Related papers: Eigenfunction Expansions and Transformation Theory

In [8] we recently proved that in our model of quantum gravity the solutions to the quantized version of the full Einstein equations or to the Wheeler-DeWitt equation could be expressed as products of spatial and temporal eigenfunctions, or…

We formulate and prove the existence and uniqueness of the generalized Fourier transform associated with the absolutely continuous part of an arbitrary selfadjoint operator on a separable Hilbert space. To this end we develop a novel method…

We consider an enlarged dimension reduction space in functional inverse regression. Our operator and functional analysis based approach facilitates a compact and rigorous formulation of the functional inverse regression problem. It also…

Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the "native" Hilbert space $\calh$ in which they are reproducing. Continuous kernels on compact domains have an expansion into…

Let $M$ be a finite module and let $I$ be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of $I$ on $M$ using the 0th local cohomology functor. We show that our definition re-conciliates with that…

We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner's…

Integral transforms $$(\mbox{\boldmath$H$}f)(x)=\int^\infty_0H^{m,n}_{\thinspace p,q} \left[xt\left|\begin{array}{c}(a_i,\alpha_i)_{1,p}\\[1mm](b_j,\beta_j)_{1,q} \end{array}\right.\right]f(t)dt$$ involving Fox's $H$-functions as kernels…

We characterize the elements of generalized Gelfand Shilov spaces in terms of the coefficients of their Fourier-Hermite expansion. The technique we use can be applied both in quasianalytic and nonquasianalytic case. The characterizations…

We study the interior problem of tomography. The starting point is the Gelfand-Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function $f$ along a collection of lines. Pick one such…

We evaluate certain multidimensional integrals in terms of the Lerch transcendent function $\Phi$, generalizing Guillera-Sondow's formulas. As an application, we get new representations of classical constants like Euler's constant $\gamma$…

Associated Legendre functions of fractional degree appear in the solution of boundary value problems in wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer,…

We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental…

The Riemann-Liouville fractional integrals and derivatives are generalized for cases when fractional exponent $d$ are functions of space and times coordinates (i.e. $d=d({\bf r}(t),t)$).

We derive transformation formulas for the generalized polarization tensors under rigid motions and scaling in three dimensions, and use them to construct an infinite number of invariants under those transformations. These invariants can be…

The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences $(1/2,1/4,1/4)$, $(1/2,1/3,1/6)$ and $(1/3,1/3,1/3)$. These form a special class of algebraic transformations of Gauss…

We rederive in a simplified version the Lehmann-Sommers eigenvalue distribution for the Gaussian ensemble of asymmetric real matrices, invariant under real orthogonal transformations, as a basis for a detailed derivation of a Pfaffian…

In this paper, we study spectral properties of generalized weighted Hilbert matrices. In particular, we establish results on the spectral norm, determinant, as well as various relations between the eigenvalues and eigenvectors of such…

The so-called equation of motion method is useful to obtain the explicit form of the eigenvectors and eigenvalues of certain non self-adjoint bosonic Hamiltonians with real eigenvalues. These operators can be diagonalized when they are…

Density Functional Theory (DFT) is one of the most used ab initio theoretical frameworks in materials science. It derives the ground state properties of a multi-atomic ensemble directly from the computation of its one-particle density \nr…

We establish a spectral representation for solutions to linear Hamilton equations with positive definite energy in a Hilbert space. Our approach is a special version of M. Krein's spectral theory of J-selfadjoint operators is the Hilbert…