Approximation orders for interpolation by surface splines to rough functions
aa r X i v : . [ m a t h . NA ] M a y Approximation Orders for Interpolation by SurfaceSplines to Rough Functions
Rob Brownlee and Will Light Department of Mathematics and Computer Science, University of Leicester, UniversityRoad, Leicester LE1 7RH, England.
Abstract
In this paper we consider the approximation of functions by radial basic function inter-polants. There is a plethora of results about the asymptotic behaviour of the error betweenappropriately smooth functions and their interpolants, as the interpolation points fill out abounded domain in IR d . In all of these cases, the analysis takes place in a natural functionspace dictated by the choice of radial basic function – the native space. In many cases, thenative space contains functions possessing a certain amount of smoothness. We address thequestion of what can be said about these error estimates when the function being inter-polated fails to have the required smoothness. These are the rough functions of the title.We limit our discussion to surface splines, as an exemplar of a wider class of radial basicfunctions, because we feel our techniques are most easily seen and understood in this setting. This author was supported by a studentship from the Engineering and Physical Sciences ResearchCouncil. The first author would like to dedicate this paper to the memory of Will Light. INTRODUCTION
The process of interpolation by translates of a basic function is a popular tool for thereconstruction of a multivariate function from a scattered data set. The setup of the problemis as follows. We are supplied with a finite set of interpolation points
A ⊂ IR d and a function f : A →
IR. We wish to construct an interpolant to f of the form( Sf )( x ) = X a ∈A µ a ψ ( x − a ) + p ( x ) , for x ∈ IR d . (1.1)Here, ψ is a real-valued function defined on IR d , and the principle ingredient of our inter-polant is the use of the translates of ψ by the points in A . The function ψ is referred to asthe basic function . The function p in Equation (1.1) is a polynomial on IR d of total degreeat most k −
1. The linear space of all such polynomials will be denoted by Π k − . Of course,for Sf to interpolate f the real numbers µ a and the polynomial p must be chosen to satisfythe system ( Sf )( a ) = f ( a ) , for a ∈ A . It is natural to desire a unique solution to the above system. However, with the presentsetup, there are less conditions available to determine Sf than there are free parametersin Sf . There is a standard way of determining the remaining conditions, which are oftencalled the natural boundary conditions : X a ∈A µ a q ( a ) = 0 , for all q ∈ Π k − . It is now essential that A is Π k − –unisolvent. This means that if q ∈ Π k − vanishes on A then q must be zero. Otherwise the polynomial term can be adjusted by any polynomialwhich is zero on A . However, more conditions are needed to ensure uniqueness of theinterpolant. The requirement that ψ should be strictly conditionally positive definite oforder k is one possible assumption. To see explanations of why these conditions arise,the reader is directed to Cheney & Light (2000). In most of the common applications thefunction ψ is a radial function. That is, there is a function φ : IR + → IR such that ψ = φ ◦| · | ,where | · | is the Euclidean norm. In these cases we refer to ψ as a radial basic function .2uchon (1976, 1978) was amongst the first to study interpolation problems of this flavour.His approach was to formulate the interpolation problem as a variational one. To do thiswe assume we have a space of continuous functions X which carries a seminorm | · | . Theso-called minimal norm interpolant to f ∈ X on A from X is the function Sf ∈ X satisfying1. ( Sf )( a ) = f ( a ), for all a ∈ A ;2. | Sf | ≤ | g | , for all g ∈ X such that g ( a ) = f ( a ) for all a ∈ A .The spaces that Duchon considers are in fact spaces of tempered distributions which he isable to embed in C (IR d ). Let S ′ be the space of all tempered distributions on IR d . Theparticular spaces of distributions that we will be concerned with are called Beppo-Levispaces. The k th order Beppo-Levi space is denoted by BL k (Ω) and defined as BL k (Ω) = n f ∈ S ′ : D α f ∈ L (Ω), α ∈ ZZ d + , | α | = k o , with seminorm | f | k, Ω = X | α | = k c α Z Ω | ( D α f )( x ) | dx ! / , f ∈ BL k (Ω) . The constants c α are chosen so that the seminorm is rotationally invariant: X | α | = k c α x α = | x | k , for all x ∈ IR d . We assume throughout the paper that 2 k > d , because this has the affect that BL k (Ω) isembedded in the continuous functions (Duchon 1976). The spaces BL k (IR d ) give rise tominimal norm interpolants which are exactly of the form given in Equation (1.1), where theradial basic function is x
7→ | x | k − d or x
7→ | x | k − d log | x | , depending on the parity of d .It is perhaps no surprise to learn that the related functions ψ are strictly conditionallypositive definite of some appropriate order. The name given to interpolants employing thesebasic functions is surface splines. This is because they are a genuine multivariate analogueof the well-loved natural splines in one dimension.It is of central importance to understand the behaviour of the error between a function f : Ω → IR and its interpolant as the set
