We observe that the exact connection coefficient relations transforming modal coefficients of one Jacobi Polynomial class to the modal coefficients of certain other classes are sparse. Because of this, when one of the classes corresponds to the Chebyshev case, the Fast Fourier Transform can be used to quickly compute modal coefficients for Jacobi Polynomial expansions of class (α, β) when 2α an...

The geometric structure of the Hamilton-Jacobi equation is analyzed by using symplectic geometry. Generating function of symplectic transform plays an important role. It will be shown that the Hamilton-Jacobi equation possesses important geometric properties such as existence condition and maximality of the stabilizing solution, which are well-known in the Riccati equation.

Hardy’s uncertainty principle states that it is impossible for a function and its Fourier transform to be simultaneously very rapidly decreasing. In this paper we prove Lp versions of this principle for the Jacobi transform and for the Fourier transform on real hyperbolic spaces.

The little q-Jacobi function transform depends on three parameters. An explicit expression as a sum of two very-well-poised 8W7-series is derived for the dual transmutation kernel relating little q-Jacobi function transforms for different parameter sets. A product formula for the dual transmutation kernel is obtained. For the inverse transform the transmutation kernel is given as a 3φ2-series, ...

The main goal of the paper is to show that wavelet transforms and packets have the multiparametric representation in the form of a product of the rotation Jacobi matrices. These representations we call the third and the fourth canonical multiparametric form. Each multiparametric wavelet transform (MPWT) depends on several free Jacobi parameters. When parameters are changed multiparametric trans...

Every Jacobi cusp form of weight k and index m over SL2(Z) Z 2 is in correspondence with 2m Dirichlet series constructed with its Fourier coefficients. The standard way to get from one to the other is by a variation of the Mellin transform. In this paper, we introduce a set of integral kernels which yield the 2m Dirichlet series via the Petersson inner product. We show that those kernels are Ja...

The main goal is to interpret the Askey-Wilson function and the corresponding transform pair on the quantum SU(1, 1) group. A weight on the C *-algebra of continuous functions vanishing at infinity on the quantum SU(1, 1) group is studied, which is left and right invariant in a weak sense with respect to a product defined using Wall functions. The Haar weight restricted to certain subalgebras a...

Abstract. We give a detailed description of the resolution of the identity of a second order q-difference operator considered as an unbounded self-adjoint operator on two different Hilbert spaces. The q-difference operator and the two choices of Hilbert spaces naturally arise from harmonic analysis on the quantum group SUq(1, 1) and SUq(2). The spectral analysis associated to SUq(1, 1) leads to...