Keywords: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
(1)
where
and
. A function
is said to generate an affine frame
(2)
of
, with frame bounds
and
, where
, if it satisfies
(3)
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
(4)
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,
(5)
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
![](https://www.scirp.org/html/htmlimages\3-1720073x\2d2755f7-e9b6-4e28-aeda-1bc0cae69ecc.png)
Doesn’t equal
.
Let
, then
forms an orthonormal basis, which is called Haar basis. Set
and ![](https://www.scirp.org/html/htmlimages\3-1720073x\fd24f02b-2407-4efd-a1ff-128880d734cd.png)
We prove that if
, then
(6)
![](https://www.scirp.org/html/htmlimages\3-1720073x\54485905-3837-4e35-93b0-dfa58471c677.png)
and
![](https://www.scirp.org/html/htmlimages\3-1720073x\408476cd-3f61-4cb9-8526-c276e92b3a5b.png)
Denote
. We have
where
![](https://www.scirp.org/html/htmlimages\3-1720073x\e02e78d7-1268-491f-8f3b-d6709ee85851.png)
![](https://www.scirp.org/html/htmlimages\3-1720073x\bf39baa4-c3f6-4c19-94bd-6bf85b0dc309.png)
![](https://www.scirp.org/html/htmlimages\3-1720073x\3de34065-b13c-4fe9-a41b-159159ef5047.png)
![](https://www.scirp.org/html/htmlimages\3-1720073x\89a2bce7-592c-49a6-8006-e4a83b54b2d9.png)
![](https://www.scirp.org/html/htmlimages\3-1720073x\ebe8340d-e49b-4b4a-bc29-c4fd45d9c4b7.png)
and
![](https://www.scirp.org/html/htmlimages\3-1720073x\89c45f2a-ad5a-47fa-8f7e-2ec77c061fba.png)
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
![](https://www.scirp.org/html/htmlimages\3-1720073x\cec33d44-d23c-46f4-8afa-4d532c24de12.png)
![](https://www.scirp.org/html/htmlimages\3-1720073x\165a2a08-c1af-4189-80a8-59951eed9e27.png)
So, we have
![](https://www.scirp.org/html/htmlimages\3-1720073x\546f5ba6-1c29-495f-b4eb-cde7d9c2ca59.png)
If
is an odd integer, we have
![](https://www.scirp.org/html/htmlimages\3-1720073x\0c562f2b-2904-4be2-a4f3-3a8ca281128d.png)
![](https://www.scirp.org/html/htmlimages\3-1720073x\167a9863-1dab-42ad-adf7-24cfca442346.png)
So, we have
![](https://www.scirp.org/html/htmlimages\3-1720073x\5d741832-18d3-469c-a97e-d0f5b2c26964.png)
Case 2. When
.
If
is an even integer, we have
![](https://www.scirp.org/html/htmlimages\3-1720073x\c26c1091-b63a-41d5-b5aa-ff4c6547cec5.png)
![](https://www.scirp.org/html/htmlimages\3-1720073x\4959731b-12f8-449c-9f50-9f4757925d22.png)
Thus
![](https://www.scirp.org/html/htmlimages\3-1720073x\b3123833-edd5-4bbf-ab0b-c1009eb04ac0.png)
If
is an odd integer, we can get
because of
As in the case
, we also have
![](https://www.scirp.org/html/htmlimages\3-1720073x\e47b28f0-793a-451f-81b6-30827d40d1b3.png)
![](https://www.scirp.org/html/htmlimages\3-1720073x\803df6e7-31f0-4fb7-bf4d-7e2ad28af390.png)
So, we get
![](https://www.scirp.org/html/htmlimages\3-1720073x\a37724dd-753a-4285-8bfd-0049102072e6.png)
Case 3. When
.
If
is an even integer. Let
![](https://www.scirp.org/html/htmlimages\3-1720073x\11c08027-3eec-4983-8c1b-b693f12175f8.png)
and
![](https://www.scirp.org/html/htmlimages\3-1720073x\44acf5e4-4fb0-4fbc-9246-dfae90169351.png)
When
, there exists an integer
satisfying
. Therefore we have
![](https://www.scirp.org/html/htmlimages\3-1720073x\679fd7e2-b398-4212-a107-13b5cc5acaaf.png)
where
. When
, we have
and
. Thus we have
![](https://www.scirp.org/html/htmlimages\3-1720073x\fd406dbb-98c9-4865-92d2-48ece905dbbb.png)
Therefore
![](https://www.scirp.org/html/htmlimages\3-1720073x\6e186b1e-575b-4447-a7a5-01572d6979d1.png)
When
, similar to the case
, we also have
![](https://www.scirp.org/html/htmlimages\3-1720073x\9effa365-df7a-45fb-b9cf-015eabe4570b.png)
So we have
![](https://www.scirp.org/html/htmlimages\3-1720073x\9865d260-0ad4-44e2-bc27-87f5e7a9f07b.png)
If
is an odd integer. We have
![](https://www.scirp.org/html/htmlimages\3-1720073x\dcd12a53-1476-43b6-bcd4-a73bbd140abb.png)
where
![](https://www.scirp.org/html/htmlimages\3-1720073x\9ca55245-4e74-45d4-920a-945626c2ef9a.png)
![](https://www.scirp.org/html/htmlimages\3-1720073x\457f5ffc-3826-4dbc-9216-686d626b008a.png)
A familiar calculation shows
![](https://www.scirp.org/html/htmlimages\3-1720073x\d8b044ad-d513-4f15-b574-6bc754cd396e.png)
Since
and
, we have
. Also when
and
, we have
![](https://www.scirp.org/html/htmlimages\3-1720073x\1285e9c8-09f3-4dff-bbd7-6c1059992347.png)
When
and
, obviously we have
![](https://www.scirp.org/html/htmlimages\3-1720073x\484f7342-6377-4495-8ee8-061bb5b7609a.png)
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.