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Analytic sampling approximation by projection operator with application in decomposition of instantaneous frequency. (English) Zbl 1279.41020

Summary: A sequence of special functions in the Hardy space \(\mathcal H^2(\mathbb T^s)\) are constructed from the Cauchy kernel on the unit disk \(\mathbb D\). Applying a projection operator of the sequence of functions leads to an analytic sampling approximation to \(f\), where \(f\) is any given function in \(\mathcal H^2(\mathbb T^s)\). That is, \(f\) can be approximated by its analytic samples in \(\mathbb D^{s}\). Under a mild condition, \(f\) is approximated exponentially by its analytic samples. By the analytic sampling approximation, a signal in \(\mathcal H^2(\mathbb T^s)\) can be approximately decomposed into components of positive instantaneous frequency. Using circular a Hilbert transform, we apply the approximation scheme in \(\mathcal H^2(\mathbb T^s)\) to \(L^{s}(\mathbb T^2)\) such that a signal in \(L^{s}(\mathbb T^2)\) can be approximated by its analytic samples on \(\mathbb C^{s}\). A numerical experiment is carried out to illustrate our results.

MSC:

41A20 Approximation by rational functions
41A25 Rate of convergence, degree of approximation
94A20 Sampling theory in information and communication theory
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