In this paper we prove that n is relatively prime to a which is also necessary.
Affine Frame; Oversampling1. Introduction
Let denote, as usual, the space of all complex-valued square integrable functions on the real line with inner product and norm. For any, we will use the notation
where and. A function is said to generate an affine frame
of, with frame bounds and, where, if it satisfies
The frame (2) of is called a tight frame, if (3) holds with, see [1] and [2]. In 1993, C. K.Chui and X. L. Shi [3] proved the following oversampling theorem:
Theorem A. Let be any positive integer and. Also, let generate a frame with frame bounds and as given by (3). Then for any positive integer which is relatively prime to, the family
remains a frame of with the same bounds. If, this result does not hold. But they only gave a countexample for the case where as in [4]. For other positive integer and which satisfy, they did not prove. The aim of this paper is to establish the inverse proposition of Theorem A, and then we following:
Theorem 1.1. Let be any positive integer and. Also, let be any affine frame of with frame bounds and. The family (4) remains a frame of with the same bounds: that is,
if and only if and are relatively prime.
2. Proofs
The sufficiency has been included in the theorem 4 of [3]. In the following we will prove the necessary part of the theorem.
Suppose for any affine frame (2) of with frame bounds and, the family (4) is also a frame of with the same bounds. Then when (1) forms an orthonormal basis, the family (4) forms a tight frame with frame bound. So we just need to prove that there exists a function such that the family (1) forms the orthonormal basis, but for any two positive integers and which satisfy, there exist two functions and such that
Doesn’t equal
.
Let, then forms an orthonormal basis, which is called Haar basis. Set
and
We prove that if, then
and
Denote. We have
where
and
In order to prove the theorem, we have three cases.
Case 1. When.
We have if. Thus, if is an even integer, we can get
So, we have
If is an odd integer, we have
So, we have
Case 2. When.
If is an even integer, we have
Thus
If is an odd integer, we can get because of As in the case, we also have
So, we get
Case 3. When.
If is an even integer. Let
and
When, there exists an integer satisfying. Therefore we have
where. When, we have and. Thus we have
Therefore
When, similar to the case, we also have
So we have
If is an odd integer. We have
where
A familiar calculation shows
Since and, we have. Also when and, we have
When and, obviously we have
When,. So we have in this case. This completes the proof of the theorem.
Acknowledgements
The authors would like to thank anonymous reviewers for their comments and suggestions. The authors are partially supported by project 11226108, 11071065, 11171306 funded by NSF of China, and project Y201225301. Project 20094306110004 funded by RFDP of high education of China.
REFERENCESReferencesC. Lee, P. Linneman and C. K. Chui, “An Introduction to Wavelets,” Academic Press, Boston, 1992.I. Daubechies, “Ten Lectures on Wavelets,” Society for Industrial and Applied Mathematics, Philadelphia, 1992.http://dx.doi.org/10.1137/1.9781611970104C. K. Chui and X. L. Shi, “Bessel Sequences and Affine Frames,” Applied and Computational Harmonic Analysis, Vol. 1, No. 1, 1993, pp. 29-49. http://dx.doi.org/10.1006/acha.1993.1003C. K. Chui and X. L. Shi, n× Oversampling preserves any tight affine frame for odd n, Proceedings of the American Mathematical Society, Vol. 121, No, 2, 1994, pp. 511-517.