω ⋆h be the set of all x ∈ Ω such that [ x, x + kh ] ⊂ Ω, [ x, x + kh ] ∩ A = ∅ for some A ∈ A with d ( A ) > kt/ ˜ c , and [ x, x + kh ] ∩ T = ∅ for all T ∈ T with d ( T ) ≥ kt/ ˜ c .Denote by V A the set of all vertices on ∂A and set B v := B ( v, kt/ ˜ c ), v ∈ V A .We next indicate how we estimate | ∆ kh S ( x ) | in different cases. Case 2 (a).
If [ x, x + kh ] ⊂ A , then ∆ kh S ( x ) = ∆ kh S ( x ) = 0 and no estimate isneeded. Case 2 (b).
If [ x, x + kh ] ⊂ ∪ v ∈V A B ( v, kt/ ˜ c ), we estimate | ∆ kh S ( x ) | trivially: | ∆ kh S ( x ) | ≤ | ∆ kh S ( x ) | + 2 k k X ℓ =0 | S ( x + ℓh ) − S ( x + ℓh ) | . Case 2 (c).
Let [ x, x + kh ] intersects the edge E =: [ w , w ] from ∂A , that isshared with A ′ ∈ A and [ x, x + kh ]
6⊂ ∪ v ∈V A B v . Let y := E ∩ [ x, x + kh ]. Evidently, | y − w j | > kt/ ˜ c , j = 1 ,
2, and in light of Lemma 4.11 we have [ x, x + kh ] ⊂ B ( y, kt ) ⊂ A ∪ A ′ . In this case we use the inequality | ∆ kh S ( x ) | ≤ | ∆ kh S ( x ) | + | ∆ kh ( S − S )( x ) |≤ | ∆ kh S ( x ) | + ct k − k D k − ν ( S − S ) k L ∞ ([ x,x + kh ]) , which follows by (4.33).The case when [ x, x + kh ] intersects an edge from ∂A that is shared with some T ∈ T is covered in Case 1 above.We proceed further similarly as in Case 1 and in the proof of Theorem 4.5 toobtain k ∆ kh S k τL τ (Ω ⋆t ) ≤ k ∆ kh S k τL τ (Ω ⋆t ) + Y + Y , (4.41) where Y := X A ∈A : d ( A ) ≥ kt/ ˜ c t τ ( k − cd ( A ) − τ ( k − − τ/p k S − S k τL p ( A ) + X A ∈A : d ( A ) ≥ kt/ ˜ c ct d ( A ) τs − k S − S k τL p ( A ) and Y := X A ∈A : d ( A ) ≤ kt/ ˜ c cd ( A ) τs k S − S k τL p ( A ) + X T ∈T : d ( T ) ≤ kt/ ˜ c cd ( T ) τs k S − S k τL p ( T ) . Case 3.
Let Ω ⋆⋆h be the set of all x ∈ Ω such that[ x, x + kh ] ⊂ ∪{ A ∈ A : d ( A ) ≤ kt/ ˜ c } ∪ { T ∈ T : d ( T ) ≤ kt/ ˜ c } . In this case we estimate | ∆ kh S ( x ) | trivially just as in (4.34). We obtain k ∆ kh S k τL τ (Ω ⋆⋆h ) ≤ k ∆ kh S k τL τ (Ω ⋆⋆h ) + X A ∈A : d ( A ) ≤ kt/ ˜ c c k S − S k τL τ ( A ) + X T ∈T : d ( T ) ≤ kt/ ˜ c c k S − S k τL τ ( T ) ≤ k ∆ kh S k τL τ (Ω ⋆⋆h ) + X A ∈A : d ( A ) ≤ kt/ ˜ c cd ( A ) τs k S − S k τL p ( A ) + X T ∈T : d ( T ) ≤ kt/ ˜ c cd ( T ) τs k S − S k τL p ( T ) . Just as in the proof of Theorem 3.3 it is important to note that in the aboveestimates only finitely many norms may overlap at a time. From above, (4.40), and(4.41) we obtain ω k ( S , t ) ττ ≤ ω k ( S , t ) ττ + A t + T t , where A t := X A ∈A : d ( A ) > kt/ ˜ c t τ ( k − cd ( A ) − τ ( k − − τ/p k S − S k τL p ( A ) + X A ∈A : d ( A ) > kt/ ˜ c ct d ( A ) τs − k S − S k τL p ( A ) + X A ∈A : d ( A ) ≤ kt/ ˜ c cd ( A ) τs k S − S k τL p ( A ) . ONLINEAR SPLINE APPROXIMATION 39 and T t := X T ∈T : δ ( T ) > kt/ ˜ c ct τ ( k − δ ( T ) − τ ( k − − τ/p k S − S k τL p ( T ) + X T ∈T : δ ( T ) > kt/ ˜ c ct τ ( k − δ ( T ) − τ ( k − − τ/p k S − S k τL p ( T ) + X T ∈T : d ( T ) > kt/ ˜ c ct τs/ d ( T ) τs/ − k S − S k τL p ( T ) + X T ∈T : d ( T ) ≤ kt/ ˜ c cd ( T ) τs k S − S k τL p ( T ) . We insert this estimate in (2.1) and interchange the order of integration andsummation to obtain | S | τB s,kτ ≤ | S | τB s,kτ + Z + Z , where Z := c X A ∈A d ( A ) − τ ( k − − τ/p k S − S k τL p ( A ) Z ˜ cd ( A ) / k t − τs + τ ( k − dt + c X A ∈A d ( A ) τs − k S − S k τL p ( A ) Z ˜ cd ( A ) / k t − τs +1 dt + c X A ∈A d ( A ) τs k S − S k τL p ( A ) Z ∞ ˜ cd ( A ) / k t − τs − dt and Z := c X T ∈T δ ( T ) − τ ( k − − τ/p k S − S k τL p ( T ) Z ˜ cδ ( T ) / k t − τs + τ ( k − dt + c X T ∈T δ ( T ) − τ ( k − − τ/p k S − S k τL p ( T ) Z ˜ cδ ( T ) / k t − τs + τ ( k − dt + c X T ∈T d ( T ) τs/ − k S − S k τL p ( T ) Z ˜ cd ( T ) / k t − τs/ dt + c X T ∈T d ( T ) sτ k S − S k τL p ( T ) Z ∞ ˜ cd ( T ) / k t − τs − dt. Observe that − τ s + τ ( k − > − s/ < k − /p which holdstrue by the hypothesis, and − τ s/ > − s < /τ = s + 2 /p whichis obvious. Therefore, all integrals above are convergent and taking into accountthat 2 − τ /p − τ s = 2 τ (1 /τ − /p − s/
2) = 0 we obtain | S | τB s,kτ ≤ | S | τB s,kτ + c X A ∈A∪T k S − S k τL p ( A ) ≤ | S | τB s,kτ + cn τ (1 /τ − /p ) (cid:16) X A ∈A∪T k S − S k τL p ( A ) (cid:17) τ/p = | S | τB s,kτ + cn τs/ k S k τL p (Ω) , where we used H¨older’s inequality. This completes the proof of Theorem 4.2. (cid:3) Acknowledgment.
We would like to give credit to Peter Petrov (Sofia University)with whom the second author discussed the theme of this article some years ago.
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