Majorants of functions with vanishing integrals over balls

We prove the existence of nontrivial functions in ℝⁿ, n ≥ 2, with vanishing integrals over balls of fixed radius and given majorant of growth. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 857–861, June, 2008.     

On some integrals over the product of three Legendre functions

The definite integrals \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2}[P_{\nu}(x)]^{3}\, \mathrm{d}x\) , \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2} [P_{\nu}(x)]^{2}P_{\nu}(-x)\, \mathrm{d}x\) , \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2}[P_{\nu+1}(x)]^{3}\,\mathrm{d}x\) , and \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2} [P_{\nu+1}(x)]^{2}P_{\nu +1}(-x)\,\mathrm{d}x \) are evaluated in closed form, where P _ν is the Legendre function of degree ν, and \(\operatorname{Re}\nu>-1\) . Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.     

A note on the numerical evaluation of integrals over strongly oscillating functions

In the present note we consider the numerical evaluation of integrals over strongly oscillating functions of the type

$\text{∫}\text{0}\text{∞}\text{f}\text{(}\text{x}\text{)?(}\text{x}\text{)}\text{dx}$

where f(x) is periodic function and where the asymptotic behaviour of ? (x) for large values of x is given by $\text{?(}\text{x}\text{) ～}\text{c}\text{x}{}^{\mathrm{\alpha }}\text{+0(}\text{1}\text{x}{}^1\text{+α}\text{)}$ with α ? 1. We show how the integral (1) can be transformed into a sum of two integrals, one of them being of the same type but with an asymptotic expansion for ? (x) given by $\text{?(}\text{x}\text{) ～}\text{c}\text{x}{}^1\text{+α}\text{+0(}\text{1}\text{x}{}^2\text{+α}\text{)}$, the other one being a proper integral.     

Positively homogeneous functions in atmospheric radiative transfer theory

Positively homogeneous functions are applied to describe absorption and scattering processes within the framework of atmospheric radiative transfer theory. Solid angle integrations are understood as surface integrations over arbitrary solid angle surfaces which are defined beyond the common picture of atmospheric radiative transfer.     

Decomposability of continuous functions from Nikol'skii classes into multiple fourier integrals

Translated from Matematicheskie Zametki, Vol. 47, No. 2, pp. 3–7, February, 1990.     

Approximation of functions with zero integrals over balls by linear combinations of the Laplace operator eigenfunctions

It is proved that the special linear combinations of Bessel functions are dense in the C ^∞-topology in the space of functions with zero integrals over balls of fixed radii on an arbitrary open domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} . Some generalizations of this result for solutions of some convolution equations of the form f * T = 0, where T is radial, are obtained. Analogous results for rank-one symmetric spaces of the noncompact type are considered.     

Operator calculations in the theory of path integrals

A method is developed in the theory of functional integration that makes it possible to use the group-theoretical formalism to calculate Feynman integrals. The method is used to consider some problems that have important applications in many branches of quantum mechanics. It particular, the formalism makes it possible to solve problems with quadratic Lagrangians with arbitrary dependence of the parameters on the time.Belarus State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 2, pp. 248–263. August, 1994.     

On the characterization of continuum structure functions

A continuum structure function is a non-decreasing mapping from the unit hypercube to the unit interval. Within the class of continuum structure functions, new axiomatic characterizations of the Natvig and the Barlow-Wu subclass are obtained.     

Product measures and convergence theorems for product integrals in integration theory for semigroup valued functions

In [11] Sion has investigated measures and integrals having values in uniform semigroups. In this paper a result which gives sufficient conditions for the existence of a unique product of semigroup valued measures is established. Moreover a convergence theorem for product integrals of semigroup valued functions is proved.     

Product measures and convergence theorems for product integrals in integration theory for semigroup valued functions

In [11] Sion has investigated measures and integrals having values in uniform semigroups. In this paper a result which gives sufficient conditions for the existence of a unique product of semigroup valued measures is established. Moreover a convergence theorem for product integrals of semigroup valued functions is proved.