×

zbMATH — the first resource for mathematics

An extension of the dyadic calculus with fractional order derivatives. Further theory and applications. (English) Zbl 0661.42018
In their second paper dealing with the extended dyadic derivative the authors complete the theory by adding operational rules for their so- called ED-derivative, e.g. a Leibnizian rule valid for Walsh functions.
They also calculate the ED-derivatives for a couple of examples: piecewise constant functions, polynomials, piecewise polynomials given as a product of a monomial and a Walsh function, Walsh and Rademacher series.
Furthermore they investigate applications in Walsh-Fourier analysis and the theory of best approximation. Finally, they deal with differential equations, especially solve the dyadic wave equation. Although not all results are in strict analogy to classical calculus, the present investigations might help to answer the questions, in which of the modern fields - like digital communication or pattern recognition - dyadic derivatives will find their interpretation.
Reviewer: W.Splettstößer

MSC:
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A36 Antidifferentiation
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] P. L. Butzer, W. Engels, and U. Wipperfürth, An extension of the dyadic calculus with fractional order derivatives. General Theory. Comput. Math. Applic. Ser. B. (to appear).
[2] Butzer, P.L.; Engels, W., Dyadic calculus and sampling theorems for functions with multi-dimensional domain. part I:general theory. part II: applications to sampling theorems., Inf. control, 52, 333-351, (1982) · Zbl 0514.42028
[3] Engels, W., On the characterization of the dyadic derivative, Act. math. acad. sci. hungar, 46, 47-56, (1985) · Zbl 0604.42032
[4] McLaughlin, J.R., Functions represented by integrated Rademacher series, Colloq. math., 20, 277-286, (1969) · Zbl 0186.10102
[5] McLaughlin, J.R., Functions represented by Rademacher series, Pac. math. J., 27, 373-378, (1968) · Zbl 0175.35701
[6] Harmuth, H.J., Transmissions of information by orthogonal functions, (1972), Springer New York
[7] Butzer, P.L.; Nessel, R.J., ()
[8] Penney, R., On the rate of growth of the Walsh antidifferentiation operator, (), 57-61 · Zbl 0319.42009
[9] Fine, N.J., On the Walsh functions, Trans. am. math. soc., 65, 372-414, (1949) · Zbl 0036.03604
[10] Morgenthaler, G.W., On the Walsh-Fourier series, Trans. am. math. soc., 84, 452-507, (1957) · Zbl 0089.27702
[11] Watari, C., Best approximation by Walsh polynomials, Tôhoku math. J., 15, 1-5, (1963) · Zbl 0111.26502
[12] Butzer, P.L.; Splettstöβer, W., Sampling principle for duration-limited signales and dyadic Walsh analysis, Inf. sci., 14, 93-106, (1978) · Zbl 0416.94002
[13] Butzer, P.L.; Scherer, K., On the fundamental approximation theorems of D. Jackson, S. N. Bernstein and theorems of M. zamansky and S. B. steckin, Aequationes math., 3, 113-115, (1969)
[14] Butzer, P.L.; Wagner, H.J., On dyadic analysis, based on the pointwise dyadic derivative, Analysis math., 1, 171-196, (1975) · Zbl 0324.42011
[15] Harmuth, H.J., Sequency theory. foundations and applications, (1977), Academic Press New York · Zbl 0521.94002
[16] Butzer, P.L.; Wagner, H.J.; Butzer, P.L.; Wagner, H.J., A calculus for Walsh functions defined on R^+, (), 75-81, Washington, D.C., April 18-20, 1973 · Zbl 0273.42011
[17] Schrödinger, E., Causality and wave mechanics, (), 1056-1068 · Zbl 1219.78099
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.