Obtain Inversion Formulas Research Articles

\u03b4-\u03b2-Gabor integral operators for a space of locally integrable generalized functions

In this article, we give a definition and discuss several properties of the δ-β-Gabor integral operator in a class of locally integrable Boehmians. We derive delta sequences, convolution products and establish a convolution theorem for the given δ-β-integral. By treating the delta sequences, we derive the necessary axioms to elevate the δ-β-Gabor integrable spaces of Boehmians. The said generalized δ-β-Gabor integral is, therefore, considered as a one-to-one and onto mapping continuous with respect to the usual convergence of the demonstrated spaces. In addition to certain obtained inversion formula, some consistency results are also given.

On two families of Funk-type transforms

We consider two families of Funk-type transforms that assign to a function on the unit sphere the integrals of that function over spherical sections by planes of fixed dimension. Transforms of the first kind are generated by planes passing through a fixed center outside the sphere. Similar transforms with interior center and with center on the sphere itself we studied in previous publications. Transforms of the second kind, or the parallel slice transforms, correspond to planes that are parallel to a fixed direction. We show that the Funk-type transforms with exterior center express through the parallel slice transforms and the latter are intimately related to the Radon–John d-plane transforms on the Euclidean ball. These results allow us to investigate injectivity of our transforms and obtain inversion formulas for them. We also establish connection between the Funk-type transforms of different dimensions with arbitrary center.

A Lorentzian inversion formula for defect CFT

We present a Lorentzian inversion formula valid for any defect CFT that extracts the bulk channel CFT data as an analytic function of the spin variable. This result complements the already obtained inversion formula for the corresponding defect channel, and makes it now possible to implement the analytic bootstrap program for defect CFT, by going back and forth between bulk and defect expansions. A crucial role in our derivation is played by the Calogero-Sutherland description of defect blocks which we review. As first applications we obtain the large-spin limit of bulk CFT data necessary to reproduce the defect identity, and also calculate one-point functions of the twist defect of the 3d Ising model to first order in the ϵ-expansion.

Bootstrapping the (A1, A2) Argyres-Douglas theory

We apply bootstrap techniques in order to constrain the CFT data of the (A1, A2) Argyres-Douglas theory, which is arguably the simplest of the Argyres-Douglas models. We study the four-point function of its single Coulomb branch chiral ring generator and put numerical bounds on the low-lying spectrum of the theory. Of particular interest is an infinite family of semi-short multiplets labeled by the spin ℓ. Although the conformal dimensions of these multiplets are protected, their three-point functions are not. Using the numerical bootstrap we impose rigorous upper and lower bounds on their values for spins up to ℓ = 20. Through a recently obtained inversion formula, we also estimate them for sufficiently large ℓ, and the comparison of both approaches shows consistent results. We also give a rigorous numerical range for the OPE coefficient of the next operator in the chiral ring, and estimates for the dimension of the first R-symmetry neutral non-protected multiplet for small spin.

The rate of convergence of truncated hypersingular integrals generated by the Poisson and metaharmonic semigroups

In harmonic analysis, an important problem is to obtain inversion formulas for the potential-type integral operators. The studies on this subject have been developed by the use of hypersingular integral technique. In this paper the families of truncated hypersingular integrals generated by the Poisson and metaharmonic semigroups and dependent on a parameter ϵ, are introduced. Then the connection between the order of smoothness of a given Lp-function ϕ and the rate of convergence of these families of truncated hypersingular integrals, which converge to ϕ when ϵ tends to 0, is obtained.

The Radon transform on the Heisenberg group and the transversal Radon transform

The Radon transform on the Heisenberg group was introduced by R. Strichartz. We regard it as a particular case of a more general transversal Radon transform that integrates functions on R m over hyperplanes meeting the last coordinate axis. The paper contains new boundedness results and explicit inversion formulas for both transforms of L p functions in the full range of the parameter p. We also show that these transforms are isomorphisms of the corresponding Semyanistyi–Lizorkin spaces of smooth functions. In the framework of these spaces we obtain inversion formulas, which are pointwise analogues of the corresponding formulas by R. Strichartz.

Infinite triangular matrices, q-Pascal matrices, and determinantal representations

We use basic properties of infinite lower triangular matrices and the connections of Toeplitz matrices with generating-functions to obtain inversion formulas for several types of q-Pascal matrices, determinantal representations for polynomial sequences, and identities involving the q-Gaussian coefficients. We also obtain a fast inversion algorithm for general infinite lower triangular matrices.

