An Integral Operator Representation of Classical Periodic Pseudodifferential Operators

Abstract

In this note we prove that every classical 1-periodic pseudodifferential operator A of order $$\begin{gathered} \alpha \in \mathbb{R} \\ mathbb{N}_0 \end{gathered}$$ can be represented in the form

where ${\alpha }_{\pm }$ and $a$ are $C^{\infty }$-smooth 1-periodic functions and ${\kappa }_{\alpha }^{\pm }$ are 1-periodic functions or distributions with Fourier coefficients ${\kappa }_{\alpha }^{+}(n)=|n{|}^{\alpha }$ and ${\kappa }_{\alpha }^{–}(n)=|n{|}^{\alpha }$ sign$(n)$ $(0\ne n\in \mathbb{Z})$ with respect to the trigonometric orthonormal basis $\{e^{in2xt}{\}}_{n\in \mathbb{Z}}$ of $L^2(0,1)$. Some explicit formulae for ${\kappa }_{\alpha }^{\pm }$ are given. The case of operators of order $\alpha \in {\mathbb{N}}_0$ is discussed, too.