Asymptotics of the Basis Functions of Generalized Taylor Series for the Class $H_{\rho,2}$

V. A. Makarichev

National Aerospace University "Kharkiv Aviation Institute"

Abstract: We study the basis functions $\varphi_{n,k}$ and $\psi_{n,p}$ of generalized Taylor series for the class $H_{\rho,2}$ and obtain asymptotic expansions of the functions $\varphi^{(l)}_{n,0}$ and $\psi^{(l)}_{n,2\cdot4^n-1}$. We prove the existence of an asymptotics for the functions $\varphi_{n,k}$ and $\psi_{n,p}$ for $k\ne 0$ and $p\ne {2\cdot4^n}-1$. The first term of the asymptotic expansions of these functions is obtained.

Keywords: generalized Taylor series, basis functions, the class of functions $H_{\rho,2}$, functional-differential equation.

UDC: 517

Received: 29.04.2009
Revised: 06.09.2010

DOI: 10.4213/mzm8844

 English version: Mathematical Notes, 2011, 89:5, 689–705

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