Strang-Fix theory for approximation order in weighted L p-spaces and Herz spaces

In this paper, we study the Strang-Fix theory for approximation order in the weighted L -spaces and Herz spaces.


Introduction
In [10], Strang and Fix considered the relation between approximation order in L 2 (R n ) of a given function and properties of its Fourier transform.In order to describe their results, we make the following definitions.Let ϕ ∈ C c (R n ), where C c (R n ) consists of all continuous functions on R n with compact support.For a sequence c on Z n , the semi-discrete convolution product ϕ * c is defined by For h > 0, σ h is the scaling operator defined by For a positive integer k , we denote the (unweighted) Sobolev space by L p k (R n ) (see section 2) and define the semi-norm by hold for some C 1 and C 2 independent of f and h.The Fourier transform f of a function f ∈ L 1 (R n ) is defined by We note that the B-spline of degree k satisfies the Strang-Fix condition of order k (for the definition of the B-spline, see the proof of Theorem 1 below).Strang and Fix proved that, when Φ consists of only one function, Φ provides controlled L 2 -approximation of order k if and only if Φ satisfies the Strang-Fix condition of order k ( [10]).Moreover, they conjectured that this equivalence holds when Φ = {ϕ 1 , • • • , ϕ N } .However, a counterexample was found by Jia in [6]. We

and
(iv) c h j (ν) = 0 whenever dist(hν, supp f ) > r hold for some C and r independent of f and h, where dist(hν, supp f ) = inf x∈supp f |hν − x|.Note that in the definition of local L p -approximation, the weights c h j depend on h.De Boor and Jia proved that Φ provides local L p -approximation of order k if and only if Φ satisfies the Strang-Fix condition of order k ( [2]).
Given a weight function w on R n , we denote the weighted L p -space and the weighted Sobolev space by L p (w) and L p k (w) (see section 2).We define the semi-norm on L p k (w) by and say that Φ hold for some C and r independent of f and h.
We denote the (homogeneous) weighted Herz space and the (homogeneous) weighted Herz-Sobolev space by Kα,p q (w 1 , w 2 ) and Kα,p q,k (w 1 , w 2 ) (see section 2).Similarly, we define the semi-norm by and say that Φ = {ϕ 1 , • • • , ϕ N } provides local Kα,p q (w 1 , w 2 )-approximation of order k if for each f ∈ Kα,p q,k (w 1 , w 2 ) there exist weights c h j (h > 0, j = 1, • • • , N) such that (iv) and hold for some C and r independent of f and h.Using the notion of multiresolution approximation, we prove the following extension of [2] to weighted L p -spaces.In the following two theorems, we denote Muckenhoupt's A p -class by A p (see section 2).
Then the following are equivalent.
(i) Φ satisfies the Strang-Fix condition of order k .(ii) For all p ∈ [1, ∞] and w ∈ A p , Φ provides local L p (w)approximation of order k .(iii) For some p ∈ [1, ∞] and w ∈ A p , Φ provides local L p (w)approximation of order k .
We also consider the result of [2] in weighted Herz spaces.
(i) Φ satisfies the Strang-Fix condition of order k .
(ii) Φ provides local Kα,p q (w 1 , w 2 )-approximation of order k .Lastly, we point out that the Strang-Fix theory for functions having noncompact support is given by, for example, Jia and Lei ([7]).

Preliminaries
This section is based on [3] and [7] (see also [5]).Given an appropriate function ϕ on R n , we define the multiresolution approximation {P h f } h>0 of a function f on R n with respect to ϕ by Let B(0, r) be the Euclidean ball of radius r centered at the origin.The Hardy-Littlewood maximal function M f of a locally integrable function f on R n is defined by where |B(0, r)| denotes the Lebesgue measure of B(0, r).
where the supremum is taken over all balls B in R n and p is the conjugate exponent of p (1/p + 1/p = 1).A p (w) is called the A p -constant of w .The class A 1 is defined by

e. and w is locally integrable ([3, p. 134]). These classes were introduced by Muckenhoupt ([8]).
C ∞ c (R n ) consists of all infinitely differentiable functions on R n with compact support.
Let w be a weight function on R n and 1 ≤ p ≤ ∞.Then the weighted L p -space L p (w) consists of all f on R n such that with the usual modification when p = ∞.
We give the definition of the weighted Sobolev space L p k (w), where 1 ≤ p ≤ ∞, k is a non-negative integer and w is a weight function on and the partial derivatives ∂ α f , taken in the sense of distributions, belong to L p (w), whenever 0 ≤ |α| ≤ k .The norm on L p k (w) is given by In particular, we denote where k ∈ Z.For a weight function w and a measurable set E , we write w(E) = E w(y) dy .Let α ∈ R, 0 < p, q < ∞, w 1 and w 2 be weight functions on R n .The (homogeneous) weighted Herz space Kα,p q (w 1 , w 2 ) consists of all f on R n such that Similarly, the weighted Herz-Sobolev space Kα,p q,k (w 1 , w 2 ) is defined by using Kα,p q (w 1 , w 2 ) instead of L p (w) ( [9]).Lastly, we give the necessary lemmas.

