In many areas of mathematics, orthonormal bases are a standard tool for representing vectors or operators. For certain problems, however, it is preferable to use an overcomplete, non-orthogonal set of vectors instead of an orthonormal basis, and thereby incorporate redundancy in the representation. Frames are overcomplete sets which are associated with redundant linear representations that have...

In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see [3]). In the case of an orthonormal basis our estimate reduces to Kadec’ optimal 1/4 re...

Several nonparametric methods in a regression model are presented. First, the most classical ones: piecewise polynomial estimators, estimation with Spline bases, kernel estimators and projection estimators on orthonormal bases (such as Fourier or wavelet bases). Since these methods suffer from the curse of dimensionality, we also present Generalized Additive Models and CART regression models. T...

The orthonormal basis generated by a wavelet of L 2 (R) has poor frequency localization. To overcome this disadvantage Coifman, Meyer, and Wickerhauser constructed wavelet packets. We extend this concept to the higher dimensions where we consider arbitrary dilation matrices. The resulting basis of L 2 (R d) is called the multiwavelet packet basis. The concept of wavelet frame packet is also gen...

This paper investigates the use of general bases with fixed poles for the purposes of robust estimation. These bases, which generalise the common FIR, Laguerre and two–parameter Kautz ones, are shown to be fundamental in the disc algebra provided a very mild condition on the choice of poles is satisfied. It is also shown, that by using a min–max criterion, these bases lead to robust estimators ...

This paper presents a new method for designing orthonormal wavelet filter banks using the Remez exchange algorithm. It is well known that orthonormal wavelet bases can be generated by paraunitary filter banks. Thus, synthesis of orthonormal wavelet bases can be reduced to the design of paraunitary filter banks. From the orthonormality and regularity of wavelets, we derive some conditions that m...

This paper investigates the mathematical background of multiresolution analysis in the specific context where the signal is represented by irregularly sampled data at known locations. The study is related to the construction of nested piecewise polynomial multiresolution spaces represented by their corresponding orthonormal bases. Using simple spline basis orthonormalization procedures involves...