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The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry…

On $\mathbb R^N$ equipped with a normalized root system $R$ and a multiplicity function $k\geq 0$ let us consider a (non-radial) kernel $K(\mathbf x)$ which has properties similar to those from the classical theory. We prove that a singular…

This article studies the problem of approximating functions belonging to a Hilbert space $\mathcal H_d$ with a reproducing kernel of the form $$\tilde K_d(\boldsymbol x,\boldsymbol t):=\prod_{\ell=1}^d…

We study integrals of the form \begin{equation*} \int_{-1}^1(C_n^{(\lambda)}(x))^2(1-x)^\alpha (1+x)^\beta\, dx, \end{equation*} where $C_n^{(\lambda)}$ denotes the Gegenbauer-polynomial of index $\lambda>0$ and $\alpha,\beta>-1$. We give…

We study the family of compact operators $B_{\alpha} = V A_{\alpha} V$, $\alpha>0$ in $L^2(\mathbb R^d)$, $d\ge 1$, where $A_{\alpha}$ is the pseudo-differential operator with symbol $a_{\alpha}(\boldsymbol\xi) = a(\alpha\boldsymbol\xi)$,…

The peculiarities of doing a canonical analysis of the first order formulation of the Einstein-Hilbert action in terms of either the metric tensor $g^{\alpha \beta}$ or the metric density $h^{\alpha \beta}= \sqrt{-g}g^{\alpha \beta}$ along…

We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials $SP_\lambda((z_1,\ldots,z_n),(w_1,\ldots,w_m);\theta)$ with respect to a natural positive semi-definite, but degenerate, Hermitian product…

In this paper we introduce the notions of $q$-Duhamel product and $q$-integration operator. We prove that the classical Wiener algebra $W(\mathbb{D})$ of all analytic functions on the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$…

If $S$ is an infinite sequence over a finite alphabet $\Sigma$ and $\beta$ is a probability measure on $\Sigma$, then the {\it dimension} of $ S$ with respect to $\beta$, written $\dim^\beta(S)$, is a constructive version of Billingsley…

The Dimensional Regularization of Bollini and Giambiags (Phys. Lett. {\bf B 40}, 566 (1972), Il Nuovo Cim. {\bf B 12}, 20 (1972). Phys. Rev. {\bf D 53}, 5761 (1996)) can not be defined for all Schwartz Tempered Distributions Explicitly…

Based on a closed formula for a star product of Wick type on $\CP^n$, which has been discovered in an earlier article of the authors, we explicitly construct a subalgebra of the formal star-algebra (with coefficients contained in the…

Let $\kappa : \mathbb D \times \mathbb D \to \mathbb C$ be a diagonal positive definite kernel and let $\mathscr H_{\kappa}$ denote the associated reproducing kernel Hilbert space of holomorphic functions on the open unit disc $\mathbb D$.…

We study singular integral operators with kernels that are more singular than standard Calder\'on-Zygmund kernels, but less singular than bi-parameter product Calder\'on-Zygmund kernels. These kernels arise as restrictions to two dimensions…

We give analytical expressions for the eigenvalues and generalized eigenfunctions of $\hat{T}_3$, the $z$-axis projection of the toroidal dipole operator, in a system consisting of a particle confined in a thin film bent into a torus shape.…

The conformal powers of the Laplacian of a Riemannian metric which are known as the GJMS-operators admit a combinatorial description in terms of the Taylor coefficients of a natural second-order one-parameter family $\H(r;g)$ of…

We introduce a polyanalytic extension of the Gaussian radial basis function (RBF) kernel by computing the action of the convolution operator on normalized Hermite functions. In particular, using the Zaremba-Bergman formula we derive an…

We study the rational Cherednik algebra attached to the complex reflection group $G(r,1,2)$. Each irreducible representation $S^\lambda$ of $G(r,1,2)$ corresponds to a standard module $\Delta(\lambda)$ for the rational Cherednik algebra. We…

For $G=\mathrm{SL}_2$ or $\mathrm{GL}_2$, we present explicit formulas for the nonabelian Fourier kernels on $G$, as conjectured by A. Braverman and D. Kazhdan. Additionally, we furnish explicit formulas for the orbital Hankel transform on…

Let $\Omega$ be an open set in $\R^d$ $(d > 1)$ and $h(\Omega)$ the Fr\'echet space of harmonic functions on $\Omega$. Given a bounded linear operator $L :h(\Omega)\to h(\Omega)$, we show that its eigenvalues $\lambda_n$, arranged in…

We define a set inner product to be a function on pairs of convex bodies which is symmetric, Minkowski linear in each dimension, positive definite, and satisfies the natural analogue of the Cauchy-Schwartz inequality (which is not implied…