# An Integral Operator Representation of Classical Periodic Pseudodifferential Operators

### Gennadi Vainikko

Tartu University, Estonia

## Abstract

In this note we prove that every classical 1-periodic pseudodifferential operator A of order $α∈RmathbbN_{0}$ can be represented in the form

$(Au)(t)=∫_{0}[κ_{α}(t−s)a_{+}(t,s)+κ_{α}(t−s)a_(t,s)+a(t,s)]u(s)ds$

where $α_{±}$ and $a$ are $C_{∞}$-smooth 1-periodic functions and $κ_{α}$ are 1-periodic functions or distributions with Fourier coefficients $κ_{α}(n)=∣n∣_{α}$ and $κ_{α}(n)=∣n∣_{α}$ sign$(n)$ $(0=n∈Z)$ with respect to the trigonometric orthonormal basis ${e_{in2xt}}_{n∈Z}$ of $L_{2}(0,1)$. Some explicit formulae for $κ_{α}$ are given. The case of operators of order $α∈N_{0}$ is discussed, too.

## Cite this article

Gennadi Vainikko, An Integral Operator Representation of Classical Periodic Pseudodifferential Operators. Z. Anal. Anwend. 18 (1999), no. 3, pp. 687–699

DOI 10.4171/ZAA/906