Nonlinear approximation theory on finite groups

Motivated by problems in signal recovery, we will investigate the distribution of the energy of the Fourier transform of a positive function on a finite group. In particular, we are able to bound from below the fraction of energy contained in various subsets of the Fourier transform of a positive function defined on a finite group. Applications to signal recovery for positive functions, as well as partial spectral analysis for data on finite groups are also presented.

[1]  Dunham Jackson,et al.  Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung , 1911 .

[2]  D. Jackson.  On approximation by trigonometric sums and polynomials , 1912 .

[3]  Paul Erdös,et al.  On a Problem of Additive Number Theory , 1956 .

[4]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[5]  E. Cheney.  Introduction to approximation theory , 1966 .

[6]  N. Vilenkin.  Special Functions and the Theory of Group Representations , 1968 .

[7]  MAJORANT PROBLEMS FOR FOURIER COEFFICIENTS , 1975 .

[8]  Jean-Pierre Serre,et al.  Linear representations of finite groups , 1977, Graduate texts in mathematics.

[9]  M. Reed,et al.  Methods of Mathematical Physics , 1980 .

[10]  B. Logan.  An interference problem for exponentials. , 1988 .

[11]  P. Diaconis.  Group representations in probability and statistics , 1988 .

[12]  D. Donoho,et al.  Uncertainty principles and signal recovery , 1989 .

[13]  P. Diaconis.  A Generalization of Spectral Analysis with Application to Ranked Data , 1989 .

[14]  D. Donoho.  Superresolution via sparsity constraints , 1992 .

[15]  B. Logan,et al.  Signal recovery and the large sieve , 1992 .

[16]  Daniel N. Rockmore,et al.  Some applications of generalized FFT's , 1997, Groups and Computation.

[17]  D. Maslen.  Efficient computation of Fourier transforms on compact groups , 1998 .