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Truncation and Duality Results for Hopf Image Algebras

100%

Banica T.

tom Vol. 62, no 2

161--179

Associated to an Hadamard matrix H∈MN(C) is the spectral measure μ∈P[0,N] of the corresponding Hopf image algebra, A=C(G) with G⊂S+N. We study a certain family of discrete measures μr∈P[0,N], coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type ∫N0(x/N)pdμr(x)=∫N0(x/N)rdνp(x), where μr,νr are the truncations of the spectral measures μ,ν associated to H,Ht. We also prove, using these truncations μr,νr, that for any deformed Fourier matrix H=FM⊗QFN we have μ=ν.

Certain new M-matrices and their properties with applications

60%

Mohan R. , Kageyama S. , Lee M. , Yang G.

Discussiones Mathematicae Probability and Statistics

2008

tom 28

nr 2

183-207

The Mₙ-matrix was defined by Mohan [21] who has shown a method of constructing (1,-1)-matrices and studied some of their properties. The (1,-1)-matrices were constructed and studied by Cohn [6], Ehrlich [9], Ehrlich and Zeller [10], and Wang [34]. But in this paper, while giving some resemblances of this matrix with a Hadamard matrix, and by naming it as an M-matrix, we show how to construct partially balanced incomplete block designs and some regular graphs by it. Two types of these M-matrices have been considered. Also we will make a mention of certain applications of these M-matrices in signal and communication processing, and network systems and end with some open problems.

A generalized Walsh system for arbitrary matrices

60%

Harding S. N. , Picioroaga G.

Demonstratio Mathematica

2019

tom Vol. 52, nr 1

40--55

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.

On the existence of complex Hadamard submatrices of the Fourier matrices

41%

Bond B. M. , Glickfield R. A. , Herr J. E.

Demonstratio Mathematica

2019

tom Vol. 52, nr 1

1--9

We use a theorem of Lam and Leung to prove that a submatrix of a Fourier matrix cannot be Hadamard for particular cases when the dimension of the submatrix does not divide the dimension of the Fourier matrix. We also make some observations on the trace-spectrum relationship of dephased Hadamard matrices of low dimension.