Wavelet analysis has become a developing branch of mathematics for over twenty years. In this paper, the notion of orthogonal nonseparable bivariate wavelet packs, which is the generalization of orthogonal univariate wavelet packs, is proposed by virtue of analogy method and iteration method. Their biorthogonality traits are researched by using time-frequency analysis approach and variable sepa...

We define a general class of (multiple) integrals of hypergeometric type associated with the Jacobi theta functions. These integrals are related to theta hypergeometric series through the residue calculus. In the one variable case, we get theta function extensions of the Meijer function. A number of multiple generalizations of the elliptic beta integral [S2] associated with the root systems An ...

Biomedical research has been empowered by tools that enable spatial and temporal control of biological systems. These have predominantly come from photocaged bioactive molecules (optochemical control; N. Ankenbruck, T. Courtney, Y. Naro, A. Deiters, Angew. Chem. Int. Ed. 2018, 57, 2768–2798) light-dependent proteins (optogenetic L. Fenno, O. Yizhar, K. Deisseroth, Annu. Rev. Neurosci. 2011, 34,...

This paper provides the details of Remark 5.4 in the author’s paper “Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group”, SIAM J. Math. Anal. 24 (1993), 795–813. In formula (5.9) of the 1993 paper a two-parameter class of Askey-Wilson polynomials was expanded as a finite Fourier series with a product of two 3phi2’s as Fourier coefficients. The proof given there use...

In this paper we analyze the bi-conjugate gradient algorithm in finite precision arithmetic, and suggest reasons for its often observed robustness. By using a tridiagonal structure, which is preserved by the finite precision bi-conjugate gradient iteration, we are able to bound its residual norm by a minimum polynomial of a perturbed matrix (i.e. the residual norm of the exact GMRES applied to ...

Simple preconditioned iterations can provide an efficient alternative to more elaborate eigenvalue algorithms. We observe that these simple methods can be viewed as forward Euler discretizations of well-known autonomous differential equations that enjoy appealing geometric properties. This connection facilitates novel results describing convergence of a class of preconditioned eigensolvers to t...

We give a brief review of the main results of the theory of elliptic hypergeometric functions — a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler’s beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic ...

A new formula connecting the elliptic 6j-symbols and the fusion of the vertex-face intertwining vectors is given. This is based on the identification of the k fusion intertwining vectors with the change of base matrix elements from Sklyanin's standard base to Rosengren's natural base in the space of even theta functions of order 2k. The new formula allows us to derive various properties of the ...

This paper is concerned with the concepts of stability and biorthogonality for a general framework of multiscale transformations. In particular, stability criteria are derived which do not make use of Fourier transform techniques but rather hinge upon classical Bernstein and Jackson estimates. Therefore they might be useful when dealing with possibly nonuniform discretizations or with bounded d...

In this paper we develop a natural generalization of Schauder basis theory, we term operator-valued basis or simply ov-basis theory, using operator-algebraic methods. We prove several results for ov-basis concerning duality, orthogonality, biorthogonality and minimality. We prove that the operators of a dual ov-basis are continuous. We also dene the concepts of Bessel, Hilbert ov-basis and obta...