A ⊂
Ω becomes “dense” in Ω. The measure of3ensity we employ is the fill-distance h = sup x ∈ Ω min a ∈A | x − a | . One might hope that forsome suitable norm k · k there is a constant γ , independent of f and h , such that k f − Sf k = O ( h γ ) , as h → . In the case of the Beppo-Levi spaces, there is a considerable freedom of choice for the normin which the error between f and Sf is measured. The most widely quoted result concernsthe norm k · k L ∞ (Ω) , but for variety we prefer to deal with the L p -norm. To do this it ishelpful to assume Ω is a bounded domain, whose boundary is sufficiently smooth. In thiscase there is a constant C >
0, independent of f and h , such that for all f ∈ BL k (Ω), k f − Sf k L p (Ω) ≤ (cid:26) Ch k − d + dp | f | k, Ω , ≤ p ≤ ∞ Ch k | f | k, Ω , ≤ p < , as h →
0. (1.2)There has been considerable interest recently in the following very natural question.What happens if the function f does not possess sufficient smoothness to lie in BL k (Ω)? Itmay well be that f lies in BL m (Ω), where 2 k > m > d . The condition 2 m > d ensuresthat f ( a ) exists for each a ∈ A , and so Sf certainly exists. However, | f | k, Ω is not defined.It is simple to conjecture that the new error estimate should be k f − Sf k L p (Ω) ≤ (cid:26) Ch m − d + dp | f | m, Ω , ≤ p ≤ ∞ Ch m | f | m, Ω , ≤ p < , as h → . (1.3)It is perhaps surprising to the uninitiated reader that this estimate is not true even withthe reasonable restrictions we have placed on k and m . We are going to describe a recentresult from Johnson (2002). To do that, we recall the familiar definition of a Sobolev space.Let W k (Ω) denote the k th order Sobolev space, which consists of functions all of whosederivatives up to and including order k are in L (Ω). It is a Banach space under the norm k f k k, Ω = k X i =0 | f | i, Ω ! / , where f ∈ W k (Ω) . We have already tacitly alluded to the Sobolev embedding theorem which states that whenΩ is reasonably regular (for example, when Ω possesses a Lipschitz continuous boundary)and k > d/
2, then the space W k (Ω) can be embedded in C (Ω) (see Adams 1978, Theorem5.4, p. 97). Now Johnson’s result is as follows.4 heorem 1.1 (Johnson) . Let Ω be the unit ball in IR d and assume d/ < m < k . For every h > , there exists an f ∈ W m (IR d ) and a sequence of sets {A n } n ∈ IN with the followingproperties:(i) each set A n consists of finitely many points contained in Ω ;(ii) the fill-distance of each set A n is at most h ;(iii) if S nk f is the surface spline interpolant to f from BL k (IR d ) associated with A n ,for each n ∈ IN , then k S nk f k L (Ω) → ∞ as n → ∞ . If the surface spline interpolation operator is unbounded, there is of course no possibilityof getting an error estimate of the kind we conjectured. Johnson’s proof uses point sets whichhave a special feature. We define the separation distance of A n as q n = min {| a − b | / a, b ∈A n , a = b } . Let the fill-distance of each A n be h n . In Johnson’s proof, the construction of A n is such that q n /h n →
0. We make this remark, because Johnson’s result in one dimensionrefers to interpolation by natural splines, and in this setting the connection between theseparation distance and the unboundedness of S nk has been known for some time. Whatis also known in the one-dimensional case is that if the separation distance is tied to thefill-distance, then a result of the type we are seeking is true. Theorem 3.5 is the definitiveresult we obtain, and is the formalisation of the conjectured bounds in Equation (1.3).Subsequent to carrying out this work, we became aware of independent work by Yoon(2002). In that paper, error bounds for the case we consider here are also offered. Becauseof Yoon’s technique of proof, which is considerably different to our own, he obtains errorbounds for functions f with the additional restriction that f lies in W k ∞ (Ω), so the resultshere have wider applicability. However, Yoon does consider the shifted surface splines, whilstin this paper we have chosen to consider only surface splines as an exemplar of what canbe achieved. At the end of Section 3 we offer some comments on the difference between ourapproach and that of Yoon.To close this section we introduce some notation that will be employed throughout thepaper. The support of a function φ : IR d → IR is defined to be the closure of the set5 x ∈ IR d : φ ( x ) = 0 } , and is denoted by supp ( φ ). The volume of a bounded set Ω is thequantity R Ω dx and will be denoted vol(Ω). We make much use of the space Π m − , so forbrevity we fix ℓ as the dimension of this space. Finally, when we write b f we mean the Fouriertransform of f . The context will clarify whether the Fourier transform is the natural oneon L (IR d ): b f ( x ) = 1(2 π ) d/ Z IR d f ( t ) e − ixt dt, or one of its several extensions to L (IR d ) or S ′ . In this section we intend to collect together a number of useful results, chiefly about the sortsof extensions which can be carried out on Sobolev spaces. We begin with the well-knownresult which can be found in many of the standard texts. Of course, the precise nature ofthe set Ω in the following theorem varies from book to book, and we have not striven herefor the utmost generality, because that is not really a part of our agenda in this paper.
