L p Bernstein Inequalities and Inverse Theorems for RBF Approximation on R d
aa r X i v : . [ m a t h . C A ] O c t L p Bernstein Inequalities and Inverse Theorems forRBF Approximation on R d ∗ John Paul Ward
Abstract
Bernstein inequalities and inverse theorems are a recent develop-ment in the theory of radial basis function (RBF) approximation. Thepurpose of this paper is to extend what is known by deriving L p Bern-stein inequalities for RBF networks on R d . These inequalities involvebounding a Bessel-potential norm of an RBF network by its corre-sponding L p norm in terms of the separation radius associated withthe network. The Bernstein inequalities will then be used to prove thecorresponding inverse theorem. When analyzing an approximation procedure, there are typically two esti-mates that one is interested in determining. The first is a direct theoremthat gives the rate at which a function may be approximated in terms of itssmoothness, and the second is an inverse estimate that guarantees a certainamount of smoothness of an approximant based on its rate of approximation.Both are equally important, and if the results match up appropriately, theycan be combined to completely characterize smoothness spaces in terms ofthe approximation procedure. In this paper, our goal will be to prove aninverse theorem for RBF approximation. The usual way one does this is byfirst deriving a Bernstein inequality, since a standard technique can then beapplied to prove the inverse theorem.Bernstein inequalities date back to 1912 when S.N. Bernstein provedthe first inequality of this type for L ∞ norms of trigonometric polynomials, ∗ This paper consists of work from the author’s dissertation written under the super-vision of Professors F. J. Narcowich and J. D. Ward at Texas A&M University. Thisresearch was supported by grant DMS-0807033 from the National Science Foundation. T of degree n satisfies (cid:13)(cid:13)(cid:13) T ( r ) (cid:13)(cid:13)(cid:13) L p ≤ n r k T k L p for 1 ≤ p ≤ ∞ . However, the first example of a Bernstein-type inequality forRBF approximants was not proved until 2001, [9]. Then in 2006, Narcowich,Ward, and Wendland derived a more standard type of Bernstein inequality,[8]. They proved L Bernstein inequalities for approximants coming from anRBF approximation space on R d where the Fourier transform of the RBFhas algebraic decay. In the same year, Mhaskar proved L p Bernstein inequal-ities for certain Gaussian networks on R d , [5], and lastly, in [6], Mhaskar,Narcowich, Prestin, and Ward were able to prove Bernstein inequalities in L p norms for a large class of spherical basis functions.In this paper we will be concerned with RBF approximants to functionsin L p ( R d ) for 1 ≤ p ≤ ∞ . The approximants will be finite linear com-binations of translates of an RBF Φ, and the translates will come from acountable set X ⊂ R d . The error of this approximation, which is measuredin a Sobolev-type norm, depends on both the function Φ and the set X .Therefore, given an RBF Φ and a set X , we define the RBF approximationspace S X (Φ) by S X (Φ) = X ξ ∈ Y a ξ Φ( · − ξ ) : Y ⊂ X, Y < ∞ ∩ L ( R d ) , and note that these approximation spaces are closely related to the onesstudied in [3, 12] , where approximation rates are derived. By choosing Φand X properly, one is able to prove results about rates of approximationas well as the stability of the approximation procedure.Our goal will be to establish L p Bernstein inequalities for certain RBF ap-proximation spaces S X (Φ), and these inequalities will take the form k g k L k,p ≤ Cq − kX k g k L p , where L k,p is a Bessel-potential space. To prove this, we willuse band-limited approximation with the bandwidth proportional to 1 /q X .Thus 1 /q X acts similarly to a Nyquist frequency, and viewing 1 /q X as afrequency, we can see the connection to the classical Bernstein inequalitiesfor trigonometric polynomials. In particular, bandwidth is playing the roleof the degree of the polynomial from the classical inequality.The RBFs that we will be concerned with have finite smoothness; exam-ples include