JAMPJournal of Applied Mathematics and Physics2327-4352Scientific Research Publishing10.4236/jamp.2014.22003JAMP-42098ArticlesPhysics&Mathematics Necessity of Oversampling Theorem for Affine Frames iquanFang1*XianliangShi2WeicaiLi1College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), Hunan Normal University, Changsha, ChinaDepartment of Mathematics, Zhejiang University of Science and Technology, Hangzhou, China* E-mail:fendui@yahoo.com(IF);1701201402021823© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In this paper we prove that n is relatively prime to a which is also necessary.

Affine Frame; Oversampling
1. 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  and . In 1993, C. K.Chui and X. L. Shi  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 . 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 . 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.