Three classes of Fourier transforms are presented: Fourier
(Laplace) transforms on the halfline, Fourier transforms of
measures with compact support and Fourier transforms of rapidly
decreasing functions (on whole line). The focus is on the behaviour
of Fourier transforms in the region of analyticity and the
distribution of their zeros. Applications of results are presented:
approximation by exponentials on the finite interval; behavior of
the nonharmonic Fourier series; MA1/4ntz-Szasz's problem of
approximation by powers on unit interval; approximation by weighted
exponentials on whole line.
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