Theorem 2.1 (Adams 1978, Theorem 4.32, p. 91) . Let Ω be an open, bounded subset of IR d satisfying the uniform cone condition. For every f ∈ W m (Ω) there is an f Ω ∈ W m (IR d ) satisfying f Ω | Ω = f . Moreover, there is a positive constant K = K (Ω) such that for all f ∈ W m (Ω) , k f Ω k m, IR d ≤ K k f k m, Ω . We remark that the extension f Ω can be chosen to be supported on any compact subsetof IR d containing Ω. To see this, we construct f Ω in accordance with Theorem 2.1, thenselect η ∈ C m (IR d ) such that η ( x ) = 1 for x ∈ Ω. Now, if we consider the compactlysupported function f Ω0 = ηf Ω ∈ W m (IR d ), we have f Ω0 | Ω = f . An elementary application ofthe Leibniz formula gives k f Ω0 k m, IR d ≤ C k f k m, Ω , where C = C (Ω , η ) . One of the nice features of the above extension is that the behaviour of the constant K (Ω) can be understood for simple choices of Ω. The reason for this is of course the choiceof Ω and the way the seminorms defining the Sobolev norms behave under dilations andtranslations of Ω. 6 emma 2.2. Let Ω be a measurable subset of IR d . Define the mapping σ : IR d → IR d by σ ( x ) = a + h ( x − t ) , where h > , and a , t , x ∈ IR d . Then for all f ∈ W m ( σ (Ω)) , | f ◦ σ | m, Ω = h m − d/ | f | m,σ (Ω) . Proof.
We have, for | α | = m ,( D α ( f ◦ σ ))( x ) = h m ( D α f )( σ ( x )) . Thus, | f ◦ σ | m, Ω = X | α | = m c α Z Ω | ( D α ( f ◦ σ ))( x ) | dx = h m X | α | = m c α Z Ω | ( D α f )( σ ( x )) | dx. Now, using the change of variables y = σ ( x ), | f ◦ σ | m, Ω = h m − d X | α | = m c α Z σ (Ω) | ( D α f )( y ) | dy = h m − d | f | m,σ (Ω) . Unfortunately, the Sobolev extension refers to the Sobolev norm. We want to work witha norm which is more convenient for our purposes. This norm is in fact equivalent to theSobolev norm, as we shall now see.
Lemma 2.3.
Let Ω be an open subset of IR d having the cone property and a Lipschitz-continuous boundary. Let b , . . . , b ℓ ∈ Ω be unisolvent with respect to Π m − . Define a normon W m (Ω) via k f k Ω = | f | m, Ω + ℓ X i =1 | f ( b i ) | ! / , f ∈ W m (Ω) . There are positive constants K and K such that for all f ∈ W m (Ω) , K k f k m, Ω ≤ k f k Ω ≤ K k f k m, Ω . Proof.
