Walsh-fourier series and the concept of a derivative†

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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".

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