An inversion formula for some convolution transforms

Inversion and convergence results are achieved for some convolution transforms whose kernel’s bilateral Laplace transform is [Π _k=1 ^∞ (1−(s ² a _k ² ))]⁻¹ wherea _k are complex.     

Integrability theorems for a class of integral transforms

It is shown under weak hypotheses that systems of 2n linear differential equations in 2n variables generate sets of identities similar in structure to the classical trigonometric identities. For clarity of exposition only the case n = 1 is actually treated, but all final equations are written in such a manner as to be directly applicable to matrix systems (n > 1). These identities allow one to avoid, in a very simple way, certain difficulties which often occur in the integration of the Riccati equations arising from application of the invariant imbedding method to two point boundary value problems associated with such linear systems. The overall usefulness of the imbedding method is thereby considerably extended. One analytical and one numerical example are given to illustrate the actual use of these identities.     

Abelian and Tauberian theorems for a class of integral transforms

Extending the Wiener-Ganelius method we give Abelian and precise Tauberian remainder results for a class of Fourier kernels which includes the Hankel transform $\text{?(x) = ∫}{}_0{}^{\mathrm{\infty }}\text{√xu J}{}_{\mathrm{r}}\text{(xu) ?(u) du, v ? ?}\text{1}\text{2}$. Further, we discuss applications to Fourier series and integrals.     

An inversion theorem for integral transforms related to singular Sturm-Liouville problems on the half line

We extend the classical theory of singular Sturm-Liouville boundary value problems on the half line, as developed by Titshmarsh and Levitan to generalized functions in order to obtain a general approach to handle many integral transforms, such as the sine, cosine, Weber, Hankel, and the K-transforms, in a unified way. This approach will lead to an inversion formula that holds in the sense of generalized functions. More precisely, for [0,) and 0<, let (x,) be a solution of the Sturm-Liouville equation

A class of integral transforms which are projections

We consider all convolution tranforms onL ²() which are projections, and determine their ranges and null-spaces; it turns out that all are orthogonal projections. By modification of the kernel, integral transforms are defined which are oblique projections, and their angle of inclination is approximated using finite dimensional spaces. Several families of such projections are treated and results for the angle of inclination as function of the parameters are displayed. In some cases an exact formula has been obtained.     

On extremal problems related to integral transforms of a class of analytic functions

For γ?0 and β<1 given, let Pγ(β) denote the class of all analytic functions f in the unit disk with the normalization f(0)=f^′(0)−1=0 and satisfying the condition

     

Real inversion and jump formula for the laplace transform,part II

A generalization of the Hirschman inversion formula and a new jump formula for the Laplace transform are proved. This paper is the second part of a paper with the same name (see this Journal v., 1, 1963, pp. 85–104). The notations used here are the same as in the first part. §1–§4 belong to the first part. This paper is to be a part of the first author’s Ph. D. thesis written under the direction of the second author at the Hebrew University, Jerusalem. The participation of the second author in this paper has been sponsored in part by the Air Force Office of Scientific Research, OAR, through the European Office of Aerospace Research, United States Air Force.     

An inversion formula for the attenuated X-ray transformation

The problem of inversion of the attenuated X-ray transformation is solved by an explicit formula. Several subsequent results are also given.     

A trace formula for Weyl transforms

A formula for the trace of a trace class Weyl transform associated to a symbol in L¹(R²ⁿ) is given.