# On Generalized Walsh Bases

@article{Dutkay2018OnGW,
title={On Generalized Walsh Bases},
author={Dorin Ervin Dutkay and Gabriel Picioroaga and Sergei Silvestrov},
journal={Acta Applicandae Mathematicae},
year={2018},
pages={1-18}
}
• Published 28 February 2018
• Mathematics
• Acta Applicandae Mathematicae
This paper continues the study of orthonormal bases (ONB) of L2[0,1]$L^{2}[0,1]$ introduced in Dutkay et al. (J. Math. Anal. Appl. 409(2):1128–1139, 2014) by means of Cuntz algebra ON$\mathcal{O}_{N}$ representations on L2[0,1]$L^{2}[0,1]$. For N=2$N=2$, one obtains the classic Walsh system. We show that the ONB property holds precisely because the ON$\mathcal{O}_{N}$ representations are irreducible. We prove an uncertainty principle related to these bases. As an application to discrete signal… Expand
3 Citations

#### Figures from this paper

A generalized Walsh system for arbitrary matrices
• Mathematics
• Demonstratio Mathematica
• 2019
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,Expand
Purely Atomic Representations of Higher-Rank Graph $$\varvec{C}^{\varvec{*}}$$C∗-Algebras
• Mathematics
• Integral Equations and Operator Theory
• 2018
We study purely atomic representations of $$C^*$$C∗-algebras associated to row-finite and source-free higher-rank graphs. We describe when purely atomic representations are unitarily equivalent andExpand
Purely atomic representations of higher-rank graph C*-algebras
• Mathematics
• 2018
We study purely atomic representations of C*-algebras associated to row-finite and source-free higher-rank graphs. We describe when purely atomic representations are unitarily equivalent and we giveExpand

#### References

SHOWING 1-10 OF 47 REFERENCES
Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scaleN
• Mathematics
• 1996
AbstractIn this paper we show how wavelets originating from multiresolution analysis of scaleN give rise to certain representations of the Cuntz algebrasON, and conversely how the wavelets can beExpand
Cuntz algebras and multiresolution wavelet analysis of scale N
• Mathematics
• 1997
In this paper we show how wavelets originating from multireso-lution analysis of scale N give rise to certain representations of the Cuntz algebras O N , and conversely how the wavelets can beExpand
Generalized Walsh Bases and Applications
• Mathematics
• 2013
We investigate convergence properties of generalized Walsh series associated with signals f∈L1[0,1]. We also show how the dependence of the generalized Walsh bases on N×N unitary matrices allows forExpand
An uncertainty principle for cyclic groups of prime order
Let $G$ be a finite abelian group, and let $f: G \to \C$ be a complex function on $G$. The uncertainty principle asserts that the support $\supp(f) := \{x \in G: f(x) \neq 0\}$ is related to theExpand
Atomic representations of Cuntz algebras
• Mathematics
• 2013
To a representation of $\O_N$ (the Cuntz algebra with $N$ generators) we associate a projection valued measure and we study the case when this measure has atoms. The main technical tool are theExpand
Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces
• Computer Science, Mathematics
• J. Complex.
• 2005
A weighted reproducing kernel Hilbert space which is based on Walsh functions is introduced and it is found that there exists a digital (t, m, s)-net, which achieves a strong tractability worst-case error bound under certain condition on the weights. Expand
Construction algorithms for polynomial lattice rules for multivariate integration
• Computer Science, Mathematics
• Math. Comput.
• 2005
An upper bound is proved on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights. Expand
A Closed Set of Normal Orthogonal Functions
A set of normal orthogonal functions {χ} for the interval 0 5 x 5 1 has been constructed by Haar†, each function taking merely one constant value in each of a finite number of sub-intervals intoExpand
Orthogonal Transforms for Digital Signal Processing
• Mathematics, Computer Science
• IEEE Transactions on Systems, Man, and Cybernetics
• 1979
The utility and effectiveness of these transforms are evaluated in terms of some standard performance criteria such as computational complexity, variance distribution, mean-square error, correlated rms error, rate distortion, data compression, classification error, and digital hardware realization. Expand
Fourier Frames for the Cantor-4 Set
• Mathematics
• 2015
The measure supported on the Cantor-4 set constructed by Jorgensen–Pedersen is known to have a Fourier basis, i.e. that it possess a sequence of exponentials which form an orthonormal basis. WeExpand