The conditions imposed on m and Ω ensure that W m (Ω) is continuously embeddedin C (Ω) (Adams 1978, Theorem 5.4, p. 97). So, given x ∈ Ω, there is a constant C suchthat | f ( x ) | ≤ C k f k m, Ω for all f ∈ W m (Ω). Thus, there are constants C , . . . , C ℓ such that k f k ≤ | f | m, Ω + ℓ X i =1 C i k f k m, Ω ≤ (cid:16) ℓ X i =1 C i (cid:17) k f k m, Ω . (2.1)7n the other hand, suppose there is no positive number K with k f k m, Ω ≤ K k f k Ω for all f ∈ W m (Ω). Then there is a sequence { f j } in W m (Ω) with k f j k m, Ω = 1 and k f j k Ω ≤ j , for j = 1 , , . . . .The Rellich selection theorem (Braess 1997, Theorem 1.9, p. 32) states that W m (Ω) iscompactly embedded in W m − (Ω). Therefore, as { f j } is bounded in W m (Ω), this sequencemust contain a convergent subsequence in W m − (Ω). With no loss of generality we shallassume { f j } itself converges in W m − (Ω). Thus { f j } is a Cauchy sequence in W m − (Ω).Next, as k f j k Ω → | f j | m, Ω →
0. Moreover, k f j − f k k m, Ω = k f j − f k k m − , Ω + | f j − f k | m, Ω ≤ k f j − f k k m − , Ω + 2 | f j | m, Ω + 2 | f k | m, Ω . Since { f j } is a Cauchy sequence in W m − (Ω), and | f j | m, Ω →
0, it follows that { f j } is aCauchy sequence in W m (Ω). Since W m (Ω) is complete with respect to k · k m, Ω , this sequenceconverges to a limit f ∈ W m (Ω). By Equation (2.1), k f − f j k ≤ (cid:16) ℓ X i =1 C i (cid:17) k f − f j k m, Ω , and hence k f − f j k Ω → j → ∞ . Since k f j k Ω →
0, it follows that f = 0. Because k f j k m, Ω = 1, j = 1 , , . . . , it follows that k f k m, Ω = 1. This contradiction establishes theresult.We are almost ready to state the key result which we will employ in our later proofs abouterror estimates. Before we do this, let us make a simple observation. Look at the unisolventpoints b , . . . , b ℓ in the statement of the previous Lemma. Since W m (Ω) can be embeddedin C (Ω), it makes sense to talk about the interpolation projection P : W m (Ω) → Π m − based on these points. Furthermore, under certain nice conditions (for example Ω being abounded domain), P is the orthogonal projection of W m (Ω) onto Π m − . Lemma 2.4.
Let B be any ball of radius h and center a ∈ IR d , and let f ∈ W m ( B ) .Whenever b , . . . , b ℓ ∈ IR d are unisolvent with respect to Π m − let P b : C (IR d ) → Π m − bethe Lagrange interpolation operator on b , . . . , b ℓ . Then there exists c = ( c , . . . , c ℓ ) ∈ B ℓ and g ∈ W m (IR d ) such that . g ( x ) = ( f − P c f )( x ) for all x ∈ B ;2. g ( x ) = 0 for all | x − a | > h ;3. there exists a C > , independent of f and B , such that | g | m, IR d ≤ C | f | m,B .Furthermore, c , . . . , c ℓ can be arranged so that c = a .Proof. Let B be the unit ball in IR d and let B = 2 B . Let b , . . . , b ℓ ∈ B be unisolventwith respect to Π m − . Define σ ( x ) = h − ( x − a ) for all x ∈ IR d . Set c i = σ − ( b i ) for i = 1 , . . . , ℓ so that c , . . . , c ℓ ∈ B are unisolvent with respect to Π m − . Take f ∈ W m ( B ).Then ( f − P c f ) ◦ σ − ∈ W m ( B ). Set F = ( f − P c f ) ◦ σ − . Let F B be constructed as anextension to F on B . By Theorem 2.1 and the remarks following it, we can assume F B issupported on B . Define g = F B ◦ σ ∈ W m (IR d ). Let x ∈ B . Since σ ( B ) = B there is a y ∈ B such that x = σ − ( y ). Then, g ( x ) = ( F B ◦ σ )( x ) = F B ( y ) = (( f − P c f ) ◦ σ − )( y ) = ( f − P c f )( x ) . Also, for x ∈ IR d with | x − a | > h , we have | σ ( x ) | >
