Wavelets on the Spectrum
Niraj K. Shukla, Shiva Mittal · Numerical Functional Analysis and Optimization · 2013
Gabardo and Nashed have introduced a generalized notion of multiresolution analysis, called nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs (Ω, Λ) in which the translation set is a spectrum Λ which is not necessarily a group nor a uniform discrete set, given by , where N ≥ 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N − 1 such that r and N are relatively prime and ℤ is the set of integers. In this article the theory of wavelets on the spectrum is developed. To describe wavelets in the nonuniform discrete setting, first we provide a characterization of an orthonormal basis for l 2(Λ) and then show that the Hilbert space l 2(Λ) can be expressed as an orthogonal decomposition in terms of countable number of its closed subspaces. In addition, we show that the wavelets associated with NUMRA are connected with the wavelets on the spectrum.