## 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.

## About this paper

##### Journal

Studia Universitatis “Babes-Bolyai” Mathematica

##### Publisher Name

Mathematica

##### Print ISSN

1843-3855

##### Online ISSN

2065-9490

google scholar link

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