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Ufimsk. Mat. Zh., 2019, Volume 11, Issue 2, Pages 99–117 (Mi ufa474)  

Azarin limiting sets of functions and asymptotic representation of integrals

K. G. Malyutina, T. I. Malyutinaa, T. V. Shevtsovab

a Kursk State University, Radischeva str. 33, 305000, Kursk, Russia
b Southwest State University, 50 October str. 94, 305040, Kursk, Russia

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.

Keywords: Riemann–Lebesgue lemma, trigonometric integral, asymptotic formula, Poisson kernel, harmonic function, Azarin limiting set.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00236_a
The reported study was funded by RFBR according to the research project no. 18-01-00236.


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English version:
Ufa Mathematical Journal, 2019, 11:2, 97–113 (PDF, 386 kB); https://doi.org/10.13108/2019-11-2-97

Bibliographic databases:

UDC: 517.53
MSC: 30E15, 31C05
Received: 18.06.2018

Citation: K. G. Malyutin, T. I. Malyutina, T. V. Shevtsova, “Azarin limiting sets of functions and asymptotic representation of integrals”, Ufimsk. Mat. Zh., 11:2 (2019), 99–117; Ufa Math. J., 11:2 (2019), 97–113

Citation in format AMSBIB
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\by K.~G.~Malyutin, T.~I.~Malyutina, T.~V.~Shevtsova
\paper Azarin limiting sets of functions and asymptotic representation of integrals
\jour Ufimsk. Mat. Zh.
\yr 2019
\vol 11
\issue 2
\pages 99--117
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\transl
\jour Ufa Math. J.
\yr 2019
\vol 11
\issue 2
\pages 97--113
\crossref{https://doi.org/10.13108/2019-11-2-97}
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