Lemma 2.1 ([3, Proposition 2.7]).
Let ϕ be a function on R n which is non-negative, radial, decreasing (as a function on (0, ∞)) and integrable.Then and k be a non-negative integer, where w 1 and w 2 satisfy either Theorem 2 (a) or Theorem 2 (b).
, where w 1 and w 2 satisfy either Theorem 2 (a) or Theorem 2 (b).Then the Hardy-Littlewood maximal operator M is bounded on Kα,p q (w 1 , w 2 ).

Lemma 2.7. Let
Lemma 2.8 ([7, Lemma 5.2]).Let k be a positive integer and Φ be a finite collection in C c (R n ).For each h > 0 , let S h (Φ) denote the linear space spanned by ϕ(•/h − ν), where ϕ ∈ Φ and ν ∈ Z n .Suppose that there is a family {u h } 0<h<1 of functions satisfying the following conditions Then Φ satisfies the Strang-Fix condition of order k .

Lemmas
where C is independent of x ∈ R n .Then for h > 0 we have that Proof.We prove only (b)'.We note that φ is in C k−1 (R n ) and there exists a constant C such that (3.1) By (3.2) and φ(0) = 1 , we have that Hence, it suffices to prove that ) is a 2πZ n -periodic function which is integrable on [−π, π) n .We show that A(x) = 0 a.e.To do this, it is enough to prove that all the Fourier coefficients are zero.By (c), we see that The proof is complete.
When we consider local L 1 (w)-approximation, the following lemma plays an important role.

Lemma 3.2. Let 1 ≤ p < ∞ and w ∈ A p . If ϕ is a function on R n
which is non-negative, radial, decreasing and integrable, then there exists a constant C such that for all f ∈ L p (w) and α ∈ R.
Proof.Since ϕ is radial, by a change of variable, we may assume α > 0. We first consider the case p = 1.We note that w ∈ A 1 if and only if there exists a constant C such that M w(x) ≤ Cw(x) a.e.([3, p.134]).By a change of variable, Fubini's theorem and ϕ(y) = ϕ(−y), we have that From Lemma 2.1, we see that Hence, since w ∈ A 1 , we get that , by Lemma 2.1 and Lemma 2.2, we can prove Lemma 3.2 when p > 1 .The proof is complete.

Strang-Fix theory in weighted L p -spaces
In this section, we prove Theorem 1.The boundedness of P h on L p (w) is given in, for example, [1].Since the proof of the boundedness of P h on L 1 (w) is omitted in [1], we give the proof.Theorem 4.1.Let 1 ≤ p < ∞, w ∈ A p , > 0 and k be a positive integer.Suppose that ϕ is a function on R n such that where C is independent of x ∈ R n .Then there exists a constant C such that for all f ∈ L p k (w) and h > 0 .
Proof.We first show that, for h > 0, P h is bounded on L p (w).Let f ∈ L p (w).From (3.1), we have that Thus, from Lemma 3.2, we can see that Hence, by the boundedness of P h on L p (w) and Lemma 2.4, it suffices to prove that . By Lemma 3.1, for almost all x ∈ R n we have that By Taylor's theorem, (3.1) and a change of variable, we see that Thus, by Minkowski's inequality for integrals and Lemma 3.2, we get that The proof is complete.
We are now ready to prove Theorem 1.If w > 0 a.e., then , Theorem 1 for p = ∞ was proved by de Boor and Jia ([2]).Hence, for the proof of Theorem 1, we consider only the case 1 ≤ p < ∞.

Lemma 3 .Lemma 3 . 1 .
1 (a) appears as [4, Chapter 5, Proposition 3.14].Let k be a positive integer and > 0 .Suppose that ϕ is a function on R n such that then the Hardy-Littlewood maximal operator M is bounded on L p (w) if and only if w ∈ A p .Let 1 ≤ p < ∞ and w ∈ A p .Then the following statements hold.
p for all balls B and measurable sets E such that E ⊂ B .(b) ([3, Corollary 7.6]) There exist constants C and δ > 0 such that δ for all balls B and measurable sets E such that E ⊂ B .Lemma 2.4 ([9, Theorem 1.1]).Let 1 ≤ p < ∞, w ∈ A p and k be a non-negative integer.Then C ∞ c (R n ) is dense in the weighted Sobolev space L p k (w).Lemma 2.5 ([9, Corollary 3.3]).Let w 1