JournalszaaVol. 18, No. 3pp. 687–699

An Integral Operator Representation of Classical Periodic Pseudodifferential Operators

  • Gennadi Vainikko

    Tartu University, Estonia
An Integral Operator Representation of Classical Periodic Pseudodifferential Operators cover
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Abstract

In this note we prove that every classical 1-periodic pseudodifferential operator A of order αRmathbbN0\alpha \in \mathbb R \\mathbb N_0 can be represented in the form

(Au)(t)=01[κα+(ts)a+(t,s)+κα(ts)a_(t,s)+a(t,s)]u(s)ds(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 α±\alpha_± and aa are CC^{\infty}-smooth 1-periodic functions and κα±\kappa_{\alpha}^± are 1-periodic functions or distributions with Fourier coefficients κα+(n)=nα\kappa_{\alpha}^+(n) = |n|^{\alpha} and κα(n)=nα\kappa_{\alpha}^–(n) = |n|^{\alpha} sign(n)(n) (0nZ)(0 \neq n \in \mathbb Z) with respect to the trigonometric orthonormal basis {ein2xt}nZ\{e^{in2xt}\}_{n \in \mathbb Z} of L2(0,1)L^2 (0,1). Some explicit formulae for κα±\kappa_{\alpha}^± are given. The case of operators of order αN0\alpha \in \mathbb 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