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Zeitschrift für Analysis und ihre Anwendungen


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Volume 18, Issue 3, 1999, pp. 687–699
DOI: 10.4171/ZAA/906

Published online: 1999-09-30

An Integral Operator Representation of Classical Periodic Pseudodifferential Operators

Gennadi Vainikko[1]

(1) Tartu University, Estonia

In this note we prove that every classical 1-periodic pseudodifferential operator A of order $\alpha \in \mathbb R \\mathbb N_0$ can be represented in the form $$(Au)(t) = \int^1_0 [\kappa_{\alpha}^+(t - s)a_+(t,s) + \kappa_{\alpha}^– (t-s)a\_(t,s) + a(t,s)]u(s)ds$$ where $\alpha_±$ and $a$ are $C^{\infty}$-smooth 1-periodic functions and $\kappa_{\alpha}^±$ are 1-periodic functions or distributions with Fourier coefficients $\kappa_{\alpha}^+(n) = |n|^{\alpha}$ and $\kappa_{\alpha}^–(n) = |n|^{\alpha}$ sign$(n)$ $(0 \neq n \in \mathbb Z)$ with respect to the trigonometric orthonormal basis $\{e^{in2xt}\}_{n \in \mathbb Z}$ of $L^2 (0,1)$. Some explicit formulae for $\kappa_{\alpha}^±$ are given. The case of operators of order $\alpha \in \mathbb N_0$ is discussed, too.

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

Vainikko Gennadi: An Integral Operator Representation of Classical Periodic Pseudodifferential Operators. Z. Anal. Anwend. 18 (1999), 687-699. doi: 10.4171/ZAA/906