On a functional equation

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.

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

functional equations; reccurence relations.

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

https://ictp.acad.ro/wp-content/uploads/2023/03/1997b-Agratini-On-a-functional-equation-Studia-Universitatis.pdf

http://www.cs.ubbcluj.ro/~studia-m/old_issues/subbmath_1997_42_04.pdf

google scholar link