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Nonlinear approximation theory on compact groups. (English) Zbl 0973.43001

The following theorem is a classical result in Fourier analysis obtained by P. Erdős and W. H. J. Fuchs [J. Lond. Math. Soc. 31, 67-73 (1956; Zbl 0070.04104)]:
Theorem 1. Suppose that \(\phi(z)=\sum_{n=0}^\infty b_nz^n\) is convergent for \(|z|<1\) and suppose that all \(b_n\) are nonnegative real numbers. Then for \(0<\alpha\leq\pi\), \(z=re^{i\theta}\) and \(0<r<1\), \[ {1\over{2\alpha}}\int_{-\alpha}^\alpha|\phi(z)|^2 d\theta \geq{1\over{6\pi}}\int_{-\pi}^\pi|\phi(z)|^2 d\theta. \tag{1} \] In the limiting case where \(r=1\) (assuming convergence) the following inequality holds true: \[ \int_{-\pi/2}^{\pi/2}|\phi(z)|^2 d\theta \geq{1\over 6}\int_{-\pi}^\pi|\phi(z)|^2 d\theta ={1\over 6}\|\phi\|^2. \tag{2} \] Besides, H. S. Shapiro [Q. J. Math., Oxf. II. Ser. 26, 9-18 (1975; Zbl 0302.42004)] and B. F. Logan [Mich. Math. J. 35, No. 3, 369-393 (1988; Zbl 0682.30017)] have shown that the constant \(1/6\) can be improved to \(1/4\) and their result is sharp in that it cannot be improved uniformly for all values of \(\alpha\) in (1).
The present article is devoted to the investigation of analogs of bounds of the form (2) in the setting of finite, and then compact, groups. In this more general setting the well-known Fourier expansion is replaced by an expansion in terms of irreducible matrix elements, and the Fourier data is precisely the coefficients of this expansion. Let us cite one theorem for the case of cyclic groups:
Theorem 2. Let \(G=\mathbb{Z}/N\mathbb{Z}\), \(n<N/2\), \(f\in L^2(G)\) and \(f\geq 0\). Then \[ \sum_{-n<j<n}|\widehat{f}(j)|^2 \geq{n\over N}\|\widehat{f}\|^2, \] or \[ {\|\widehat{f}\|_n\over{\|f\|}} \geq\sqrt{{n\over N}}, {(3)} \] where \(\|\widehat{f}\|\) denotes the projection of \(f\) onto the frequencies between \(-n\) and \(n\).
The sharpness of the estimates obtained is also investigated. In particular, the authors have shown that the lower bound for the ratio (3) cannot be increased uniformly. This is analogous to the tightness of the \(1/4\) bound in (1) provided by H. S. Shapiro and B. F. Logan.
The results obtained are applied to the problem of recovering a positive function (signal) from a subset of its Fourier data, and in particular, its low frequency information. A priori analysis indicates a connection between the ratio of the energy contained in the low frequency portion of the power spectrum energy to the total energy and the stability of recovering the original function from this noisy low passed version via standard optimization techniques. The authors include several numerical experiments. Further applications to data analysis on nonabelian groups are also discussed.
The article contains also several open questions suggested by the obtained results.
A comprehensive bibliography of the article reflects several results obtained in this area.

MSC:

43-04 Software, source code, etc. for problems pertaining to abstract harmonic analysis
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A29 Approximation with constraints
42A10 Trigonometric approximation
43A77 Harmonic analysis on general compact groups
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References:

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