An Integral Operator Representation of Classical Periodic Pseudodifferential Operators

In this note we prove that every classical 1-periodic pseudodifferential operator A of order a € R \ No can be represented in the form (Au)(t) = f [(t s)a(t,$) + K(t s)a(t,$) + a(t,$)J(s)ds where Oj and a are C'-smooth 1-periodic functions and ?C are 1-periodic functions or distributions with Fourier coefficients k(n) = n i 0 and (n) = n°sign(n) (0 $ n € 7L) with respect to-the trigonometric orthonormal basis {e"''},Ez of L 2 (0, 1). Some explicit formulae for are given. The case of operators of order a E No is discussed, too.