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Gaussian radial basis functions can be an accurate basis for multivariate interpolation. In practise, high accuracies are often achieved in the flat limit where the interpolation matrix becomes increasingly ill-conditioned. Stable…
We introduce \`a la Vasilevski the weighted poly-Bergman spaces in the unit disc and provide concrete orthonormal basis and give close expression of their reproducing kernel. The main tool in the description if these spaces is the so-called…
We present explicit expressions for the Mellin transforms of Laguerre and Hermite functions in terms of a variety of special functions. We show that many of the properties of the resulting functions, including functional equations and…
We use generating functions to express orthogonality relations in the form of $q$-beta integrals. The integrand of such a $q$-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal
We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding…
The aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the…
In this article, we introduce inhomogeneous variable Triebel-Lizorkin spaces, $F_{p(\cdot),q(\cdot)}^{\alpha(\cdot),H}(\mathbb R^n)$, associated with the Hermite operator $H:=-\Delta+|x|^2$, where $\Delta$ is the Laplace operator on…
We prove a sufficient condition for frame-type wavelet series in $L^p$, the Hardy space $H^1$, and BMO. For example, functions in these spaces are shown to have expansions in terms of the Mexican hat wavelet, thus giving a strong answer to…
The Fredholm-Hammerstein integral equations (FHIEs) with weakly singular kernels exhibit multi-point singularity at the endpoints or boundaries. The dense discretized matrices result in high computational complexity when employing numerical…
We represent low dimensional quantum mechanical Hamiltonians by moderately sized finite matrices that reproduce the lowest O(10) boundstate energies and wave functions to machine precision. The method extends also to Hamiltonians that are…
Generalised Hermite-Gaussian modes (gHG modes), an extended notion of Hermite-Gaussian modes (HG modes), are formed by the summation of normal HG modes with a characteristic function $\alpha$, which can be used to unite conventional HG…
The spherical Fourier transform on a harmonic Hadamard manifold $(X^n, g)$ of positive volume entropy is studied. If $(X^n, g)$ is of hypergeometric type, namely spherical functions of $X$ are represented by the Gauss hypergeometric…
It is well known that for any wave function $\psi(x,z)$ of the KP hierarchy, there is another wave function called its ''adjoint'' such that the path integral of their product with respect to $z$ around any sufficiently large closed path is…
Generalized eigenfunctions may be regarded as vectors of a basis in a particular direct integral of Hilbert spaces or as elements of the antidual space $\Phi^\times$ in a convenient Gelfand triplet…
We study integration in a class of Hilbert spaces of analytic functions defined on the $\mathbb{R}^s$. The functions are characterized by the property that their Hermite coefficients decay exponentially fast. We use Gauss-Hermite…
Wigner functions generically attain negative values and hence are not probability densities. We prove an asymptotic expansion of Wigner functions in terms of Hermite spectrograms, which are probability densities. The expansion provides…
We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions $H_n$. Let $h= (H_0, H_1, ..., H_n)$ be the vector of the first $n+1$ Hermite functions. We give a complete characterization of all…
We show that every biorthogonal wavelet determines a representation by operators on Hilbert space satisfying simple identities, which captures the established relationship between orthogonal wavelets and Cuntz-algebra representations in…
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…
The standard Radon transform of holomorphic functions is not always well defined, as the integration of such functions over planes may not converge. In this paper, we introduce new Radon-type transforms of co-(real)dimension $2$ for…