the Sobolev splines and thin-plate splines. The sets X will bediscrete subsets of R d with no accumulation points. For the inverse theorem,2e will additionally require that there do not exist arbitrarily large regionswith no point from X . The basic strategy that we will use is the following, which is the same asthe one used in [6]. Given g = P ξ ∈ X a ξ Φ( · − ξ ) ∈ S X (Φ), we choose anappropriate band-limited approximant g σ , and we have k g k L k,p ≤ k g σ k L k,p + k g − g σ k L k,p . We then split the second term into two ratios. k g k L k,p ≤ k g σ k L k,p + k a k ℓ p k g k L p k g − g σ k L k,p k a k ℓ p ! k g k L p . (1)The term k a k ℓ p / k g k L p will be bounded by a stability ratio R S,p that isindependent of the function g . We will then need to bound the error ofapproximating g by band-limited functions. Combining these results witha Bernstein inequality for band-limited functions, we will be able to provethe Bernstein inequality for all functions in S X (Φ), and afterward the cor-responding inverse theorem will follow. For any approximation procedure, one would like to determine the error ofthe approximation and the stability of the procedure. When considering anRBF approximation space S X (Φ), these quantities are bounded in terms ofcertain measurements of the set X . The error of approximation is given interms of the fill distance h X = sup x ∈ R d inf ξ ∈ X | x − ξ | , which measures how far a point in R d can be from X , and the stability ofthe approximation is determined by the separation radius q X = 12 inf ξ,ξ ′∈ Xξ = ξ ′ (cid:12)(cid:12) ξ − ξ ′ (cid:12)(cid:12) , which measures how close two points in X may be. In order to balance therate of approximation with the stability of the procedure, approximation is3estricted to sets X for which h X is comparable to q X , and sets for whichthe mesh ratio ρ X := h X /q X is bounded by a constant will be called quasi-uniform.Many of the results in this paper will be proved by working in the Fourierdomain, and we will use the following form of the Fourier transform in R d : b f ( ω ) = 1(2 π ) d/ Z R d f ( x ) e − iω · x dx. If f is a radial function, then there is a function ϕ : (0 , ∞ ) → R such that f ( x ) = ϕ ( | x | ), and in this case, the Fourier transform of f is given by b f ( ω ) = | ω | − ( d − / Z ∞ ϕ ( t ) t d/ J ( d − / ( | ω | t ) dt, where J ( d − / denotes the order ( d − / L k,p ( R d ), which coincide with the standard Sobolev spaces W k,p ( R d ) when k is a positive integer and 1 < p < ∞ , cf. [10, Section 5.3].The Bessel potential spaces are defined by L k,p = { f : b f = (1 + |·| ) − k/ b g, g ∈ L p ( R d ) } for 1 ≤ p ≤ ∞ , and they are equipped with the norm k f k L k,p = k g k L p . For the extremal cases p = 1 , ∞ , the relationships between the spaces L k,p and W k,p are more complex. For d = 1 and k even, the spaces are equivalent;however, when k is odd, neither function space is contained in the other forany d , cf. [10, Section 5.6]. In order to prove the Bernstein inequalities, we will need to require certainproperties of the RBFs involved. As much of the work will be done inthe Fourier domain, we state the constraints in terms of the RBFs’ Fouriertransforms. Given a radial function Φ : R d → R with (generalized) Fouriertransform b Φ, let φ : (0 , ∞ ) → R be the function defined by b Φ( ω ) = φ ( | ω | ).We will say a function Φ is admissible of order β if there exist constants C , C > β > d such that for all σ ≥ l ≤ l d := ⌈ ( d + 3) / ⌉ ,the function F ( t ) := φ ( t )(1 + t ) β/ satisfies4i) C ≤ F ( t ) ≤ C for t ≥ / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ddt (cid:19) l F ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C t − l for t ≥ / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ddt (cid:19) l F ( σt ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C for t ≥ / . Two particular classes of admissible functions are the Sobolev splinesand the thin-plate splines. The Sobolev spline Φ of order β > d is given byΦ = C |·| ( β − d ) / K ( d − β ) / ( |·| ) , where K is a modified Bessel function of the third kind. This functionpossesses the Fourier transform b Φ = (1 + |·| ) − β/ . With this definition, one can verify that the Sobolev splines of order β arethe canonical example of admissible functions of order β since F ( t ) = φ ( t )(1 + t ) β/ = 1 . They fit the theory particularly well as they are Green’s functions for theBessel potential differential operators, which we use to measure smoothness.For a positive integer m > d/
