Summability of Laguerre expansions

В РАБОтЕ пОлУЧЕНы УсИ лЕНИь РЕжУльтАтОВ МА РкЕттА О сУММИРУЕМОстИ МОДИФ ИцИРОВАННых РАжлОжЕНИИ лАгЕРРА. У стАНОВлЕНО, ЧтОα=1/6 Ест ь кРИтИЧЕскИИ ИНДЕкс Д ль сУММИРУЕМОстИ пО ЧЕжАРО. ДОкАжАНО, Чт О пРИα=1/6 сРЕДНИЕ ЧЕжАР О схОДьтсь пОЧтИ ВсУДУ. пОлУЧЕН тАкжЕ АНАлОг тЕОРЕМы ФЕИЕРА-лЕБЕгА.     

Divergent Laguerre series

We prove failure of a.e. convergence of partial sums of Laguerre expansions of functions for 4$">. The idea which is used goes back to Stanton and Tomas. We follow Meaney's paper (1983), where divergence results were proved in the Jacobi polynomial case.

     

Summability of orthogonal series with speed

Summary Some sufficient conditions are found for summability of orthogonal series with speed.     

Summability of Lagrange type interpolation series

We consider certain aspects of the theory of interpolation via entire functions of exponential type, which include sampling node sequences χ which can be highly irregular and data sequences {y(_x)}_xn∈χ which are not necessarily bounded. Under appropriate conditions, we show that there is an entire function of a suitable exponential type which uniquely interpolates the data and indicate the validity of certain summabilty methods for the corresponding Lagrange type interpolation series. Some of our results significantly extend the work of Schoenberg,Cardinal interpolation and spline functions VII: The behavior of cardinal spline interpolation as their degree tends to infinity, J. Analyse Math.27 (1974), 205–229.     

Summability of Fourier series with the method of lacunary arithmetical means at the Lebesgue points

The existence of the `rare' sequence of partial sums summable with the method of arithmetical means at each Lebesgue point is proved in the paper. The proof is based on the strategy of random choice.

     

Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in

We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions

on the standard simplex in . It is proved that such an expansion is uniformly summable on the simplex for any continuous function if and only if . Moreover, it is shown that means define a positive linear polynomial identity, and the index is sharp in the sense that means are not positive for .

     

Summability of orthogonal series by some linear methods

Classes of matrix methods are given which sum the orthogonal series satisfying classical coefficient conditions of Men'shov-Kaczmarz, of Zygmund and of Alexits. The literature contains seven titles.Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 753–762, June, 1969.In conclusion the author expresses his thanks to the referee whose valuable remarks have helped to produce a better work.     

Summability of Fourier orthogonal series for Jacobi weight on a ball in

Fourier orthogonal series with respect to the weight function
on the unit ball in are studied. Compact formulae for the sum of the product of orthonormal polynomials in several variables and for the reproducing kernel are derived and used to study the summability of the Fourier orthogonal series. The main result states that the expansion of a continuous function in the Fourier orthogonal series with respect to is uniformly summable on the ball if and only if .

     

Inversion of the multi-dimensional Laplace transform-expansion by Laguerre series

In this paper an algorithm to numerically invert two-dimensional Laplace transform known in closed form as an analytic function is presented. The method is based on expanding the inverse function in a series of products of (generalized) Laguerre polynomials. It is based on the method by Weeks (1966) and the generalized version presented by Piessens and Branders (1971) for the one-dimensional case.     

Summability of multiple trigonometric Fourier series by linear methods

The paper deals with the problem of estimation of deviations of functions of several variables from linear means of their multiple trigonometric Fourier series. An approach of reducing this problem to the corresponding problem for functions of single variable is developed.     

Laguerre series direct method for variational problems

A direct method for solving variational problems via Laguerre series is presented. First, an operational matrix for the integration of Laguerre polynomials is introduced. The variational problems are reduced to the solution of algebraic equations. An illustrative example is given.