Integral transforms play an important rôle in Analytic number theory, the part of Number theory where problems of a number-theoretic nature are solved by the use of various methods from Analysis. The most common integral transforms that are used are: Mellin transforms (Robert Hjalmar Mellin, 18541933), Laplace transforms (Pierre-Simon, marquis de Laplace, 1749-1827) and Fourier transforms (Jose...

This thesis contains three articles. The first two concern inversion and local injectivity of the weighted Radon transform in the plane. The third paper concerns two of the key results from compressive sensing. In Paper A we prove an identity involving three singular double integrals. This is then used to prove an inversion formula for the weighted Radon transform, allowing all weight functions...

this article is devoted to study of the autoconvolution equations and generalized mittag-leffler functions. these types of equations are given in terms of the laplace transform convolution of a function with itself. we state new classes of the autoconvolution equations of the first kind and show that the generalized mittag-leffler functions are solutions of these types of equations. in view of ...

To FrançoisRouvì ere on his 60 th birthday. Abstract The X-ray transform on a compact symmetric space M is here inverted by means of an explicit inversion formula. The proof uses the conjugacy of the minimal closed geodesics in M and of the maximally curved totally geodesic spheres in M , proved in Math.

The finite symmetric trilinear integral transform is extended to distributions by using quite different technique than Zemanian (1968) and Dube (1976) and an inversion formula is established using Parseval's identity. The operational calculus generated is applied to find the temperature inside an equilateral prism of semi-infinite length.

In this paper we explain with simple geometric arguments the appearance of diffraction hyperbolas in RADAR images. Moreover, we outline an inversion formula from integral geometry, which is derived using only elementary properties of the Fourier transform, to reconstruct the refractive index of a medium from RADAR data. The theoretical results of this paper are supported by real data from an in...

The fundamental solution (Green function) for the Cauchy problem of the space-time fractional diffusion equation is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. Then, by using the Mellin transform, a general representation of the Green function in terms of Mellin-Barnes integrals in the complex plane is derived. This allow...

The optical Mellin transform has recently been success­ fully realized and used to construct useful scale invariant optical pattern recognition systems, in Doppler signal pro­ cessing, and in the correlation of nonvertical imagery, among other applications. The Mellin transform of f(x) can be synthesized optically by applying a coordinate transformation ξ = lnx to the input coordinate x and an ...

Gaussian measure is constructed for any given hyperplane in an infinite dimensional Hilbert space, and this is used to define a generalization of the Radon transform to the infinite dimensional setting, using Gauss measure instead of Lebesgue measure. An inversion formula is obtained and a support theorem proved.