2. Since F B is supported on B , g ( x ) = 0 for | x − a | > h . Hence, g satisfies properties and . By Theorem 2.1 there is a K , independent of f and B , such that k F B k m,B = k F B k m, IR d ≤ K k F k m,B . We have seen in Lemma 2.3 that if we endow W m ( B ) and W m ( B ) with the norms k v k B i = | v | m,B i + ℓ X i =1 | v ( b i ) | ! / , i = 1 , , then k · k B i and k · k m,B i are equivalent for i = 1 ,
2. Thus, there are constants K and K ,independent of f and B , such that k F B k B ≤ K k F B k m,B ≤ K K k F k m,B ≤ K K K k F k B . Set C = K K K . Since F B ( b i ) = F ( b i ) = ( f − P c f )( σ − ( b i )) = ( f − P c f )( c i ) = 0 for i =1 , . . . , ℓ , it follows that | F B | m,B ≤ C | F | m,B . Thus, | g ◦ σ − | m, IR d ≤ C | ( f − P c f ) ◦ σ − | m,B .Now, Lemma 2.2 can be employed twice to give | g | m, IR d = h d/ − m | g ◦ σ − | m, IR d ≤ Ch d/ − m | ( f − P c f ) ◦ σ − | m,B = C | f − P c f | m,B . | f − P c f | m,B = | f | m,B to complete the first part of the proof. Theremaining part follows by selecting b = 0 and choosing b , . . . , b ℓ accordingly in the aboveconstruction. Lemma 2.5 (Duchon 1978) . Let Ω be an open, bounded, connected subset of IR d havingthe cone property and a Lipschitz-continuous boundary. Let f ∈ W m (Ω) . Then thereexists a unique element f Ω ∈ BL m (IR d ) such that f Ω | Ω = f , and amongst all elementsof BL m (IR d ) satisfying this condition, | f Ω | m, IR d is minimal. Furthermore, there exists aconstant K = K (Ω) such that, for all f ∈ W m (Ω) , | f Ω | m, IR d ≤ K | f | m, Ω . We arrive now at our main section, in which we derive the required error estimates. Ourstrategy is simple. We begin with a function f in BL m (IR d ). We want to estimate k f − S k f k for some suitable norm k · k , where S k is the minimal norm interpolation operator from BL k (IR d ), and k > m . We suppose that we already have an error bound using the norm k · k for all functions g ∈ BL k (IR d ). Our proof now proceeds as follows. Firstly, we adjust f in a somewhat delicate manner, obtaining a function F , still in BL m (IR d ), and withseminorm in BL m (IR d ) not too far away from that of f . We then smooth F by convolvingit with a function φ ∈ C ∞ (IR d ). The key feature of the adjustment of f to F is that( φ ∗ F )( a ) = f ( a ) for every point a in our set of interpolation points. It then follows that F ∈ BL k (IR d ). We then use the usual error estimate in BL k (IR d ). A standard procedure(Lemma 3.1) then takes us back to an error estimate in BL m (IR d ). Lemma 3.1.
Let m ≤ k and let φ ∈ C ∞ (IR d ) . For each h > , let φ h ( x ) = h − d φ ( x/h ) for x ∈ IR d . Then there exists a constant C > , independent of h , such that for all f ∈ BL m (IR d ) , | φ h ∗ f | k, IR d ≤ Ch m − k | f | m, IR d . Furthermore, we have | φ h ∗ f | k, IR d = o ( h m − k ) as h → . roof. The chain rule for differentiation gives ( D γ φ h )( x ) = h − ( d + | γ | ) ( D γ φ )( x/h ) for all x ∈ IR d , and γ ∈ ZZ d + . Thus, for β ∈ ZZ d + with | β | = m we have Z IR d | ( D γ φ h ∗ D β f )( x ) | dx = Z IR d (cid:12)(cid:12)(cid:12)(cid:12)Z IR d ( D γ φ h )( x − y )( D β f )( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) dx = h − d + | γ | ) Z IR d (cid:12)(cid:12)(cid:12)(cid:12)Z IR d ( D γ φ ) (cid:16) x − yh (cid:17) ( D β f )( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) dx = h − | γ | Z IR d (cid:12)(cid:12)(cid:12)(cid:12)Z IR d ( D γ φ )( t )( D β f )( x − ht ) dt (cid:12)(cid:12)(cid:12)(cid:12) dx = h − | γ | Z IR d (cid:12)(cid:12)(cid:12)(cid:12)Z K ( D γ φ )( t )( D β f )( x − ht ) dt (cid:12)(cid:12)(cid:12)(cid:12) dx, (3.1)where K = supp ( φ ). An application of the Cauchy-Schwartz inequality gives Z IR d | ( D γ φ h ∗ D β f )( x ) | dx ≤ h − | γ | Z IR