tailieunhanh - ON DIFFERENCE EQUATION WITH GENERALIZED DILATION PAVEL PLASCHINSKY Received 22 July 2004; Revised 24

ON DIFFERENCE EQUATION WITH GENERALIZED DILATION PAVEL PLASCHINSKY Received 22 July 2004; Revised 24 January 2005; Accepted 27 January 2005 We investigate the functional equation with generalized dilation in the special weighted functional spaces. We provide some sufficient conditions for the existence of the inversion operator in the same form and consider several examples. Copyright © 2006 Pavel Plaschinsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Consider the functional equation with generalized dilation: ∞ a(n,x)nντ f nτ x =. | ON DIFFERENCE EQUATION WITH GENERALIZED DIlAtION PAVEL PLASCHINSKY Received 22 July 2004 Revised 24 January 2005 Accepted 27 January 2005 We investigate the functional equation with generalized dilation in the special weighted functional spaces. We provide some sufficient conditions for the existence of the inversion operator in the same form and consider several examples. Copyright 2006 Pavel Plaschinsky. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Consider the functional equation with generalized dilation 00 a n x nVTf nTx g x x G 0 00 T V G R n 1 where a n x is bounded almost everywhere on 0 0 for arbitrary natural n and the sequence an of their Lo-norms belongs to 11. The equations of this type are used in many areas of physics 4 5 for example irradiation of black bodies. But in physics there were no rigorous proofs rather it was the idea of using the method of the Dirichlet convolution inverse we will call it here the discrete Mellin convolution . One can find the expansive bibliography and history of the algebraic approach to the integral and difference equations with transformed argument in for example 1-3 8 . The traditional use of the integral transforms in the case of constant coefficients does not work in Lp Banach spaces and we apply the method of the reciprocal sequences. In 6 the functional operator Ma T on the left-hand side of was completely investigated in the case of constant coefficients a n . In 7 it was shown that the operator Ma T is bounded in LV p that is in the Banach space of functions f x such that f x xV-1 p G Lp with the corresponding norm. In addition sufficient conditions for the existence of the inversion operator of the same form as in were found 1 a n x a1 a2 am T n x 2 II ak 111 ess inf ak 1 x esssup ak 1 x k 1 . m. Hindawi Publishing .