Discrete Kontorovich–Lebedev transforms

The Ramanujan Journal - Discrete analogs of the classical Kontorovich–Lebedev transforms are introduced and investigated. It involves series with the modified Bessel function or Macdonald...     

On the Fourier-Sine and Kontorovich–Lebedev Generalized Convolution Transforms and Their Applications

Ukrainian Mathematical Journal - We study generalized convolutions for the Fourier sine and Kontorovich–Lebedev transforms $$ left(hunderset{F_s,K}{ast }fright)(x) $$ in a two-parameter...     

New index transforms of the Lebedev–Skalskaya type

New index transforms, involving the real part of the modified Bessel function of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution of the initial value problem for the second-order partial differential equation, involving the Laplacian, is obtained. It is noted that the corresponding operators with the imaginary part of the modified Bessel function of the first kind lead to the familiar Kontorovich–Lebedev transform and its inverse.     

A Convolution Operator Related to the Generalized Mehler–Fock and Kontorovich–Lebedev Transforms

In this paper we study a generalization of an index integral involving the product of modified Bessel functions and associated Legendre functions. It is applied to a convolution construction associated with this integral, which is related to the classical Kontorovich–Lebedev and generalized Mehler–Fock transforms. Mapping properties and norm estimates in weighted L _p -spaces, 1 ≤ p ≤ 2, are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L ₂.     

Poincaré like inequalities for semigroups, cosine and sine operator functions

Here we present Poincaré type general L _p inequalities regarding semigroups, cosine and sine operator functions.     

On the dominated convergence theorem for the Kurzweil–Stieltjes integral

This paper is concerned with a version of the Lebesgue dominated convergence theorem (DCT) which has been stated for the Kurzweil–Stieltjes integral of real functions. Our objective in this work is to analyze the extension of this result to include vector functions with values in Banach spaces. We establish that the mentioned convergence theorem for the Kurzweil–Stieltjes integral can be formulated in weaker versions for reflexive and separable Banach spaces, and spaces having the Schur property, nonetheless it is not verified in the general case.     

Stability and Fourier–Mukai transforms on elliptic fibrations

We systematically develop Bridgeland's [7] and Bridgeland–Maciocia's [10] techniques for studying elliptic fibrations, and identify criteria that ensure 2-term complexes are mapped to torsion-free sheaves under a Fourier–Mukai transform. As an application, we construct an open immersion from a moduli of stable complexes to a moduli of Gieseker stable sheaves on elliptic threefolds. As another application, we give various 1–1 correspondences between fibrewise semistable torsion-free sheaves and codimension-1 sheaves on Weierstrass surfaces.     

On the Ostrowski’s inequality for Riemann-Stieltjes integral and applications

An Ostrowski type integral inequality for the Riemann-Stieltjes integral ∫_a^b ƒ (t) du (t), where ƒ is assumed to be of bounded variation on [a,b] andu is ofr-H- H?lder type on the same interval, is given. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.