Solution of certain weakly singular integral equations of the first kind with indefinite parameters

In this paper we study a two-dimensional weakly singular integral equation of the first kind with logarithmic kernel. We construct a pair of spaces of the desired elements and the right-hand sides, where we prove the correctness of the problem under consideration and obtain inversion formulas for the integral operator.

Inversion of the Kipriyanov–Radon transform via fractional derivatives in a one-dimensional parameter

This paper considers the Kipriyanov–Radon transform constructed as a special Radon transform adopted for dealing with singular Bessel differential operators of the corresponding indices acting on a part of the variables. The authors obtain inversion formulas generalizing the classical formulas for the Radon transform of axially-symmetric functions and relating to the integro-differentiation of fractional order in a one-dimensional parameter.

Singular value decomposition for the 2D fan-beam Radon transform of tensor fields

In this article we study the fan-beam Radon transform Dm of symmet- rical solenoidal 2D tensor fields of arbitrary rank m in a unit disc D as the operator, acting from the object space L2(D ; Sm) to the data space L2((0, 2π) × (0, 2π)). The orthogonal polynomial basis s (±m) n,k of solenoidal tensor fields on the disc D was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator Dm was obtained. The inversion formula for the fan-beam tensor transform Dm follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.

Singular value decomposition for the 2D fan-beam Radon transform of tensor fields

In this article we study the fan-beam Radon transform of symmetrical solenoidal 2D tensor fields of arbitrary rank m in a unit disc as the operator, acting from the object space L 2 (; S m ) to the data space L 2 ([0, 2π) × [0, 2π)). The orthogonal polynomial basis of solenoidal tensor fields on the disc was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator was obtained. The inversion formula for the fan-beam tensor transform follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.

The Segal-Bargmann "Coherent State" Transform for Compact Lie Groups

Let K be an arbitrary compact, connected Lie group. We describe on K an analog of the Segal-Bargmann "coherent state" transform, and we prove (Theorem 1) that this generalized coherent state transform maps L2(K) isometrically onto the space of holomorphic functions in L2(G, μ), where G is the complexification of K and where μ is an appropriate heat kernel measure on G. The generalized coherent state transform is defined in terms of the heat kernel on the compact group K, and its analytic continuation to the complex group G. We also define a "K-averaged" version of the coherent state transform, and we prove a similar result for it (Theorem 2). In the Appendix we describe the "coherent states" which motivate the definitions of these transforms. In Section 9, we obtain inversion formulas for both the coherent state transform and the K-averaged coherent state transform (Theorems 3, 4). As a consequence, we obtain formulas for certain Bergman kernels on the complex group G (Theorems 5, 6). In Section 10, we derive a general "Paley-Wiener" theorem for K, and we exhibit a family of convolution transforms, each of which is an isometric isomorphism of L2(K) onto a certain L2-space of holomorphic functions on G. These transforms include the K-averaged coherent state transform as a special case. We also discuss the analogous results for R1. (The results of Section 10 are Theorems 7-10.) Finally, in Section 11, we discuss the case of quotient spaces. We prove (Theorem 11) that both the coherent state transform and the K-averaged coherent state transform pass in a natural way from K to K/H, where H is a closed, connected subgroup of K.

Dual operators and Lagrange inversion in several variables

The original Lagrange inversion formula, which gives explicitly the inverse under composition of a formal series, was obtained by Lagrange in 1770 through formal computations involving logarithms of inlmite products. See Lagrange [ 121. There exists an extensive literature on various versions of the inversion formula obtained by algebraic, analytic, and combinatorial methods. Comtet [3] gives a survey and an ample bibliography. During the last ten years there have appeared several papers on generalizations and new proofs of the inversion formula. In 1974 Abhyankar obtained an inversion formula using algebraic methods. See Bass, Connell, and Wright [ 11. Several authors have related Lagrange inversion with Rota’s Finite Operator Calculus [ 151, for example, Joni [8], Garsia and Joni [4], Hofhauer [7], and Roman and Rota [14]. Joyal and Labelle used a combinatorial theory of formal series to obtain inversion formulas (see [9-l I] ) and Viskov [ 161 used ideas from Lie algebra theory to prove formulas similar to Abhyankar’s. In the present paper we use operator methods to obtain inversion formulas in several variables that generalize the formulas of Abhyankar, Joni, and Viskov. We get our results studying an algebra with involution of linear operators on the algebra of formal Laurent series in several indeterminates. We also show that by means of an anti-isomorphism of operator algebras, which we call the operator Bore1 transform, the study of finite operator calculus can be reduced to the study of certain groups of operators on the algebra of formal Laurent series. Our approach may be considered as an algebraic analogue of the complex variable methods used by Good [5] to obtain the Lagrange inversion formula as a consequence of the change of variables theorem for the