2, the thin-plate splines of order 2 m takethe form Φ = ( |·| m − d , d odd |·| m − d log |·| , d even , and possess the generalized Fourier transforms b Φ = C |·| − m . Admissibility of order 2 m can be verified by analyzing the function F ( t ) = φ ( t )(1 + t ) m = t − m (1 + t ) m . F ( l ) can then be bounded as follows: (cid:12)(cid:12)(cid:12) F ( l ) ( t ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X k =0 C k (cid:18) ddt (cid:19) l − k t − m (cid:18) ddt (cid:19) k (1 + t ) m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C l X k =0 t − m − ( l − k ) t m − k ≤ Ct − l . One of the essential results for proving the Bernstein inequalities is a boundof a stability ratio for S X (Φ). We define the L p stability ratio R S,p associatedwith this collection by R S,p = sup S X (Φ) ∋ g =0 k a k ℓ p k g k L p , where g = P ξ ∈ X a ξ Φ( ·− ξ ). The goal of this section is to bound the stabilityratio by Cq d/p ′ − βX for some C independent of a and X , where p ′ is theconjugate exponent to p . For this section, we will assume we are workingwith a fixed countable set X ⊂ R d with 0 < q X < β .To begin, fix Y = { ξ j } Nj =1 ⊂ X and g = P Nj =1 a j φ ( · − ξ j ). We willderive a bound for k a k ℓ p / k g k L p and show the bound is independent of Y and a . The strategy for proving this is as follows. Let K be a smoothfunction and define b K σ ( ω ) = b K ( ω/σ ). We will then consider the convo-lutions K σ ∗ g ( x ) = P Nj =1 a j K σ ∗ Φ( x − ξ j ). For an appropriate choice of σ , the interpolation matrix ( A σ ) i,j = K σ ∗ Φ( ξ i − ξ j ) will be invertible,and the norm of its inverse will be bounded. Then a = A − σ ( K σ ∗ g ) | Y and k a k ℓ p ≤ (cid:13)(cid:13) A − σ (cid:13)(cid:13) ℓ p k K σ ∗ g | Y k ℓ p . We will then be left with bounding k K σ ∗ g | Y k ℓ p in terms of q X and k g k L p . We now define the class of smooth functions with which we will convolve g .Consider a Schwartz class functions K : R d → R that satisfies:(i) There is a κ : [0 , ∞ ) → [0 , ∞ ) such that b K ( ω ) = κ ( | ω | ).6ii) κ ( r ) = 0 for r ∈ [0 ,
1] and κ is non-vanishing on an open set.Given such a K , we define the related family { K σ } σ ≥ by b K σ ( ω ) = b K ( ω/σ ).Note that property (ii) requires each function K σ to have a Fourier transformwhich is 0 in a neighborhood of the origin, and as σ increases, so does thisneighborhood. The convolution Φ ∗ K σ will retain this property and allowus to obtain diagonal dominance in A σ .Before moving on, we will need to determine certain bounds on thefunctions K σ . First we need an L ∞ bound. | K σ ( x ) | ≤ C d Z R d b K σ ( ω ) dω ≤ C d σ d Z ∞ κ ( t ) t d − dt, so | K σ ( x ) | ≤ C d σ d . (2)Next we will need a bound on K σ for r = | x | >
0. Writing K σ as aFourier integral, we see | K σ ( x ) | = r − ( d − / (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ σ κ ( t/σ ) t d/ J ( d − / ( rt ) dt (cid:12)(cid:12)(cid:12)(cid:12) , and by a change of variables, we have | K σ ( x ) | = σ d/ r − ( d − / (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ κ ( t ) t d/ J ( d − / ( σrt ) dt (cid:12)(cid:12)(cid:12)(cid:12) . Since K is a Schwartz class function, κ is also smooth. Additionally, κ isrequired to be 0 in a neighborhood of the origin. Therefore, κ satisfies theconditions of Proposition A.2, and | K σ ( x ) | ≤ C d σ d/ − l d r ( d − / l d . (3)Using (2) and (3), we will prove a bound of L p ′ norms of linear combi-nations of translates of K σ . Proposition 2.1.
Let T : R N → L ( R d ) ∩ L ∞ ( R d ) be the linear operatordefined by T ( γ ) = N X j =1 γ j K σ ( x − ξ j ) . hen k T ( γ ) k L p ′ ≤ C d σ d/p (cid:18) σq X ) ( d − / l d (cid:19) /p k γ k ℓ p ′ . Proof.
