# On a functional equation

## Abstract

This is a survey paper devoted to the following functional equation
$\sum_{k\in Z}p_{k}u\left( x-k\right) =v\left( x\right) ,\ \ \ \ \ x\in \mathbb{R}$
which is in connection with the notion of wavelets. If $$v\left( k\right)$$ vanishes for $$k\in\mathbb{Z}$$ and if $$p_{k}=0$$ for $$k<0$$ and $$k\geq m+1$$, then, for $$x=n$$, the above equation leads us to the well-known general $$m^{th}$$-order linear recurrence relation. For $$v\left( x\right) =u\left(2x\right) ,\ \ x\in\mathbb{R}$$, we present how this equation appears as a necessity in the field of mathematics. We also indicate three properties which must be fulfilled by the function and the sequence so that these equations admit solutions. When the sequence $$\left( p_{k}\right) _{k\in\mathbb{Z}}$$ has a compact support other properties are revealed and the technique to obtain solutions is described.

## Authors

Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

## Keywords

functional equations; reccurence relations.

## Paper coordinates

O. Agratini, On a functional equation, Studia Universitatis Babes-Bolyai Mathematica, 42 (1997) no. 4, pp. 5-8.

## PDF

##### Journal

Studia Universitatis “Babes-Bolyai” Mathematica

Mathematica

1843-3855

2065-9490