Azarin limiting sets of functions and asymptotic representation of integrals

Abstract

In the paper we consider integrals of form $$\int\limits_a^b f(t)\exp[i\varphi(rt)\ln(rt)]\,dt\,,$$ where $\varphi(r)$ is a smooth increasing function on the semi-axis $[0,\infty)$ such that $\lim\limits_{r\to+\infty}\varphi(r)=\infty\,.$ We find a precise information on their asymptotic behavior and we prove an analogue of Riemann-Lebesgue lemma for trigonometric integrals. By applying this lemma, we succeed to obtain the asymptotic formulae for integrals with an absolutely continuous function. The proposed method of obtaining asymptotic formulae differs from classical method like Laplace method, applications of residua, saddle-point method, etc. To make the presentation more solid, we mostly restrict ourselves by the kernels $\exp[i\ln^p(rt)]$. Appropriate smoothness conditions for the function $f(t)$ allow us to obtain many-terms formulae. The properties of the integrals and the methods of obtaining asymptotic estimates differ in the cases $p\in(0,1)$, $p=1$, $p>1$. As $p\in(0,1)$, the asymptotic expansions are obtained by another method, namely, by expanding the kernel into a series. We consider the cases, when as an absolutely continuous function $f(t)$, we take a product of a power function $t^\rho$ and the Poisson kernel or the conjugate Poisson kernel for the half-plane and as the integration set, the imaginary semi-axis serves. The real and imaginary parts of these integrals are harmonic functions in the complex plane cut along the positive semi-axis. We find the Azarin limiting sets for such functions.