Related papers: Fourier transform on the Lobachevsky plane and ope…
The present paper, is devoted to investigation of operator--valued Fourier multiplier theorems from $B_{q_{1},r}^{s}$ to $B_{q_{2},r}^{s}$, optimal embedding of Besov spaces, the separability and positivity of differential operators. Here,…
We construct a version of Fourier transform for a class of non-commutative algebras over abelian varieties which include algebras of twisted differential operators generalizing the previous construction of Laumon (alg-geom/9603004) and of…
We highlight the important role of the Fourier transform in deriving inversion formulas for the integral transforms of tomographic imaging. We demonstrate this principle by deriving inversion formulas for the divergent beam transform and…
The truncated Fourier operator $\mathscr{F}_{\mathbb{R^{+}}}$, $$ (\mathscr{F}_{\mathbb{R^{+}}}x)(t)=\frac{1}{\sqrt{2\pi}} \int\limits_{\mathbb{R^{+}}}x(\xi)e^{it\xi}\,d\xi\,,\ \ \ t\in{}{\mathbb{R^{+}}}, $$ is studied. The operator…
In this article we investigate the Fourier series and transforms for the functions defined on the [-pi, pi]^ d or on the R^d and belonging to the (Bilateral) Grand Lebesgue Spaces. As a particular case we obtain some results about Fourier's…
In this work, we define a new class of fractional analytic functions containing functional parameters in the open unit disk. By employing this class, we introduce two types of fractional operators, differential and integral. The fractional…
We show an exact (i.e. no smooth error terms) Fourier inversion type formula for differential operators over Riemannian manifolds. This provides a coordinate free approach for the theory of pseudo-differential operators.
This paper introduces a factorization for the inverse of discrete Fourier integral operators that can be applied in quasi-linear time. The factorization starts by approximating the operator with the butterfly factorization. Next, a…
In this paper we consider logarithmic operators in two different contexts: the adapted to (continuous) Schr\"odinger operators and the classical discrete setting. The Schr\"odinger operator $\mathcal L_V$ on $\mathbb R^d$ is defined as…
We analyze the parabolic Dirac operator $D \pm i\partial_t$ in a biquaternionic setting, characterizing its kernel via generalized div-curl systems and Cauchy-Riemann-type relations between the real and imaginary parts. Using the machinery…
The quaternion Fourier transform (QFT), a generalization of the classical 2D Fourier transform, plays an increasingly active role in particular signal and colour image processing. There tends to be an inordinate degree of interest placed on…
The transformation property of the Caputo fractional derivative operator of a scalar function under rotation in two dimensional space is derived. The study of the transformation property is essential for the formulation of fractional…
We consider the image of the operator inducing the determinantal point process with the confluent hypergeometric kernel. The space is described as the image of $L_2[0, 1]$ under a unitary transform, which generalizes the Fourier transform.…
Polynomials in differentiation operators are considered. The Darboux transformations covariance determines non-Abelian entries to form the coefficients of the polynomials. Joint covariance of a pair of such polynomials (Lax pair) as a…
The notion of invariant operators, or Fourier multipliers, is discussed for densely defined operators on Hilbert spaces, with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. As a consequence,…
In an earlier paper (A. N. Kochubei, {\it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirov's fractional differentiation operator $D^\alpha$, $\alpha >0$, to radial functions on a non-Archimedean…
Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space $D=d^n$ with $d$ an odd integer, and is inspired by the Cooley-Tukey formalism. The `large Fourier…
Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements $D$ of the associated Lie algebras as linear differential operators gives relations among the functions in a…
The Fourier transform operation is an important conceptual as well as computational tool in the arsenal of every practitioner of physical and mathematical sciences. We discuss some of its applications in optical science and engineering,…
Derivations play a fundamental role in the definition of vertex (operator) algebras, sometimes regarded as a generalization of differential commutative algebras. This paper studies the role played by the integral counterpart of the…