d Z K | ( D γ φ )( t ) | dt ! Z K | ( D β f )( x − ht ) | dt ! dx, and so, Z IR d | ( D γ φ h ∗ D β f )( x ) | dx ≤ h − | γ | Z IR d | ( D γ φ )( t ) | dt Z IR d Z K | ( D β f )( x − ht ) | dtdx. (3.2)The Parseval formula together with the relation ( D α ( φ h ∗ f )) b = ( i · ) α ( φ h ∗ f ) b provide uswith the equality X | α | = k c α Z IR d | ( D α ( φ h ∗ f ))( x ) | dx = X | α | = k c α Z IR d | ( ix ) α ( φ h ∗ f ) b ( x ) | dx = Z IR d X | α | = k c α x α | ( φ h ∗ f ) b ( x ) | dx. (3.3)Now, when Equation (3.3) is used in conjunction with the relation X | α | = k c α x α = | x | k = | x | m + k − m ) = X | β | = m c β x β X | γ | = k − m c γ x γ ,
11e obtain X | α | = k c α Z IR d | ( D α ( φ h ∗ f ))( x ) | dx = Z IR d X | β | = m c β x β X | γ | = k − m c γ x γ | ( φ h ∗ f ) b ( x ) | dx = X | β | = m c β Z IR d X | γ | = k − m c γ x γ | ( ix ) β ( φ h ∗ f ) b ( x ) | dx = X | β | = m c β Z IR d X | γ | = k − m c γ x γ | ( D β ( φ h ∗ f )) b ( x ) | dx = X | β | = m c β X | γ | = k − m c γ Z IR d | ( ix ) γ ( D β ( φ h ∗ f )) b ( x ) | dx = X | β | = m c β X | γ | = k − m c γ Z IR d | ( D γ ( D β ( φ h ∗ f ))) b ( x ) | dx = X | β | = m c β X | γ | = k − m c γ Z IR d | ( D γ ( D β ( φ h ∗ f )))( x ) | dx. Since the operation of differentiation commutes with convolution, we have that X | α | = k c α Z IR d | ( D α ( φ h ∗ f ))( x ) | dx = X | β | = m c β X | γ | = k − m c γ Z IR d | ( D γ φ h ∗ D β f )( x ) | dx. (3.4)Combining Equation (3.2) with Equation (3.4) we deduce that X | α | = k c α Z IR d | ( D α ( φ h ∗ f ))( x ) | dx ≤ X | β | = m c β X | γ | = k − m c γ h − | γ | Z IR d | ( D γ φ )( t ) | dt Z IR d Z K | ( D β f )( x − ht ) | dtdx = h m − k ) | φ | k − m, IR d X | β | = m c β Z IR d Z K | ( D β f )( x − ht ) | dtdx. Fubini’s theorem permits us to change the order of integration in the previous inequality.Thus, X | α | = k c α Z IR d | ( D α ( φ h ∗ f ))( x ) | dx ≤ h m − k ) | φ | k − m, IR d X | β | = m c β Z K Z IR d | ( D β f )( x − ht ) | dxdt. Finally, a change of variables in the inner integral above yields X | α | = k c α Z IR d | ( D α ( φ h ∗ f ))( x ) | dx ≤ h m − k ) | φ | k − m, IR d X | β | = m c β Z K Z IR d | ( D β f )( z ) | dzdt. Setting C = | φ | k − m, IR d p vol( K ) we conclude that | φ h ∗ f | k, IR d ≤ Ch m − k | f | m, IR d as required.To deal with the remaining statement of the lemma, we observe that for γ = 0 we have Z K ( D γ φ )( t ) dt = Z IR d ( D γ φ )( t ) dt = ( d D γ φ )(0) = (( i · ) γ b φ )(0) = 0 . | β | = m , Z IR d | ( D γ φ h ∗ D β f )( x ) | dx = h − | γ | Z IR d (cid:12)(cid:12)(cid:12)(cid:12)Z K ( D γ φ )( t )(( D β f )( x − ht ) − ( D β f )( x )) dt (cid:12)(cid:12)(cid:12)(cid:12) dx. Now, if we continue in precisely the same manner as before, we obtain X | α | = k c α Z IR d | ( D α ( φ h ∗ f ))( x ) | dx ≤ h m − k ) | φ | k − m, IR d X | β | = m c β Z K Z IR d | ( D β f )( x − ht ) − ( D β f )( x ) | dxdt. Since D β f ∈ L (IR d ) for each β ∈ ZZ d + with | β | = m , it follows that for almost all t, x ∈ IR d , | ( D β f )( x − ht ) − ( D β f )( x ) | → , as h → g ( x, t ) = 2 | ( D β f )( x − ht ) | + 2 | ( D β f )( x ) | , for almost all x, t ∈ IR d ,we see that | ( D β f )( x − ht ) − ( D β f )( x ) | ≤ g ( x, t ) , for almost all x, t ∈ IR d and each h >
0. It follows by calculations similar to those usedabove that Z K Z IR d g ( x, t ) dxdt = 4vol( K ) Z IR d | ( D β f )( x ) | dx < ∞ . Applying Lebesgue’s dominated convergence theorem, we obtain Z K Z IR d | ( D β f )( x − ht ) − ( D β f )( x ) | dxdt → , as h → m ≤ k , | φ h ∗ f | k, IR d = o ( h m − k ) as h → Lemma 3.2.
