Orthonormal bases generated by Cuntz algebras. (English) Zbl 1309.42039
The Cuntz algebra \(\mathcal{O}_N\) is the \(C^*\)-algebra generated by \(N\) isometries \(S_i,\,\, i = 0, \cdots, N - 1\), satisfying \(S_i^*S_j=\delta_{ij}\) and \(\sum_{i=1}^{N-1}S_iS_i^*\). O. Bratteli and P. E. T. Jørgensen showed how one can construct orthonormal wavelet bases from various choices of quadrature mirror filters; see [O. Bratteli and P. E. T. Jørgensen, Integral Equations Oper. Theory 28, No.4, 382–443 (1997; Zbl 0897.46054)] and [O. Bratteli and P. E. T. Jørgensen, Wavelets through a looking glass. The world of the spectrum. Basel: Birkhäuser (2002; Zbl 1012.42023)]. In the paper under review, the authors show how some orthonormal bases including Fourier bases on fractal measures, generalized Walsh bases on the unit interval and piecewise exponential bases on the middle third Cantor set, can be generated by representations of the Cuntz algebra using some unitary matrix valued functions.
Reviewer: Mohammad Sal Moslehian (Mashhad)
MSC:
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 46L05 | General theory of \(C^*\)-algebras |
Keywords:
wavelet base; Cuntz algebras; fractal; Fourier basis; Hadamard matrix; quadrature mirror filterReferences:
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