A generalized Walsh system for arbitrary matrices

• Autorzy: Harding Steven N. , Picioroaga Gabriel

Identyfikatory

DOI

10.1515/dema-2019-0006

• Języki publikacji: EN

Abstrakty

EN

In this paper we study in detail a variation of the orthonormal bases (ONB) of L2 [0,1] introduced in [Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409(2),1128-1139] by means of representations of the Cuntz algebra ON on L2 [0,1]. For N = 2 one obtains the classic Walsh system which serves as a discrete analog of the Fourier system. We prove that the generalized Walsh system does not always display periodicity, or invertibility, with respect to function multiplication. After characterizing these two properties we also show that the transform implementing the generalized Walsh system is continuous with respect to filter variation. We consider such transforms in the case when the orthogonality conditions in Cuntz relations are removed. We show that these transforms which still recover information (due to remaining parts of the Cuntz relations) are suitable to use for signal compression, similar to the discrete wavelet transform.

Słowa kluczowe

EN

Cuntz algebras; Walsh basis; Hadamard matrix

PL

algebry Cuntza; funkcja Walsha; macierz Hadamarda

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2019
• Tom: Vol. 52, nr 1
• Strony: 40--55
• Opis fizyczny: Bibliogr. 8 poz., rys., wykr.

Twórcy

autor

Harding Steven N.

• Iowa State University, Department of Mathematics, 411 Morrill Road, Ames, IA 50011, U.S.A.

autor

Picioroaga Gabriel

• University of South Dakota, Department of Mathematical Sciences, 414 E. Clark Street, Vermillion, SD 57069, U.S.A.

Bibliografia

• [1] Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409(2),1128–1139
• [2] Walsh J. L., A closed set of normal orthogonal functions, Amer. J. Math, 1923, 45(1), 5–24
• [3] Vilenkin N., On a class of complete orthonormal systems, Bull. Acad. Sci. Math. [Izv. Akad. Nauk SSSR Ser. Mat.], 1947, 11(4),363–400
• [4] Chrestenson H. E., A class of generalized Walsh functions, Pacic. J. Math.,1955, 5(1), 17–31
• [5] Dutkay D. E., Picioroaga G., Silvestrov S., On generalized Walsh bases, Acta Appl. Math., 2018,1–18, https://doi.org/10.1007/s10440-018-0214-x
• [6] Harding S. N., Generalized Walsh transforms, Cuntz algebras representations and applications in signal processing, Uni-versity of South Dakota, Master Thesis, 2015, Copyright - Database copyright ProQuest LLC
• [7] Dutkay D. E., Picioroaga G., Generalized Walsh bases and applications, Acta Appl. Math., 2014, 133(1), 1–18, https://doi.org/10.1007/s10440-013-9856-x
• [8] Jorgensen P. E. T., Analysis and Probability: Wavelets, Signals, Fractals, Springer-Verlag New York, 2006

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).