Non-oscillating Paley-Wiener functions (Q1884394)

Let \(W\) be the class of real entire functions of exponential type belonging to \(L_2({\mathbb R})\). The authors investigate this class (they do not use the same denotation). Let \(p\) be a real function of the class \(C^\infty \) with compact support. For the functions \(f(z) = \int _0^\infty p(s) \sin (sz) \,ds\), \(p(0) \neq 0 ,\) and \(f(z) = \frac{1}{z} \int_{-\infty}^\infty (1-e^{isz})p(s)\,ds\), \(\int_{-\infty}^\infty p(s)\,ds\neq 0, \) the authors prove asymptotical formulas of the type \(f^{(n)}(x) = \frac{a_n}{x^{n+1}}(1+o(1))\), \(x\to \infty\), \(a_n\neq0\), \(n=0, 1, \ldots \) (\(x\) is real). Thus \(f\in W\) and the functions \(f^{(n)}(z)\) have finite many of real zeros. Such functions are called non-oscillating \(PW\)-functions. The question of existence of ones was open. Then the authors construct non-oscillating \(PW\)-functions with special decreasing on the real line. Fourier transform of non-oscillating \(PW\)-functions is in \(C^\infty({\mathbb R}\backslash \{ 0 \} )\). This proposition we see among other results of the paper.