An Integral Operator Representation of Classical Periodic Pseudodifferential Operators

G. Vainikko

Abstract: In this note we prove that every classical 1-periodic pseudodifferential operator $A$ of order $\alpha \in \R \setminus \N_0$ can be represented in the form $$ (Au)(t) = \int\limits_0^1 \Big[\kappa_\alpha^{\sc+}(t - s)a_{\sc+}(t,s) + \kappa_\alpha^{\sc-}(t - s)a_{\sc-}(t,s) + a(t,s)\Big]u(s)\,ds $$ where $\alpha_{\sc\pm}$ and $a$ are $C^\infty$-smooth 1-periodic functions and $\kappa_\alpha^{\sc\pm}$ are 1-periodic functions or distributions with Fourier coefficients $\hat\kappa_\alpha^{\sc+}(n) = |n|^\alpha$ and $\hat \kappa_\alpha^{\sc-}(n) = |n|^\alpha{\rm sign}(n) \ \ (0 \ne n \in \Z)$ with respect to the trigonometric orthonormal basis $\{e^{in2\pi t}\}_{n\in\Z}$ of $L^2(0,1)$. Some explicit formulae for $\kappa_\alpha^{\sc\pm}$ are given. The case of operators of order $\alpha \in \N_0$ is discussed, too.

Keywords: classical periodic pseudodifferential operators, periodic integral operators, asymptotic expansions

Classification (MSC2000): 47G30, 47G10, 58G15

Full text of the article:

• Compressed DVI file (22 kilobytes)
• Compressed PostScript file (93 kilobytes)
• PDF file (218 kilobytes)