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We present a construction of dual windows associated with Gabor frames with compactly supported windows. The size of the support of the dual windows is comparable to that of the given window. Under certain conditions, we prove that there exist dual windows with higher regularity than the canonical dual window. On the other hand, there are cases where no differentiable dual window exists, even in the overcomplete case. As a special case of our results, we show that there exists a common smooth dual window for an interesting class of Gabor frames. In particular, for any value of

A frame

It is known that a frame

Just to give the reader an impression of the results to come, consider the B-spline

The results will be based on a construction of dual windows, as presented in Section

Note that a complementary approach to duality for Gabor frames that also deals with the issue of regularity is considered by Laugesen [

In the literature, various characterizations of the pairs of dual Gabor frames are available. For general frames, Li gave a characterization in [

Two Bessel sequences

We will use the following to apply Lemma

Let

Consider

We will now present a general result about the existence of frames with a dual window of a special form. Note that, in contrast with most results from the literature, we do not need to assume that the integer translates of the window function form a partition of unity.

Associated to a function

Let

The function

Take

Then,

To prove (i), we note that by (

Before we start the general analysis of the dual windows in Theorem

Let

If

By the assumption (

An example of a smooth function

As noted in the introduction, the possibility of constructing a smooth dual window is a significant improvement compared to the use of the canonical dual window, which might not even be continuous. We return to this point in Example

Another interesting feature of the construction in Theorem

Let

By the choice of the function

It is known that the partition of unity condition (

Based on Theorem

Given a function

Let

If

Assume that

Then, for any

Note that (

Consider the function

Note that the conclusion in Example

Consider

Let us now provide the details for the example mentioned in Section

Consider the B-spline

On the other hand, consider the scaled B-spline

We will now present the general version of Theorem

Let

(1) If

Assume that, for some constant

(2) Assume that

Then,

Note that the conditions in Theorem

For nonnegative functions, the conditions in Theorem

Let

Assume that the set

Then,

We check condition (2) in Theorem

Let

The full proof of Theorem

the zeroset of

We leave the obvious modifications to the general case to the reader.

We use the following abbreviation:

(1) We will prove the contrapositive result, so suppose that a dual window

(2) We construct a real-valued differentiable function

Let

We define

We now extend

The authors would like to thank the reviewers for their useful comments and suggestions. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2010-0007614).