Abstract
We investigate convergence properties of generalized Walsh series associated with signals f∈L 1[0,1]. We also show how the dependence of the generalized Walsh bases on N×N unitary matrices allows for applications in signal encoding and encryption, provided the signals are piece-wise constant on N-adic subintervals of [0,1].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmed, N., Rao, K.R.: Orthogonal Transforms for Digital Signal Processing. Springer, Berlin (1975)
Beauchamp, K.G.: Walsh Functions and Their Applications. Academic Press, New York (1975)
Buchberger, B.: Gröbner bases: an algorithmic method in polynomial ideal theory. In: Recent Trends in Multidimensional Systems Theory. Reider ed., Bose (1985)
Chrestenson, H.E.: A class of generalized Walsh functions. Pac. J. Math. 5, 17–31 (1955)
Delfs, H., Knebl, H.: Introduction to Cryptography, Principles and Applications, 2nd edn. Springer, Berlin (2007)
Dick, J., Kuo, F.Y., Pillichshammer, F., Sloan, I.H.: Construction algorithms for polynomial lattice rules for multivariate integration. Math. Comput. 74(252), 1895–1921 (2005)
Dick, J., Pillichshammer, F.: Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces. J. Complex. 21(2), 149–195 (2005)
Dutkay, D., Picioroaga, G., Song, M.S.: Orthonormal bases generated by Cuntz algebras. J. Math. Anal. Appl., in press. arXiv:1212.4134
Fine, N.J.: On the Walsh functions. Trans. Am. Math. Soc. 65, 372–414 (1949)
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003)
Harmuth, H.J.: Transmissions of Information by Orthogonal Functions. Springer, Berlin (1969)
Harmuth, H.J.: Sequency Theory. Foundations and Applications. Academic Press, New York (1977)
Kaczmarz, S., Steinhaus, H.: Le systeme orthogonal de M.Rademacher. Stud. Math. 2, 231–247 (1930)
Li, H., Torney, D.C.: A complete system of orthogonal step functions. Proc. Am. Math. Soc. 132(12), 3491–3502 (2004) (electronic)
Morgenthaler, G.W.: On Walsh-Fourier series. Trans. Am. Math. Soc. 84, 472–507 (1957)
Vilenkin, N.: On a class of complete orthonormal systems. Bull. Acad. Sci. URSS Ser. 11, 363–400 (1947)
Walsh, J.L.: A closed set of normal orthogonal functions. Am. J. Math. 45(1), 5–24 (1923)
Acknowledgements
This work was partially supported by a grant from the Simons Foundation (#228539 to Dorin Dutkay). The second named author would like to thank Professors Jose Flores for many discussions about Maple, and Catalin Georgescu for helpful insights related to Gröbner bases.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dutkay, D.E., Picioroaga, G. Generalized Walsh Bases and Applications. Acta Appl Math 133, 1–18 (2014). https://doi.org/10.1007/s10440-013-9856-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-013-9856-x