Abstract
In this paper we discuss the definition of Gibbs derivatives on finite, not necessarily Abelian, groups in terms of the partial Gibbs derivatives. We consider the matrix representation of Gibbs derivatives defined in this way, which enables us to disclose FFT-like algorithms for the calculation of the values of Gibbs derivatives of functions on finite groups into fields admitting the existence of a Fourier transform on groups.
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Gibbs, J.E., Walsh Functions as Solutions of a Logical Differential Equation, National Physical Lab., Teddingon, England, DES Report, No. 1, 1969.
Gibbs, J.E., Simpson, J., Differentiation on Finite Abelian Groups, National Physical Lab. Teddington, England, DES Rept, No. 14, 1974.
Butzer, P.L., Wagner, H.J., Walsh-Fourier Series and the Concept of a Derivative, Applicable Analysis, Vol. 3, pp. 29–46, 1973.
Onneweer, C.W., Fractional Differentiation on the Group of Integers of a p-adic or p-series field. Anal. Math., Vol. 3, pp. 119–130, 1977.
Moraga, C., Introduction to Linear p-adic Invariant Systems, Cybernetics and Systems Research, Vol. 2, pp. 121–124, Ed., R. Trappl, Vienna: Electronic Science Publ., 1984.
Cohn-Sfetcu, S., Gibbs, J.E., Harmonic Differential Calculus and Filtering in Galois Fields, Proc. 1976 IEEE Conf. on Acoustrics, Speech and Singal Processing, Philadelphia, Pa., pp. 148–153, 1976.
Stankovic, R.S., A Note on differential Operators on Finite non-Abelian Groups, Applicable Analysis, Vol. 21, pp. 31–41, 1986.
Stankovic, R.S., A Note on Differential Operators on Finite Abelian Groups. Cybernetics and Systems, 18, pp. 221–231, 1987.
Stankovic, R.S., A Note on Spectral Theory on Finite non-Abelian Groups, Theory and Application of Spectral Techniques, Ed., C. Moraga, Forschungsbereicht 268, ISSN 0933-6192, Dortmund University, 1988.
He Zelin, The Derivatives and Integrals of Fractional Order in Walsh-Fourier Analysis, with Applications to Approximation Theory, J. Approx. Theory, 39, 361–373, 1983.
Stankovic, R.S., Stojic, M.R., A Note on the Discrete Haar Derivative, Colloquia Mathematica Societatis Janos Bolyai 49. Alfred Haar Memorial Conference, Budapest, Hungary, pp. 897–907. 1985.
Stankovic, R.S., Stojic, M.R., A Note on the Generalized Discrete Haar Derivative, Automatika, 28, 3–4, pp. 117–122, 1987.
Nussbaumer, H.J., Fast Fourier Transform and Convolution Algorithms, Springer-Verlag, Berlin, Heidelberg, New York, 1981.
Karpovsky, M.G., Tranchtenberg, E.A., Fourier Transform over Finite Groups for Error Detection and error correction in Computation Channels, Inform. Control, Vol. 40, No. 3, pp. 335–358, March, 1979.
Stankovic, R.S., Matrix Interpretation of Fast Fourier Transform on Finite non-Abelian Groups, Res. Rept. in Appl. Math., YU ISSN 0353-6491, Ser. Fourier Analysis, Rept, No. 3, April 1990, p. 1–31 ISBN 86-81611-03-8.
Karpovsky, M.G., Fast Fourier Transforms on Finite non-abelian Groups, IEEE Trans. Comput., Vol. C-26, No. 10, pp. 1028–1030, Oct. 1977.
Gibbs, J.E., Stankovic, R.S., Matrix Interpretation of Gibbs Derivatives on Finite Groups, private communication.
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Stankovic, R.S. Fast algorithms for calculation of Gibbs derivatives on finite groups. Approx. Theory & its Appl. 7, 1–19 (1991). https://doi.org/10.1007/BF02845188
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DOI: https://doi.org/10.1007/BF02845188