On an integral transform with generalized associated Legendre functions

We study some new properties of generalized associated Legendre functions of first and second kind P _k ^m,n (z) and Q _k ^m,n (z). Applying these functions, we introduce an integral transform that can be used in solving boundary-value problems of mathematical physics.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 33–43.     

On a generalized Hammerstein-type integral equation

Theorems on the existence, uniqueness and continuous dependence of the solution of a generalized Hammerstein-type integral equation are given. The well-known Banach fixed-point theorem is employed to establish the results.     

On a generalized Hammerstein-type integral equation

Theorems on the existence, uniqueness and continuous dependence of the solution of a generalized Hammerstein-type integral equation are given. The well-known Banach fixed-point theorem is employed to establish the results.     

The obrechkoff integral transform: properties and relation to a generalized fractional calculus

This survey is devoted to one of the most general Laplace-type integral transforms, the so-called Obrechkoff integral transform, introduced and studied for the first time by Obrechkoff[25]. It has been modified by Dimovski [5],[6] and used as a basis of a Mikusinski-type operational calculus for the hyper-Bessel differential operators of arbitrary order. Later, in a series of papers Dimovski and Kiryakova [8],[9],[10] have found operational properties, complex and real inversion formulas, Abel-type theorems for the Obrechkoff transform. This theory has been further developed by Kiryakova [16],[17],[18] using the tools of the Meijer's G-functions and of the fractional calculus. Namely, a new definition as a G-transform has been given for the Obrechkoff transform. The hyper-Bessel operators themselves, have given rise to a new generalized fractional calculus and further extensive use of the G-functions. Many other generalized differentiation and integration operators happen to be special cases in this calculus, too. Special cases of the Obrechkoff transform have been "rediscovered" later by many authors. We give examples how their results could be derived from the general ones surveyed here.     

关于广义Hardy-Hilbert积分不等式及其应用

通过引入参数λ(1-q/p＜λ≤2,p≥q＞1)及两个非负且在(0, ∞)递增的可微函数u(x)和v(x)建立了一种广义带权的Hardy-Hilbert积分不等式.特别,当p=2时,得到经典Hilbert积分不等式的各种推广.作为应用,当u(x)和v(x)是幂函数、指数函数和对数函数时,建立了若干重要不等式.     

An underdetermined problem of integral geometry for the generalized radon transform

Under study is a new problem of integral geometry. All kinds of planes are considered in the three-dimensional Euclidean space. The available data are given by the integrals over all these planes of an unknown piecewise-smooth function depending both on the spatial variables and the variables characterizing the planes. The sought object is a first kind discontinuity surface of the integrand. The uniqueness theorem of a desired surface is proved. The above results are related to one of the aspects of the theory of testing unknown media by various physical signals.     

Inversion of the Weyl integral transform and the radon transform on Rn using generalized wavelets

We invert the Weyl integral transform by means of a generalized continuous wavelet transform on the half line associated with the Bessel operatorL , >–1/2. Next, we use the connection between radial classical wavelets onR ⁿ and generalized wavelets associated with the Bessel operatorL( _n–2)/2 to derive new inversion formulas for the Radon transform onR ⁿ ,n2.     

On Fourier transform of generalized Brownian functionals

Let $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ and $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ be the spaces of generalized Brownian functionals of the white noises ? and ?, respectively. A Fourier transform from $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ into $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ is defined by ??(?) = ∫$\text{S}$¹: exp[?i ∫_$\text{R}$?(t) ?(t) dt]: ${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}\text{?(}\text{B}\text{?}\text{) dμ(}\text{B}\text{?}$), where : :${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}$ denotes the renormalization with respect to ? and μ is the standard Gaussian measure on the space $\text{S}$¹ of tempered distributions. It is proved that the Fourier transform carries ?(t)-differentiation into multiplication by i?(t). The integral representation and the action of?? as a generalized Brownian functional are obtained. Some examples of Fourier transform are given.     

On the use of integral transform methods: a new perspective

Integral transforms are usually used to ‘solve’ renewal‐type equations. These transforms typically require inversion to be of use to a decision maker. In this paper it is demonstrated that in certain situations, stochastic quantities of interest are expressible in terms of integral transforms, thus avoiding the problems associated with inversion.     

On generalized inverses of matrices over integral domains

It is proved that a matrix A over an integral domain admits a 1-inverse if and only if a linear combination of all the r × r minors of A is equal to one, where r is the rank of A. Some results on the existence of Moore-Penrose inverses are also obtained.