A hypergeometric basis for the Alpert multiresolution analysis
Provides a theoretical foundation for Alpert wavelets using hypergeometric functions, but is purely mathematical with no immediate application or empirical validation.
The authors construct an explicit orthonormal basis of piecewise hypergeometric polynomials for the Alpert multiresolution analysis, expressing Fourier transforms and wavelet-scaling function matrix entries in terms of hypergeometric functions, leading to new identities and orthogonality relations.
We construct an explicit orthonormal basis of piecewise ${}_{i+1}F_{i}$ hypergeometric polynomials for the Alpert multiresolution analysis. The Fourier transform of each basis function is written in terms of ${}_2F_3$ hypergeometric functions. Moreover, the entries in the matrix equation connecting the wavelets with the scaling functions are shown to be balanced ${}_4 F_3$ hypergeometric functions evaluated at $1$, which allows to compute them recursively via three-term recurrence relations. The above results lead to a variety of new interesting identities and orthogonality relations reminiscent to classical identities of higher-order hypergeometric functions and orthogonality relations of Wigner $6j$-symbols.