Interpolatory and Orthonormal Trigonometric Wavelets

The aim of this paper is the detailed investigation of trigono-metric polynomial spaces as a tool for approximation and signal analysis. Sample spaces are generated by equidistant translates of certain de la Vall ee Poussin means. The diierent de la Vall ee Poussin means enable us to choose between better time-or frequency-localization. For nested sample spaces and corresponding wavelet spaces, we discuss diierent bases and their transformations. x1 Introduction Trigonometric polynomials and the approximation of periodic functions by poly-nomials play an important role in harmonic analysis. Here we are interested in constructing time-localized bases for certain spaces of trigonometric polynomi-als. We use de la Vall ee Poussin means of the usual Dirichlet kernel, which allow the investigation of simple projections onto these spaces. Interpolation and orthogonal projection are discussed. With the diierent de la Vall ee Poussin means, the operator norms of these projections as well as the time-frequency localization of the basis functions can be controlled. In order to improve the time-localization, the side oscillations are reduced by including more frequencies and averaging the highest ones. Adapting basic ideas of wavelet theory, inter-polatory and orthonormal bases are employed, both of which are constructed from equidistant shifts of a single polynomial. One of our main goals is the investigation of basis transformations in a form that facilitates fast algorithms. We observe that the corresponding transformation matrices have a circulant structure and can be diagonalized by Fourier matrices. The resulting diagonal matrices contain the eigenvalues, which are computed explicitly. So, the algorithms can be easily realized using the Fast Fourier Transform (FFT). Further, we consider the nesting of the sample spaces to obtain multiresolu-tion analyses (MRA's). For the resulting orthogonal wavelet spaces, we proceed as above and nd wavelet bases consisting of translates of a single polynomial. Again, interpolatory and orthonormal bases are constructed, which show the same time-frequency behaviour as the sample bases do. The basis transformations can be described analogously by circulant matrices. Most important for practical reasons are the decomposition of signals in frequency bands, which correspond to the wavelet spaces, and their reconstruction.