Suppose φ ∈ C ∞ (IR d ) is supported on the unit ball and satisfies Z IR d φ ( x ) dx = 1 and Z IR d φ ( x ) x α dx = 0 , for all < | α | ≤ m − . For each ε > and x ∈ IR d , let φ ε ( x ) = ε − d φ ( x/ε ) . Let B be any ball of radius h and center a ∈ IR d . For a fixed p ∈ Π m − let f be a mapping from IR d to IR such that f ( x ) = p ( x ) forall x ∈ B . Then ( φ ε ∗ f )( a ) = p ( a ) for all ε ≤ h . roof. Let B denote the unit ball in IR d . We begin by employing a change of variables todeduce ( φ ε ∗ f )( a ) = Z IR d φ ε ( a − y ) f ( y ) dy = ε − d Z IR d φ (cid:16) a − yε (cid:17) f ( y ) dy = Z IR d φ ( x ) f ( a − xε ) dx = Z B φ ( x ) f ( a − xε ) dx. Then, for x ∈ B , | ( a − xε ) − a | ≤ ε ≤ h . Thus, f ( a − xε ) = p ( a − xε ) for all x ∈ B .Moreover, there are numbers b α such that p ( a − xε ) = p ( a ) + P < | α |≤ m − b α x α . Hence,( φ ε ∗ f )( a ) = Z B φ ( x ) p ( a − xε ) dx = Z IR d φ ( x ) (cid:18) p ( a ) + X <α ≤ m − b α x α (cid:19) dx = p ( a ) . Definition 3.3.
Let Ω be an open, bounded subset of IR d . Let A be a set of points in Ω .The quantity sup x ∈ Ω inf a ∈A | x − a | = h is called the fill-distance of A in Ω . The separationof A is given by the quantity q = min a,b ∈A a = b | a − b | . The quantity h/q will be called the mesh-ratio of A . Theorem 3.4.
Let A be a finite subset of IR d of separation q > and let d < m ≤ k .Then for all f ∈ BL m (IR d ) there exists an F ∈ BL k (IR d ) such that1. F ( a ) = f ( a ) for all a ∈ A ;2. there exists a C > , independent of f and q , such that | F | m, IR d ≤ C | f | m, IR d and | F | k, IR d ≤ Cq m − k | f | m, IR d .Proof. Take f ∈ BL m (IR d ). For each a ∈ A let B a ⊂ IR d denote the ball of radius δ = q/ a . For each B a let g a be constructed in accordance with Lemma 2.4. That is,for each a ∈ A take c ′ = ( c , . . . , c ℓ ) ∈ B ℓ − a and g a ∈ W m (IR d ) such that1. a, c , . . . , c ℓ are unisolvent with respect to Π m − ;14. g a ( x ) = ( f − P ( a,c ′ ) f )( x ) for all x ∈ B a ;3. P ( a,c ′ ) f ∈ Π m − and ( P ( a,c ′ ) f )( a ) = f ( a );4. g a ( x ) = 0 for all | x − a | > δ ;5. there exists a C >
0, independent of f and B a , such that | g a | m, IR d ≤ C | f | m,B a .Note that if a = b , then supp ( g a ) does not intersect supp ( g b ), because if x ∈ supp ( g a ) then | x − b | > | b − a | − | x − a | ≥ q − δ = 6 δ. Using the observation above regarding the supports of the g a ’s it follows that (cid:12)(cid:12)(cid:12)(cid:12)X a ∈A g a (cid:12)(cid:12)(cid:12)(cid:12) m, IR d = X | α | = m c α Z IR d (cid:12)(cid:12)(cid:12)(cid:12)X a ∈A ( D α g a )( x ) (cid:12)(cid:12)(cid:12)(cid:12) dx = X | α | = m c α X b ∈A Z supp ( g b ) (cid:12)(cid:12)(cid:12)(cid:12)X a ∈A ( D α g a )( x ) (cid:12)(cid:12)(cid:12)(cid:12) dx = X | α | = m c α X b ∈A Z supp ( g b ) | ( D α g b )( x ) | dx = X a ∈A | g a | m, IR d . Applying Condition 5 to the above equality we have (cid:12)(cid:12)(cid:12)(cid:12)X a ∈A g a (cid:12)(cid:12)(cid:12)(cid:12) m, IR d ≤ C X a ∈A | f | m,B a ≤ C | f | m, IR d . Now set H = f − P a ∈A g a . It then follows from Condition 2 above that H ( x ) = ( P ( a,c ′ ) f )( x )for all x ∈ B a , and from Condition 3 that H ( a ) = f ( a ) for all a ∈ A . Let φ ∈ C ∞ (IR d ) besupported on the unit ball and enjoy the properties Z IR d φ ( x ) dx = 1 and Z IR d φ ( x ) x α dx = 0 , for all 0 < | α | ≤ m − . Now set F = φ δ ∗ H . Using Lemma 3.1, there is a constant C >
0, independent of q and f , such that | F | k, IR d ≤ C δ m − k ) (cid:12)(cid:12)(cid:12)(cid:12) f − X a ∈A g a (cid:12)(cid:12)(cid:12)(cid:12) m, IR d ≤ C δ m − k ) (cid:18) | f | m, IR d + (cid:12)(cid:12)(cid:12)(cid:12)X a ∈A g a (cid:12)(cid:12)(cid:12)(cid:12) m, IR d (cid:19) ≤ C (1 + C ) δ m − k ) | f | m, IR d . C >
0, independent of q and f , such that | F | m, IR d ≤ C (cid:12)(cid:12)(cid:12)(cid:12) f − X a ∈A g a (cid:12)(cid:12)(cid:12)(cid:12) m, IR d ≤ C (1 + C ) | f | m, IR d . Thus | F | k, IR d ≤ Cq m − k | f | m, IR d and | F | m, IR d ≤ C | f | m, IR d for some appropriate constant C >
0. Since F = φ δ ∗ H and H | B a ∈ Π m − for each a ∈ A , it follows from Lemma 3.2 that F ( a ) = H ( a ) = f ( a ) for all a ∈ A . Theorem 3.5.
