Related papers: An integral formula for L^2-eigenfunctions of a fo…
We analyze a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well potential and $\varepsilon$ is a small parameter. We construct…
We present a new formula of Cauchy type for the nonsymmetric Macdonald polynomials which are joint eigenfunctions of q-Dunkl operators. This gives the explicit formula for a reproducing kernel on the polynomial ring of n variables.
In the present paper, we introduce a method to construct two variable biorthogonal polynomial families with the help of one variable biorthogonal and orthogonal polynomial families. By using this new technique, we define 2D Hermite…
Fortran 77 programs for the computation of modified Bessel functions of purely imaginary order are presented. The codes compute the functions $K_{ia}(x)$, $L_{ia}(x)$ and their derivatives for real $a$ and positive $x$; these functions are…
The $k$-Cauchy-Fueter operators, $k=0,1,\ldots$, are quaternionic counterparts of the Cauchy-Riemann operator in the theory of several complex variables. The weighted $L^2$ method to solve Cauchy-Riemann equation is applied to find the…
We define an analogue of the classical Mittag-Leffler function which is applied to two variables, and establish its basic properties. Using a corresponding single-variable function with fractional powers, we define an associated fractional…
In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre \cite{CS1} are extended to those for the integro-differential operators associated with symmetric,…
In hep-th/0202087 it was argued that the operator L_0 is bad defined in kappa-basis as a kernel operator. Indeed, we show that L_0 is a difference operator. We also find a representation of L_1 and L_{-1} in a class of difference operators.
The purpose of this work is to study spectral methods to approximate the eigenvalues of nonlocal integral operators. Indeed, even if the spatial domain is an interval, it is very challenging to obtain closed analytical expressions for the…
We present a method to derive asymptotics of eigenvalues for trace-class integral operators $K:L^2(J;d\lambda)\circlearrowleft$, acting on a single interval $J\subset\mathbb{R}$, which belong to the ring of integrable operators \cite{IIKS}.…
We provide an algorithm to compute generators of the orthogonal group of the discriminant group associated to an integral quadratic lattice over the integers. We give a closed formula for its order.
We obtain global pointwise estimates for kernels of the resolvents $(I-T)^{-1}$ of integral operators \[Tf(x) = \int_{\Omega} K(x, y) f(y) d \omega(y)\] on $L^2(\Omega, \omega)$ under the assumptions that $||T||_{L^2(\omega) \rightarrow L^2…
New index transforms of the Lebedev type are investigated. It involves the real part of the product of the modified Bessel functions as the kernel. The boundedness and invertibility are examined for these operators in the Lebesgue weighted…
We work out instances of a general conjecture on congruences between Hecke eigenvalues of induced and cuspidal automorphic representations of a reductive group, modulo divisors of certain critical L-values, in the case that the group is a…
We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear…
For any Hermitian Lie group G of tube type we construct a Fock model of its minimal representation. The Fock space is defined on the minimal nilpotent K_C-orbit X in p_C and the L^2-inner product involves a K-Bessel function as density.…
We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin…
We introduce and systematically develop two classes of discrete integrable operators: those with $2\times 2$ matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A…
We provide a general condition on the kernel of an integro-differential operator so that its associated quadratic form satisfies a coercivity estimate with respect to the $H^s$-seminorm.
In the paper we consider self-adjoint partial integral operators of Fredholm type $T$ with a degenerate kernel on the space $L_2([a,b]\times[c,d]).$ Essential and discrete spectra of $T$ are described.