Spectral asymptotic behavior of a class of integral operators

Integral operators of the type $$(Tf)(x) = \int_0^1 {\frac{{x^\beta y^\gamma }}{{(x + y)^\alpha }}} f(y)dy,$$ the kernels of which have a singularity at a single point, are discussed. H. Widom's method and some of his results are used to show that, if α>0, β, γ>?1/2, ρ=β+γ?α+1>0, then we have for the distribution function of the singular numbers of the operator, $$\mathop {\lim }\limits_{\varepsilon \to 0} N(\varepsilon ,T)ln^{ - 2} {\textstyle{1 \over \varepsilon }} = {\textstyle{1 \over {2\pi ^2 \varepsilon }}}.$$      

An inversion formula for a class of integral transforms,II

In this note we give a procedure for inverting the integral transform f(x) = ∫₀^∞k(xt) φ(t) dt, where the functions f(x) and k(x) are known and φ(x) is to be found. The inversion is accomplished in two steps: by first defining a transforming function, which is an integral, followed by the application of an infinite order differential operator.     

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

     

Exact asymptotic behavior of singular values of a class of integral operators

We find an exact asymptotic formula for the singular values of the integral operator of the form , a Jordan measurable set) where and L is slowly varying function with some additional properties. The formula is an explicit expression in terms of L and T.     

Generating functions via integral transforms

In this paper, we use some integral transforms to derive, for a polynomial sequence {Pn(x)}_n?0, generating functions of the type , starting from a generating function of type , where {γn}_n?0 is a real numbers sequence independent on x and t. That allows us to unify the treatment of a generating function problem for many well-known polynomial sequences in the literature.     

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.     

Strictly positive definite functions by integral transforms

We study the continuity and strict positive definiteness of positive definite functions on quasi-metric spaces given by integral transforms. We apply some of our findings to positive definite functions on the Euclidean space $R^m$ which are given by cosine transforms ($m=1$) and Fourier–Bessel transforms ($m>1$). We also apply the results to positive definite functions on a general quasi-metric space realized as extensions of certain real Laplace transforms defined by conditionally negative definite functions on the quasi-metric space itself.     

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.     

Large deviations and asymptotic efficiency of a statistic of integral type. II

Part I has been published in the collection Studies in the Theory of Probability Distributions. IV (Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., Vol. 85), Leningrad, 1979, pp. 175–187. With the aid of the methods of the branching theory of nonlinear equations, one finds a coarse asymptotics of the probabilities of large deviations for integral statistics of the form, which are generalizations of the Cramér-von Mises-Smirnov statistic, and also for the twosample variants of these statistics. The obtained results allow us to compute the local exact Bahadur relative asymptotic efficiency. One establishes that the latter coincides with both the Bahadur approximate and the Pitman efficiencies.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 151–175, 1980.In conclusion the author expresses his gratitude to P. Groeneboom for sending him a preprint of [11] and to H. S. Wieand for the possibility of getting acquainted with [30].     

Comparison theorems and asymptotic behavior of correlation estimators in spaces of continuous functions. II

This paper is the second part of [12]. Using the comparison theorems which were proved in the first part, the asymptotic normality of the estimator — in a model of a series of several samples — of the correlation function of a stationary Gaussian random process in spaces of continuous functions with weights is established. A method for constructing functional confidence intervals for an unknown correlation function in these spaces is described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 579–583, May, 1991.