Let Ω be an open, bounded, connected subset of IR d satisfying the coneproperty and having a Lipschitz-continuous boundary. Suppose also d < m ≤ k . For each h > , let A h be a finite, Π k − –unisolvent subset of Ω with fill-distance h . Assume alsothat there is a quantity ρ > such that the mesh-ratio of each A h is bounded by ρ for all h > . For each mapping f : A h → IR , let S hk f be the minimal norm interpolant to f on A h from BL k (IR d ) . Then there exists a constant C > , independent of h , such that for all f ∈ BL m (Ω) , k f − S hk f k L p (Ω) ≤ (cid:26) Ch m − d + dp | f | m, Ω , ≤ p ≤ ∞ Ch m | f | m, Ω , ≤ p < , as h → . Proof.
Take f ∈ BL m (Ω). By Duchon (1976), f ∈ W m (Ω). We define f Ω in accordancewith Lemma 2.5. For most of this proof we wish to work with f Ω and not f , so forconvenience we shall write f instead of f Ω . Construct F in accordance with Theorem 3.4and set G = f − F . Then F ( a ) = f ( a ) and G ( a ) = 0 for all a ∈ A h . Furthermore, there isa constant C >
0, independent of f and h , such that | F | k, IR d ≤ C (cid:18) hρ (cid:19) m − k | f | m, IR d , (3.5) | G | m, IR d ≤ | f | m, IR d + | F | m, IR d ≤ (1 + C ) | f | m, IR d . (3.6)Thus S hk f = S hk F and S hm G = 0, where we have adopted the obvious notation for S hm .Hence, k f − S hk f k L p (Ω) = k f − S hk F k L p (Ω) = k F + G − S hk F k L p (Ω) ≤ k F − S hk F k L p (Ω) + k G − S hm G k L p (Ω) . C > C >
0, independent of h and f , such that k f − S hk f k L p (Ω) ≤ C h β ( k ) | F | k, Ω + C h β ( m ) | G | m, Ω , as h → β ( j ) = (cid:26) j − d + dp , ≤ p ≤ ∞ j, ≤ p < . Finally, using the bounds in Equations (3.5) and (3.6) we have k f − S hk f k L p (Ω) ≤ C h β ( m ) | f | m, IR d , as h → C >
0. To complete the proof we remind ourselves that we havesubstituted f Ω with f , and so an application of Lemma 2.5 shows that we can find C > k f − S hk f k L p (Ω) ≤ C h β ( m ) | f Ω | m, IR d ≤ C C h β ( m ) | f | m, Ω , as h → φ used in the proof of The-orem 3.5. However, Yoon’s approach is simply to smooth at this stage, obtaining theequivalent of our function F in the proof of Theorem 3.5. Because there is no preprocessingof f to H , Yoon’s function F does not enjoy the nice property F ( a ) = f ( a ) for all a ∈ A . Itis this property which makes the following step, where we treat G = f − F , a fairly simpleprocess. Correspondingly, Yoon has considerably more difficulty treating his function G .Our method also yields the same bound as that in Yoon, but for a wider class of functions.Indeed we would suggest that BL m (Ω) is the natural class of functions for which one wouldwish an error estimate of the type given in Theorem 3.5.17 eferences Adams, R. A. (1978),
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