After proving the bound in the cases p = 1 and p = ∞ , the resultwill then follow by the Riesz-Thorin theorem (cf. [11, Chapter 5] ). First,by Proposition B.1 we have, N X j =1 | K σ ( x − ξ j ) | ≤ k K σ k L ∞ + X | x − ξ j |≥ q | K σ ( x − ξ j ) |≤ C d σ d + σ d/ − l d q ( d − / l d X ! . Simplifying the previous expression, we obtain N X j =1 | K σ ( x − ξ j ) | ≤ C d σ d (cid:18) σq X ) ( d − / l d (cid:19) . (4)Therefore k T ( γ ) k L ∞ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X j =1 γ j K σ ( x − ξ j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X j =1 | K σ ( x − ξ j ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ k γ k ℓ ∞ ≤ C d σ d (cid:18) σq X ) ( d − / l d (cid:19) k γ k ℓ ∞ Now in the case p = ∞ , k T ( γ ) k L = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X j =1 γ j K σ ( x − ξ j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ≤ N X j =1 | γ j | k K σ k L ≤ C k γ k ℓ . .2 Interpolation matrices Here, the interpolation matrices ( A σ ) i,j = K σ ∗ Φ( ξ i − ξ j ) will be shown tobe invertible by the following lemma. In fact, the function K σ ∗ Φ is positivedefinite, even in the case where Φ is only conditionally positive definite.Furthermore, the lemma will provide a bound for the ℓ p norm of A − σ . Lemma 2.2 ([6, Lemma 5.2]) . Given an n × n matrix A , denote its diagonalpart by D , and let F = A − D . If D is invertible and (cid:13)(cid:13) D − F (cid:13)(cid:13) ℓ < , then A is invertible and (cid:13)(cid:13) A − (cid:13)(cid:13) ℓ < (cid:13)(cid:13) D − (cid:13)(cid:13) ℓ (1 − (cid:13)(cid:13) D − F (cid:13)(cid:13) ℓ ) − . The diagonal entries of A σ are equal to K σ ∗ Φ(0), and the off diagonalabsolute column sums are of the form P i = j | K σ ∗ Φ( ξ i − ξ j ) | . In order toapply the lemma, we must bound the former from below and the latter fromabove. First, K σ ∗ Φ(0) = C d Z ∞ σ κ ( t/σ ) φ ( t ) t d − dt = C d σ d − β Z ∞ κ ( t ) t β − d +1 ( σt ) β φ ( σt ) dt ≥ C Φ ,d σ d − β Z ∞ κ ( t ) t β − d +1 dt. The last inequality can be verified by considering the representation( σt ) β φ ( σt ) = ( σt ) β (1 + ( σt ) ) β/ (cid:16) φ ( σt )(1 + ( σt ) ) β/ (cid:17) and applying the definition of admissibility. It now follows that K σ ∗ Φ(0) ≥ C Φ ,d σ d − β . (5)Next, we need a bound on | K σ ∗ Φ( x ) | for x = 0. Since K σ ∗ Φ has aradial Fourier transform in L ( R d ), we can write it as a one dimensionalintegral. Note that in the following integral r = | x | . | K σ ∗ Φ( x ) | = C d r − ( d − / (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ σ κ ( t/σ ) φ ( t ) t d/ J ( d − / ( rt ) dt (cid:12)(cid:12)(cid:12)(cid:12) = C d σ ( d +2) / − β r ( d − / (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ κ ( t ) t β ( σt ) β φ ( σt ) t d/ J ( d − / ( σrt ) dt (cid:12)(cid:12)(cid:12)(cid:12) . κ and its derivatives implythat the integrand satisfies the conditions of Proposition A.2, so | K σ ∗ Φ( x ) | ≤ C Φ ,d σ ( d +2) / − β r ( d − / ( σr ) l d . With this estimate we can bound the off diagonal absolute column sums of A σ . Using Proposition B.1 we have X i = j | K σ ∗ Φ( ξ i − ξ j ) | ≤ C Φ ,d σ d − β ( σq X ) ( d − / l d . (6)We are now ready to apply the lemma. Define M = max ( , (cid:18) C ,d C ,d (cid:19) / (( d − / l d ) ) , where the constants C ,d and C ,d are from (5) and (6) respectively. Wethen define σ = M/q X , so( K σ ∗ Φ(0)) − X i = j | K σ ∗ Φ( ξ i − ξ j ) | ≤ . Therefore (cid:13)(cid:13) A − σ (cid:13)(cid:13) ℓ ≤ C Φ ,d σ β − d , and in terms of q X , (cid:13)(cid:13) A − σ (cid:13)(cid:13) ℓ ≤ C Φ ,d q d − βX . (7)As A σ is self-adjoint the same bound holds for (cid:13)(cid:13) A − σ (cid:13)(cid:13) ℓ ∞ . The Riesz-Thorin interpolation theorem can then be applied to get (cid:13)(cid:13) A − σ (cid:13)(cid:13) ℓ p ≤ C Φ ,d q d − βX (8)for 1 ≤ p ≤ ∞ . To finish the bound of the stability ratio we require a bound of a discretenorm by a continuous one. To accomplish this, we can use an argumentsimilar to the proof of [7, Theorem 1].
