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This paper is concerned with analyzing the mathematical properties, such as the regularity and stability of nonstationary biorthogonal wavelet systems based on exponential B-splines. We first discuss the biorthogonality condition of the nonstationary refinable functions, and then we show that the refinable functions based on exponential B-splines have the same regularities as the ones based on the polynomial B-splines of the corresponding orders. In the context of nonstationary wavelets, the stability of wavelet bases is not implied by the stability of a refinable function. For this reason, we prove that the suggested nonstationary wavelets form Riesz bases for the space that they generate.

For the last two decades, the wavelet transforms have become very useful tools in a variety of applications such as signal and image processing and numerical computation. The construction of classical wavelets is now well understood thanks to such pioneering works as [

Exponential B-splines and polynomials have been found to be quite useful in a number of applications such as computer-aided geometric design, shape-preserving curve fitting, and signal interpolation [

One natural and convenient way to introduce wavelets is to follow the notion of multiresolution analysis. However, because the refinement masks we are interested in are non-stationary (i.e., scale dependent), we use the structure of non-stationary multiresolution analysis as introduced in [

the set

The nested embedding of the spaces

A generalization of the biorthogonal wavelets of Cohen-Daubechies-Feuveau [

This paper is organized as follows. In Section

Given a set of complex numbers

It is well known that the integer translates

Given refinable functions

Let the polynomials

Assume that

By virtue of Proposition

Let

This is a direct consequence of [

For the given Laurent polynomials

Let

Let

Let

Recalling the definition of

This result proves that

Let

Due to Lemma

For a given

Assume that the Laurent polynomial

The Laurent polynomial

It is known (e.g., see [

For each

In the sequel, we will use the notation

Let

With the refinement masks

Assume that

Since

The above proposition discusses the stability of wavelet functions at each fixed level. The real problem is the global stability of the set

Let

Let

We now arrive at the central results of this section.

Let

Let

Since

Let

For a given function

The authors are grateful to the anonymous referees for their valuable suggestions on this paper. The work of Y. J. Lee was supported by Basic Science Research Program (2009-0068156) and J. Yoon was supported by Mid-Career Researcher Program (2009-0084583) and Basic Science Research Program (2010-0016257), through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology.