Walsh-fourier series and the concept of a derivative^†

Abstract

Abstract In the theory of Walsh-Fourier series on the dyadic group G developed so far, such essential concepts as continuity, Lipschitz conditions modulus of continuity ,etc., have been employed. However, with the exception of several articles by J. E. Gibbs, no attempt has been made to define a derivative for functions f given on G. In this paper such a derivative D [1] f is defined which has most properties in common with the ordinary derivative. D [1] is a linear, closed operator, the inverse operator or integral is defined, the fundamental theorem of the calculus is valid for these two notions, The derivative enables one to estimate the order of magnitude of the Walsh-Fourier coefficients, the degree of approximation of f by the partial sums of the Walsh-Fourier series of f, etc. Moreover the Walsh functions are the non-trivial eigensolutions of a first-order linear differential equation. The proofs depend upon a Walsh-Fourier "coefficient method". †Dedicated to Eberhard Hopf on the occasion of his 70th birthday †The Contribution of this named author was supported by grant No.B/1-4858 from the Minister für Forschung des Landes Nordrhein Westfalen-Landesamt für Forschung. The authors wish to thank the landesamt for permission to publish the results here †Dedicated to Eberhard Hopf on the occasion of his 70th birthday †The Contribution of this named author was supported by grant No.B/1-4858 from the Minister für Forschung des Landes Nordrhein Westfalen-Landesamt für Forschung. The authors wish to thank the landesamt for permission to publish the results here Notes †Dedicated to Eberhard Hopf on the occasion of his 70th birthday †The Contribution of this named author was supported by grant No.B/1-4858 from the Minister für Forschung des Landes Nordrhein Westfalen-Landesamt für Forschung. The authors wish to thank the landesamt for permission to publish the results here