HOMOGENEOUS APPROXIMATION PROPERTY FOR VARIOUS DIRECTIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In the theory of signal reconstruction, the homogeneous approximation property (HAP) for wavelets is useful. In this paper, we consider the HAP for the continuous wavelet transform with matrix dilation. When the dilation of wavelet is different in various directions, we show that the HAP holds in L2(Rd). The HAP also holds in L∞(Rd) when we add some conditions.


INTRODUCTION
In 1952, Duffin and Schaeffer proposed the frame of Hilbert space in [5]. Frames have become an important tool in many other disciplines, because they can provide many different expression of vectors. Frame theory plays an important role in signal processing and many other fields. In recent years, more and more scholars are interested in frame theory, especially Gabor frames and wavelet frames.
Wavelet frames are a class of important frames and have many useful properties [1,2,4,6,7,15] during the development. Through limits a, b to discrete values, we can makes wavelet transform τ(a, b)ψ forms a frame. There have been plenty of results on various properties of wavelet frames, including necessary and sufficient conditions for wavelet systems to be frames.
The wavelet transform of f ∈ L 2 (R d ) with respect to ψ ∈ L 2 (R d ) is defined by Here G := {(a, b) : a > 0, b ∈ R d } is a group and the action on it is defined by (a, b)(s,t) = (as, b + at).
The homogeneous approximation property (HAP) is an interesting properties of wavelet frames. If the wavelet frame has good generators, then it has the HAP. The HAP has been studied in [1,8,9,10,11,14,16,17]. In addition, the HAP for wavelet frame with nice wavelet function and arbitrary expansive dilation matrix also was studied in [16,17] recently.
The following results in recent years is the HAP for the continuous wavelet transform. In [12,13], they show that every pair of admissible wavelets possesses the HAP in L 2 -sense, while it is not true in general whenever pointwise convergence is considered. But if we add some conditions on the wavelets and function to be reconstructed, then the HAP holds in L ∞ (R). In the case of multivariate, this result is still true in L ∞ (R d ), but the condition that wavelets and the function to be reconstructed are all compactly supported on R d \0 is just a sufficient condition, not the necessary condition [13].
In this paper, we consider the HAP for the continuous wavelet transform with matrix dilation.
When the dilation of wavelet is different in various directions, we show that the HAP holds in The HAP also holds in L ∞ (R d ) when we add some conditions.

NOTATIONS AND PRELIMINARY RESULTS
In this section, we will introduce some notations and definitions. Definef where |M| = |detM|. It is easy to see that where M T denotes the transpose of M.
Similar, we call a pair of function (ψ 1 , ψ 2 ) admissible about the matrix functions Now for (ψ 1 , ψ 2 ), we give the definitions of homogeneous approximation property (HAP) for various directions in L 2 (R d ) and in L ∞ (R d ) for continuous matrix wavelet transforms.
Definition 2.1. We call that (ψ 1 , ψ 2 ) possess the homogeneous approximation property about Now we give an important proposition.
From this proposition, we can see that the continuous wavelet transform can reconstruct a function f as following: where the convergence is in the weak sense. Here (ψ 1 , ψ 2 ) is a pair of admissible function. The following is a useful lemma in this paper.
then we have Proof. For any x ∈ R d , we have Hence, f A 1 ,A 2 is well defined on R d .
For any x, x ∈ R d , similar arguments show that Next we prove (6). By (7), for any By Fubini's Theorem, we have

HAP FOR VARIOUS DIRECTIONS IN
In this section, we first consider HAP for various directions in L 2 (R d ). Proof. Let A 2 > A 1 > 0 and B > 0 be constants to be determined later. Suppose that A 2 > A 2 and 0 < A 1 ≤ A 1 . Then for any f ∈ L 2 (R) and (s,t) ∈ G , we have By Proposition 2.3, we can make E A 1 ,A 2 ;B arbitrary small by choosing A 2 and B large enough and A 1 small enough. This completes the proof.
Now we show that the HAP for various directions also holds in L ∞ (R d ).
which satisfy s ≥ s 0 > 0, and any Proof. By Lemma 2.4, it is easy to see that for any f ∈ L 2 (R d ), we have First, we estimate I. By substituting M −1 α (s)ω for ω, we have Since both ψ 1 and ψ 2 are admissible about the matrix function M α (a), M α (a) separately, we a.e. and lim A 1 →0 By the dominated convergence theorem, we have Hence, we can choose some A 1 > 0 such that for any s ≥ s 0 and 0 < A 1 ≤ A 1 , Next we estimate II. Using Hölder's inequality twice, we have Putting (10) and (11) together, we get the conclusion.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.