Fractional Multiresolution Analysis and Associated Scaling Functions in L2(R)L^2(\mathbb R)

In this paper, we show how to construct an orthonormal basis from Riesz basis by assuming that the fractional translates of a single function in the core subspace of the fractional multiresolution analysis form a Riesz basis instead of an orthonormal basis. In the definition of fractional multiresolution analysis, we show that the intersection triviality condition follows from the other conditions. Furthermore, we show that the union density condition also follows under the assumption that the fractional Fourier transform of the scaling function is continuous at $0$. At the culmination, we provide the complete characterization of the scaling functions associated with fractional multiresolutrion analysis.Comment: 19 pages, 0 figures. arXiv admin note: text overlap with arXiv:2008.0896

Convolution Based Special Affine Wavelet Transform and Associated Multi-resolution Analysis

In this paper, we study the convolution structure in the special affine Fourier domain to combine the advantages of the well known special affine Fourier and wavelet transforms into a novel integral transform coined as special affine wavelet transform and investigate the associated constant Q-property in the joint time-frequency domain. The preliminary analysis encompasses the derivation of the fundamental properties, orthogonality relation, inversion formula and range theorem. Finally, we extend the scope of the present study by introducing the notion of multi-resolution analysis associated with special affine wavelet transform and the construction of orthogonal special affine wavelets. We call it special affine multi-resolution analysis. The necessary and sufficient conditions pertaining to special affine Fourier domain under which the integer shifts of a chirp modulated functions form a Riesz basis or an orthonormal basis for a multi-resolution subspace is established.Comment: 25 pages, 4 figure