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On the existence of complex Hadamard submatrices of the Fourier matrices

Bond Bailey Madison, Glickfield R. Alexander, Herr John E.

Demonstratio Mathematica

2019

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.

A generalized Walsh system for arbitrary matrices

Harding Steven N., Picioroaga Gabriel

Demonstratio Mathematica

2019

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.

Truncation and Duality Results for Hopf Image Algebras

Banica T.

Bulletin of the Polish Academy of Sciences. Mathematics

2014

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.