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Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms. (English) Zbl 0814.45001

The convolution equation \(s = f* \mu\) of the first kind is considered, which has an applied physical sense. Here \(f\) is the input signal, \(s\) is the output signal and \(\mu\) is the system of impulse. The input signal \(f\) belongs to one of the spaces \(L^ p (\mathbb{R})\), \(1 \leq p \leq \infty\) (locally) or \(C(\mathbb{R})\).
The authors construct a general scheme which allows them to find the input signal \(f\) (deconvolution) from the system \(s_ i = f* \mu_ i\), \(i = 1, 2, \dots, n\). In particular, by this way wavelet or Gabor coefficients of \(f\) may be found. They also give some applications of these methods to concrete realizations \(\mu\) (f.e. \(\mu_ i\) are characteristic functions of intervals). The multidimensional case is also discussed.

MSC:

45F15 Systems of singular linear integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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