Some estimates of special classes of integrals

Abstract

We study the integrals f^b _a f(t) exp(i| ln rt|^σ) dt and obtain asymptotic formula for these functions of non‐regular growth. This is a peculiar kind of the theory asymptotic expansions. In particular, we get asymptotic formulae for different entire functions of non‐regular growth. Asymptotic formulas for Levin‐Pfluger entire functions of completely regular growth are well‐known [1]. Our formulas allow to find limiting Azarin's [2] sets for some subharmonic functions. The kernel exp(i| ln rt|^σ) contains arbitrary parameter σ > 0. The integrals for σ ∈(0, 1), σ = 1, σ > 1 essentially differ. Our arguments can apply to more general kernels. We give a new variant of the classic lemma of Riemann and Lebesgue from the theory of the transformation of Fourier.

Darbe nagrinejami integralai f^b _a f(t) exp(i| ln r|^σ) dt ir tiriamos šiu nereguliaraus augimo greičiu funkciju asimptotines formules. Gautos naujos asimptotines formules, leidžiančios rasti Azarino aibes kai kurioms subharmoninems funkcijoms. Branduolys exp(i| ln rt|^σ) priklauso nuo vieno parametro σ > 0. Trys atvejai, kai 0 < σ < 1, σ = 1 ir σ > 0, yra esminiai skirtingi. Darbo metodika gali būti naudojama ir bendresniems branduoliu atvejams. Irodytas naujas Rimano ir Lebego lemos varijantas, kuris naudojamas Furje transformacijos teorijoje.

Notes

The author expresses gratitude to A.F. Grishin for valuable remarks made in the process of this work.