Affiliation:
1. Iowa State University, Department of Mathematics, 411 Morrill Road, Ames , IA 50011, U.S.A
2. University of South Dakota, Department of Mathematical Sciences, 414 E. Clark Street, Vermillion , SD 57069, U.S.A.
Abstract
Abstract
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.
Reference8 articles.
1. [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. [2] Walsh J. L., A closed set of normal orthogonal functions, Amer. J. Math, 1923, 45(1), 5-24
3. [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. [4] Chrestenson H. E., A class of generalized Walsh functions, Pacic. J. Math., 1955, 5(1), 17-31
5. [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