Proposition 2.3. If ≤ p ≤ ∞ and f ∈ L p , then k K σ ∗ f | Y k ℓ p ≤ C d q − d/pX k f k L p . roof. Let p ′ be the conjugate exponent to p , i.e. 1 /p + 1 /p ′ = 1. Thensince ℓ p ′ is dual to ℓ p , there is a vector γ ∈ R n such that k γ k ℓ p ′ = 1 and k K σ ∗ f | Y k ℓ p = N X j =1 γ j K σ ∗ f ( ξ j ) . An explicit construction of γ can be found in [4, Proposition 6.13]. Writingthe convolution as an integral, we get k K σ ∗ f | Y k ℓ p = N X j =1 γ j Z R d K σ ( ξ j − y ) f ( y ) dy ≤ Z R d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j =1 γ j K σ ( ξ j − y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | f ( y ) | dy We can now apply H¨older’s inequality to get k K σ ∗ f | Y k ℓ p ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X j =1 γ j K σ ( · − ξ j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ′ k f k L p , and finally applying Proposition 2.1 gives the result k K σ ∗ f | Y k ℓ p ≤ C d σ d/p (cid:18) σ q X ) ( d − / l d (cid:19) /p k γ k ℓ p ′ k f k L p ≤ C d q − d/pX k f k L p . We are now in a position to prove the bound on the stability ratio for p ∈ [1 , ∞ ]. Recall X is a countable subset of R d with 0 < q X <
1, and Φ isan admissible function of order β . Theorem 2.4.
Let R S,p be the stability ratio associated with S X (Φ) . Then R S,p = sup S X (Φ) ∋ g =0 k a k ℓ p k g k L p ≤ C Φ ,d q d/p ′ − βX . roof. It has been shown that the interpolation matrix ( A σ ) i,j = K σ ∗ Φ( ξ i − ξ j ) is invertible. Therefore k a k ℓ p ≤ (cid:13)(cid:13) A − σ (cid:13)(cid:13) ℓ p k K σ ∗ g | Y k ℓ p . Using(8), we get k a k ℓ p ≤ C Φ ,d q d − βX k K σ ∗ g | Y k ℓ p . Finally, applying the M-Z inequality gives the result.
As in the previous section, X will be a fixed countable set with 0 < q X < β . At this point, we are leftwith bounding the two remaining terms of (1). This will require choosingband-limited functions that approximate the elements of S X (Φ) and satisfythe Bernstein inequality as well. In particular, given g ∈ S X (Φ) we need tofind a band-limited function g σ so that k g − g σ k L k,p k a k ℓ p ≤ Cq β − k − d/p ′ X for 1 ≤ p ≤ ∞ . Since most of the work will be done in the Fourier domain,we will impose the condition k < β − d . We begin by defining a class of band-limited functions. A function g ∈ S X (Φ) will be convolved with one of these functions in order to define itsband-limited approximant. Consider a Schwartz class function K : R d → R that satisfies the following properties:(i) There is a non-increasing κ : [0 , ∞ ) → [0 , ∞ ) such that b K ( ω ) = κ ( | ω | )(ii) κ ( ω ) = 1 for ω ≤ , and κ ( ω ) = 0 for ω ≥ K is different from the one introduced in Section 2, and inthis section K will be of the form described above. Given such a K , wedefine the family of functions { K σ } σ ≥ by b K σ ( ω ) = b K ( ω/σ ). Band-limitedapproximants to g ∈ S X (Φ) are then defined by g σ = K σ ∗ g . The first thingwe must check is that g σ satisfies the Bernstein inequality. The followinglemma addresses this issue. Lemma 3.1.
Let f ∈ L p ( R d ) , then k f ∗ K σ k L m,p ≤ C d σ k f ∗ K σ k L m − ,p for ≤ p ≤ ∞ and any positive integer m . roof. First, notice that we can write K σ ∗ f = K σ ∗ ( K σ ∗ f ) , so h (1 + |·| ) m/ b K σ b f i ∨ = h (1 + |·| ) / b K σ (1 + |·| ) ( m − / b K σ b f i ∨ = h (1 + |·| ) / b K σ i ∨ ∗ h (1 + |·| ) ( m − / b K σ b f i ∨ . As K is a Schwartz class function, the first function in the convolution is in L . Likewise, h (1 + |·| ) ( m − / b K σ i ∨ is also in L . Additionally, f being in L p implies that the second function of the convolution is in L p . ThereforeYoung’s inequality implies k f ∗ K σ k L m,p = (cid:13)(cid:13)(cid:13)(cid:13)h (1 + |·| ) m/ b K σ b f i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L p ≤ (cid:13)(cid:13)(cid:13)(cid:13)h (1 + |·| ) / b K σ i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L (cid:13)(cid:13)(cid:13)(cid:13)h (1 + |·| ) ( m − / b K σ b f i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L p = (cid:13)(cid:13)(cid:13)(cid:13)h (1 + |·| ) / b K σ i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L k f ∗ K σ k L m − ,p . Now it is known that there exist finite measures ν and λ such that(1 + | x | ) / = b ν ( x ) + 2 π | x | b λ ( x ) , cf. [10, Chapter 5]. We therefore have (cid:13)(cid:13)(cid:13)(cid:13)h (1 + |·| ) / b K σ i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L = (cid:13)(cid:13)(cid:13)(cid:13)hb ν b K σ + 2 π |·| b λ b K σ i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L ≤ k ν ∗ K σ k L + 2 π (cid:13)(cid:13)(cid:13)(cid:13) λ ∗ h |·| b K σ i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L ≤ C (cid:18) (cid:13)(cid:13)(cid:13)(cid:13)h |·| b K σ i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L (cid:19) , (cid:13)(cid:13)(cid:13)(cid:13)h |·| b K σ i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L ≤ Cσ . First (cid:13)(cid:13)(cid:13)(cid:13)h |·| b K σ i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ≤ C d Z R d | ω | b K σ ( ω ) dω = C d Z σ tκ (cid:18) t σ (cid:19) t d − dt = C d σ d Z κ (cid:18) t (cid:19) t d dt, and hence (cid:13)(cid:13)(cid:13)(cid:13)h |·| b K σ i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ≤ C d σ d . (9)Now, for | x | = r > (cid:12)(cid:12)(cid:12)(cid:12)h |·| b K σ i ∨ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = r − ( d − / (cid:12)(cid:12)(cid:12)(cid:12)Z σ tκ (cid:18) t σ (cid:19) t d/ J ( d − / ( rt ) dt (cid:12)(cid:12)(cid:12)(cid:12) = r − ( d − / σ d/ (cid:12)(cid:12)(cid:12)(cid:12)Z tκ (cid:18) t (cid:19) t d/ J ( d − / ( σrt ) dt (cid:12)(cid:12)(cid:12)(cid:12) , so by Proposition A.3 (cid:12)(cid:12)(cid:12)(cid:12)h |·| b K σ i ∨ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C d r − ( d − / σ d/ ( σr ) l d . (10)Utilizing inequalities 9 and 10 gives (cid:13)(cid:13)(cid:13)(cid:13)h |·| b K σ i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L ≤ C d σ − d (cid:13)(cid:13)(cid:13)(cid:13)h |·| b K σ i ∨ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ + Z | x |≥ σ (cid:12)(cid:12)(cid:12)(cid:12)h |·| b K σ i ∨ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ C d σ + C d Z ∞ σ r − ( d − / σ d/ ( σr ) l d r d − dr ≤ C d σ. Corollary 3.2.
Define σ to be /q X . Then for all g ∈ S X (Φ) , k g ∗ K σ k L k,p ≤ C Φ ,d q − kX k g k L p . .2 Approximation Analysis Now that we know the band-limited approximants to the elements of S X (Φ)satisfy the Bernstein inequality, we must bound the error of approximationin L k,p . We will begin by bounding the approximation error in L k, and L k, ∞ and then use interpolation to obtain the result for all other values of p . Inboth extremal cases, this reduces to bounding the error of approximating theRBF by band-limited functions. For p = 1 this is straightforward; however,the case p = ∞ is more involved.In order to simplify some expressions, we define the functions E Φ ,k := (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) (1 + |·| ) k/ (Φ − Φ ∗ K σ ) ∧ (cid:17) ∨ (cid:12)(cid:12)(cid:12)(cid:12) ,F ( t ) := φ ( t )(1 + t ) β/ . If we are to bound the error of approximating Φ by band-limited functions,we will certainly need a point-wise bound of E Φ ,k . Let us begin with an L ∞ bound. E Φ ,k ( x ) ≤ Z R d (1 − b K σ ( ω )) b Φ( ω )(1 + | ω | ) k/ dω ≤ C d σ d − β + k Z ∞ / (1 − κ ( t )) F ( σ t ) t d − ((1 /σ ) + t ) ( β − k ) / dt. Therefore E Φ ,k ( x ) ≤ C β,d σ d − β + k . (11)Next, for | x | = r > E Φ ,k ( x ) = r − ( d − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ σ / (1 − κ ( t/σ )) F ( t )(1 + t ) ( β − k ) / t d/ J ( d − / ( rt ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = σ d/ − β + k r ( d − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ / (1 − κ ( t )) F ( σ t )((1 /σ ) + t ) ( β − k ) / t d/ J ( d − / ( σ rt ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Therefore by Proposition A.2 E Φ ,k ( x ) ≤ C Φ ,d r − ( d − / σ d/ − β + k ( σ r ) l d . (12)With these results we are now able to bound the error of approximation. Theorem 3.3.
