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/;, which is formulated for fractional orders, is roughly the Euler summation process applied to the Fourier-Walsh series off after it has first 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 presents 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.

[1]  C. W. Clenshaw,et al.  The special functions and their approximations , 1972 .

[2]  K. R. Rao,et al.  Orthogonal Transforms for Digital Signal Processing , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[3]  J. Walsh.  A Closed Set of Normal Orthogonal Functions , 1923 .

[4]  Ein Differential- und Integralkalkül in der Walsh-Fourier-Analysis mit Anwendungen , 1973 .

[5]  O. Morgenthaler,et al.  Proceedings of the Conference , 1930 .

[6]  Walsh Spectral Analysis , 1981 .

[7]  C. K. Yuen,et al.  Walsh Functions and Their Applications , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[8]  Sampling representation of sequency-band-limited nonstationary random processes. , 1980 .

[9]  M. Maqusi.  Truncation error bounds for sampling expansions of sequency-band-limited signals , 1978 .

[10]  K. Knopp,et al.  Theory and Applications of Infinite Series , 1972 .

[11]  C. W. Onneweer.  Fractional differentiation on the group of integers of ap-adic orp-series field , 1977 .

[12]  H. D. Frankel,et al.  Applications of Walsh Functions , 1971 .

[13]  On the characterization of the dyadic derivative , 1985 .

[14]  N. Fine.  On the Walsh functions , 1949 .

[15]  W. Splettstober.  Error Analysis in the Walsh Sampling Theorem , 1980, 1980 IEEE International Symposium on Electromagnetic Compatibility.

[16]  P. L. Butzer,et al.  Dyadic Calculus and Sampling Theorems for Functions with Multidimensional Domain. I: General Theory , 1982, Inf. Control..

[17]  N. Ahmed,et al.  Sequency Theory: Foundations and Applications , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[18]  On the rate of growth of the Walsh antidifferentiation operator , 1976 .

[19]  E. Hansen.  A Table of Series and Products , 1977 .

[20]  P. Butzer,et al.  Walsh-fourier series and the concept of a derivative † , 1973 .

[21]  P. L. Butzer,et al.  Sampling principle for duration-limited signals and dyadic Walsh analysis , 1978, Inf. Sci..

[22]  Absolute summability of Walsh-Fourier series , 1976 .

[23]  Fractional differentiation and Lipschitz spaces on local fields , 1980 .

[24]  W. Engels,et al.  A note on Maqusi's proofs and truncation error bounds in the dyadic(Walsh)sampling theorem , 1982 .

[25]  He Zelin,et al.  The derivatives and integrals of fractional order in Walsh-Fourier analysis, with applications to approximation theory , 1983 .

[26]  J. S. Wang.  Statistical Theory of Superlattices with Long-Range Interaction. I. General Theory , 1938 .

[27]  Generalization of some results concerning Walsh series and the dyadic derivative , 1979 .

[28]  Wolfgang Engels,et al.  On Walsh differentiable dyadically stationary random processes , 1982, IEEE Trans. Inf. Theory.

[29]  Differentiation on a p-Adic Or p-Series Field , 1978 .

[30]  On term by term dyadic differentiability of Walsh series , 1976 .

[31]  On linear methods of summation of Fourier series with respect to multiplicative systems , 1968 .

[32]  P. L. Butzer,et al.  On the definition of dyadic differentiation , 1979 .

[33]  J. Pál,et al.  On a generalization of the concept of derivative , 1977 .

[34]  P. Butzer,et al.  On dyadic analysis based on the pointwise dyadic derivative , 1975 .