An extension of the dyadic calculus with fractional order derivatives: General theory

A new dyadic calculus is set up, principally based upon that introduced by Butzer and Wagner (1972/1975), and Zelin He (1983). The extended dyadic derivative of a function f, which is formulated for fractional orders, is roughly the Euler summation process applied to the Fourier-Walsh series of f after it has been equipped with a certain multiplicative factor. This extended calculus is not only applicable to piecewise constant functions (as is the classical dyadic derivative) but also to piecewise polynomials. This paper present the full general theory of this extended dyadic calculus: introduction and justification of the dyadic derivative, its fundamental properties and those of its eigenvalues; the corresponding anti-differentiation operator, a type of counterpart of the fundamental theorem of the Newton-Leibniz calculus in the frame of Walsh analysis. Applications and further theory are dealt with in the second paper in the series.