Let Y = { ξ i } Ni =1 be a finite subset of a quasi-uniform set X .Given g = P Nj =1 a j Φ( · − ξ j ) ∈ S X (Φ) , we have g − g ∗ K σ k L k,p ≤ C Φ ,d q β − k − d/p ′ X k a k ℓ p for ≤ p ≤ ∞ .Proof. We will show that this holds when p = 1 and p = ∞ , and the resultwill follow from the Riesz-Thorin interpolation theorem. In this case, theoperator being interpolated is T : ℓ p ( Y ) → L p ( R d ), where T ( a ) = (Id − ∆) k/ N X j =1 a j Φ( · − ξ j ) − K σ ∗ N X j =1 a j Φ( · − ξ j ) . Letting g σ = g ∗ K σ , we have k g − g σ k L k, ∞ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X j =1 a j Φ( · − ξ j ) − N X j =1 a j Φ ∗ K σ ( · − ξ j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L k, ∞ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X j =1 a j (cid:16) (1 + |·| ) k/ (Φ − Φ ∗ K σ ) ∧ (cid:17) ∨ ( · − ξ j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ , and using the notation for E Φ ,k above k g − g σ k L k, ∞ ≤ k a k ℓ ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X j =1 E Φ ,k ( · − ξ j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ Now by (11), (12), and Proposition B.1 N X j =1 E Φ ,k ( x − ξ j ) ≤ k E Φ ,k k L ∞ + X | x − ξ j |≥ q Y E Φ ,k ( x − ξ j ) ≤ C Φ ,d σ d − β + k (cid:18) σ q X ) ( d − / l d (cid:19) . For p = 1, k g − g σ k L k, ≤ k a k ℓ k E Φ ,k k L , and k E Φ ,k k L ≤ C d q dX k E Φ ,k k L ∞ + Z | x |≥ q X E Φ ,k ( x ) dx ≤ C Φ ,d σ k − β + C Φ ,d Z ∞ q σ d/ − β + k − l d r ( d − / l d − d +1 dr ≤ C Φ ,d σ k − β . Bernstein Inequalities and Inverse Theorems
In approximation theory, there are a variety of applications for Bernsteininequalities. While they are most commonly associated with the derivationof inverse theorems, they can also be useful in proving direct theorems.For example, a Bernstein inequality for multivariate polynomials is used incertain RBF approximation error estimates, cf. [13, Chapter 11]. However,in this paper, we will only address the Bernstein inequalities themselves andtheir matching inverse theorems.With the bound of the stability ratio and the band-limited approximationestimate in hand, we are in a position to prove the Bernstein inequalities.
Theorem 4.1.
Let X be a countable set with < q X < , and let Φ be anadmissible function of order β . If k < β − d , ≤ p ≤ ∞ , and g ∈ S X (Φ) ,then k g k L k,p ≤ C Φ ,d q − kX k g k L p Proof.
Let g σ be the previously defined approximant of g . Then k g k L k,p ≤ k g σ k L k,p + k g − g σ k L k,p = k g σ k L k,p + k a k ℓ p k g k L p k g − g σ k L k,p k a k ℓ p ! k g k L p Applying Theorem 2.4, Corollary 3.2, and Theorem 3.3 k g k L k,p ≤ k g σ k L k,p + k a k ℓ p k g k L p k g − g σ k L k,p k a k ℓ p ! k g k L p ≤ C Φ ,d q − kX k g k L p + (cid:18) C Φ ,d q dp ′ − βX (cid:19) (cid:18) C Φ ,d q β − k − dp ′ X (cid:19) k g k L p ≤ C Φ ,d q − kX k g k L p . Having established Bernstein inequalities for S X (Φ), we can now provethe corresponding inverse theorem. Theorem 4.2.
Let { X n } n ≥ be a nested sequence ( X n ⊂ X n +1 ) of countablesets in R d satisfying: ρ X n ≤ C for some constant C > and < h X n , q X n < − n . Furthermore, suppose ≤ p ≤ ∞ , f ∈ L p ( R d ) , and Φ is admissible f order β . If there is a constant c f > , independent of n , and a positiveinteger l such that inf g ∈ S Xn (Φ) k f − g k L p ( R d ) ≤ c f h ln , then f ∈ L k,p for every ≤ k < min { β − d, l } .Proof. Let f n ∈ S X n be a sequence of functions satisfying k f − f n k L p ≤ c f h ln . Note that f n ∈ S X m for m > n because the sets X n are nested.Using the notation h n = h X n and q n = q X n , we have k f n +1 − f n k L k,p ≤ C Φ ,d (cid:18) h n +1 q n +1 (cid:19) k h − kn +1 k f n +1 − f n k L p ≤ C Φ ,d h − kn +1 ( k f n +1 − f k L p + k f − f n k L p ) ≤ C Φ ,d,f h − kn +1 ( h ln +1 + h ln ) , and since h n < − n , it follows that k f n +1 − f n k L k,p ≤ C Φ ,d,f − ( l − k ) n . This shows f n is a Cauchy sequence in L k,p . Since L k,p is complete, f n converges to some function e f ∈ L k,p . Since f n converges to both f and e f in L p , f = e f a.e., and therefore f ∈ L k,p . A Bessel Functions and Fourier Integrals A d -dimensional Fourier integral of a radial function reduces to a one-dimensional integral involving a Bessel function of the first kind. Here,we list some of the properties of these Bessel functions and prove boundsfor the corresponding Fourier integrals. Proposition A.1 ([13, Proposition 5.4 & Proposition 5.6]) . (1) ddz { z ν J ν ( z ) } = z ν J ν − ( z ) (2) J / ( z ) = q πz sin( z ) , J − / ( z ) = q πz cos( z ) (3) J ν ( r ) = q πr cos( r − νπ − π ) + O ( r − / ) for r → ∞ and ν ∈ R (4) J l/ ( r ) ≤ l +2 πr for r > and l ∈ N lim r → r − l J l/ ( r ) = l Γ ( l/ for l ∈ N The next proposition makes use of integration by parts in order to boundthe Fourier integral of a function whose support lies outside of a neighbor-hood of the origin.
