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Polynomials --- Series --- 517.52 --- 681.3*G12 --- Algebra --- Mathematics --- Processes, Infinite --- Sequences (Mathematics) --- Series and sequences --- Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- Polynomials. --- Series. --- 681.3*G12 Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- 517.52 Series and sequences

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Wavelets (Mathematics) --- 517.518.8 --- 519.6 --- 681.3*G12 --- 681.3*I4 --- 681.3*I4 Image processing: image displays image processing software (Computing methododologies) --- Image processing: image displays image processing software (Computing methododologies) --- 681.3*G12 Approximation: chebyshev elementary function least squares linear approximation minimax approximation and algorithms nonlinear and rational approximation spline and piecewise polynomial approximation (Numerical analysis) --- Approximation: chebyshev elementary function least squares linear approximation minimax approximation and algorithms nonlinear and rational approximation spline and piecewise polynomial approximation (Numerical analysis) --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- 517.518.8 Approximation of functions by polynomials and their generalizations --- Approximation of functions by polynomials and their generalizations --- Wavelet analysis --- Harmonic analysis --- 681.3*I4 Image processing: image displays; image processing software (Computing methododologies) --- Image processing: image displays; image processing software (Computing methododologies) --- 681.3*G12 Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- Wavelets (Mathematics).

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The aim of this work is to present several topics in time-frequency analysis as subjects in abelian group theory. The algebraic point of view pre­ dominates as questions of convergence are not considered. Our approach emphasizes the unifying role played by group structures on the development of theory and algorithms. This book consists of two main parts. The first treats Weyl-Heisenberg representations over finite abelian groups and the second deals with mul­ tirate filter structures over free abelian groups of finite rank. In both, the methods are dimensionless and coordinate-free and apply to one and multidimensional problems. The selection of topics is not motivated by mathematical necessity but rather by simplicity. We could have developed Weyl-Heisenberg theory over free abelian groups of finite rank or more generally developed both topics over locally compact abelian groups. However, except for having to dis­ cuss conditions for convergence, Haar measures, and other standard topics from analysis the underlying structures would essentially be the same. A re­ cent collection of papers [17] provides an excellent review of time-frequency analysis over locally compact abelian groups. A further reason for limiting the scope of generality is that our results can be immediately applied to the design of algorithms and codes for time­ frequency processing.

Signal processing --- Time-series analysis --- Frequency spectra --- Mathematics --- 517.518 --- 681.3*G12 --- Metric theory of functions --- Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- 681.3*G12 Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- 517.518 Metric theory of functions --- Abelian groups --- Analysis of time series --- Autocorrelation (Statistics) --- Harmonic analysis --- Mathematical statistics --- Probabilities --- Spectra, Frequency --- Spectrum, Frequency --- Spectrum analysis --- Commutative groups --- Group theory --- Computer mathematics. --- Signal processing. --- Image processing. --- Speech processing systems. --- Applied mathematics. --- Engineering mathematics. --- Computational Mathematics and Numerical Analysis. --- Signal, Image and Speech Processing. --- Computational Science and Engineering. --- Applications of Mathematics. --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Engineering --- Engineering analysis --- Mathematical analysis --- Computational linguistics --- Electronic systems --- Information theory --- Modulation theory --- Oral communication --- Speech --- Telecommunication --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Computer mathematics --- Electronic data processing --- Abelian groups. --- Frequency spectra. --- Signal processing - Mathematics