Related papers: Vector valued theta functions associated with bina…
Weyl-orbit functions have been defined for each simple Lie algebra, and permit Fourier-like analysis on the fundamental region of the corresponding affine Weyl group. They have also been discretized, using a refinement of the coweight…
This is a sequel to [SIGMA 9 (2013), 007, 23 pages, arXiv:1210.1177], in which there is a construction of a $2\times2$ positive-definite matrix function $K (x)$ on $\mathbb{R}^{2}$. The entries of $K(x)$ are expressed in terms of…
The Wigner functions on the one dimensional lattice are studied. Contrary to the previous claim in literature, Wigner functions exist on the lattice with any number of sites, whether it is even or odd. There are infinitely many solutions…
In this paper matrix quantitative weighted estimates on spaces of homogeneous type, such as endpoint estimates, strong type estimates are provided. To that end we extend some earlier results on convex body domination due to Nazarov,…
Borcherds lift for an even lattice of signature (p,q) is a lifting from weakly holomorphic modular forms of weight (p-q)/2 for the Weil representation. We introduce a new product operation on the space of such modular forms and develop a…
Special functions are often defined as a Fourier or Laplace transform of a positive measure, and the positivity of the measure manifests as positive definiteness of certain matrices. The purpose of this expository note is to give a sample…
We describe an approach to express the geometric side of the Arthur-Selberg trace formula in terms of zeta integrals attached to prehomogeneous vector spaces. This will provide explicit formulas for weighted orbital integrals and for the…
We give a new method for constructing vector-valued modular forms from singular scalar-valued ones. As an application we prove the identity between two remarkable spaces of vector-valued modular forms which seem to be unrelated at a first…
Tent spaces of vector-valued functions were recently studied by Hyt\"onen, van Neerven and Portal with an eye on applications to H^\infty-functional calculi. This paper extends their results to the endpoint cases p = 1 and p = \infty along…
We study the spectral zeta functions associated to the radial Schr\"odinger problem with potential V(x)=x^{2M}+alpha x^{M-1}+(lambda^2-1/4)/x^2. Using the quantum Wronskian equation, we provide results such as closed-form evaluations for…
Let $\mathcal{B}^{\omega}(B_d)$ denote the $\omega$-weighted Bloch space in the unit ball $B_d$ of $\mathbb{C}^d$, $d\ge 1$. We show that the quadratic integral $$ \int_x^1 \frac{\omega^2(t)}{t}\, dt,\quad 0<x<1, $$ governs the radial…
Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb{Q}$. We…
Given a bivariate weight function defined on the positive quadrant of $\mathbb{R}^2$, we study polynomials in two variables orthogonal with respect to varying measures obtained by special modifications of this weight function. In…
We derive the Weil-Petersson measure on the moduli space of hyperbolic surfaces with defects of arbitrary opening angles and use this to compute its volume. We conjecture a matrix integral computing the corresponding volumes and confirm…
We study the composition of two set-valued functions defined on locally convex topological linear spaces. We assume that these functions map into certain complete lattices of sets that have been used to establish a conjugation theory for…
The notion of the genus of a quadratic form is generalized to vertex operator algebras. We define it as the modular braided tensor category associated to a suitable vertex operator algebra together with the central charge. Statements…
We prove certain general forms of functional relations among Witten multiple zeta-functions in several variables (or zeta-functions of root systems). The structural background of those functional relations is given by the symmetry with…
To a finite quadratic module, that is, a finite abelian group D together with a non-singular quadratic form Q:D --> Q/Z, it is possible to associate a representation of either the modular group, SL(2,Z), or its metaplectic cover, Mp(2,Z),…
We introduce and investigate an infinite family of functions which are shown to have generalised quantum modular properties. We realise their "companions" in the lower half plane both as double Eichler integrals and as non-holomorphic theta…
We develop a theory of weights for a quantum analogue of the symmetric pair (gl4,gl2 x gl2) realised as a quantum symmetric pair subalgebra. Based on Letzter's triangular decomposition we define Verma modules. Using magical operators that…