Proposition A.2.
Let α ≥ , and let f ∈ C n ([0 , ∞ )) for some naturalnumber n > . Also, assume there are constants C, ǫ > such that f = 0 on [0 , ] and (cid:12)(cid:12) f ( j ) ( t ) (cid:12)(cid:12) ≤ Ct − d − ǫ for j ≤ n and t > / . Then there is aconstant C ǫ such that (cid:12)(cid:12)(cid:12)R ∞ / f ( t ) t d/ J ( d − / ( αt ) dt (cid:12)(cid:12)(cid:12) ≤ C ǫ α − n Proof.
We first define a sequence of functions arising when integrating byparts. Let f = f , f = f ′ , and f j = (cid:16) f j − t (cid:17) ′ for j ≥
2. Note that when j ≥
2, there are constants c j,l such that f j ( t ) = P jl =1 c j,l f ( l ) ( t ) t − j + l +1 ,and therefore | f j ( t ) | ≤ Ct − d − j +1 − ǫ for t > /
2. Applying the DominatedConvergence Theorem, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ / f ( t ) t d/ J ( d − / ( αt ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = lim b →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b / f ( t ) t d/ J ( d − / ( αt ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . After integrating by parts and taking the limit we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ / f ( t ) t d/ J ( d − / ( αt ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ / f ( t ) t t d/ J d/ ( αt ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Integrating by parts j times, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ / f ( t ) t d/ J ( d − / ( αt ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 α j lim b →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b / f j ( t ) t t d/ j J d/ j − ( αt ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 α j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ / f j +1 ( t ) t t d/ j +1 J d/ j ( αt ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ / f ( t ) t d/ J ( d − / ( αt ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α n Z ∞ / (cid:12)(cid:12)(cid:12)(cid:12) f n ( t ) t t d/ n J d/ n − ( αt ) (cid:12)(cid:12)(cid:12)(cid:12) dt. Proposition A.3.
Let α ≥ , and let f be a function in C n ([0 , ∞ )) forsome natural number n > . Also, assume f = 1 in a neighborhood of and f ( t ) = 0 for t > . Then there is a constant C such that (cid:12)(cid:12)(cid:12)(cid:12)Z tf ( t ) t d/ J ( d − / ( αt ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cα − n . Proof.
We first define a sequence of functions arising when integrating byparts. Let f ( t ) = tf ( t ), f = f ′ , and f j = (cid:16) f j − t (cid:17) ′ for j ≥
2. Note thatwhen j ≥ f j ( t ) = O ( t − j +2 ) as t →
0. After integrating by parts n times,we have (cid:12)(cid:12)(cid:12)(cid:12)Z f ( t ) t d/ J ( d − / ( αt ) dt (cid:12)(cid:12)(cid:12)(cid:12) = 1 α n (cid:12)(cid:12)(cid:12)(cid:12)Z f n ( t ) t t d/ n J d/ n − ( αt ) dt (cid:12)(cid:12)(cid:12)(cid:12) . B Sums of Function Values Over Discrete Sets
Here we provide a bound for sums of function values taken from discretesets in R d . This result is important for showing that the constant in theBernstein inequality does not depend on the number of centers. Proposition B.1.
Let X ⊂ R d be a countable set with q X > , and let Y = { y j } Nj =1 be a subset of X such that | y j | ≥ q X for ≤ j ≤ N . If f : R d → R is a function with | f ( x ) | ≤ C | x | − d − ǫ for some C, ǫ > , then N X j =1 | f ( y j ) | ≤ d (1 + 1 /ǫ )( Cq − d − ǫX ) . Proof.
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