Asymptotic Representation of Orthogonal Polynomials

Golinskii, B. L.

doi:10.1070/rm1980v035n02abeh001637

Cited by 6 publications

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“…In the papers of Horup [182] and Golinskii [57,62,63] one gives, under certain conditions on the weight p(0), the asymptotic formula…”

Section: I-vd--a~ (Oo) '-J Bn (Oo) ?=mentioning

confidence: 99%

V. A. Steklov's problem in the theory of orthogonal polynomials

Suetin¹

1979

J Math Sci

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One considers the asymptotic properties of orthogonal polynomials of various types depending onthe properties and the singularities of the weight function and of the orthogonality line. One gives conditions for the boundedness of the orthogonal polynomials on some set or on the entire orthogonality line, asymptotic formulas for them, and various growth estimates in the case of singularities of tim weight function and of the contour. One presents some methods for the investigation of the asymptotic properties of the orthogonal polynomials. IntroductionIf on a finite interval [a, b] there is given a bounded nondecreasing function >{x} with an infinite set of points of increase, then this function uniquely determines a system of polynomials {On(X)}, orthonormal on the interval [a, b] with the integral weight p(x), i.e., satisfying the condition b l P,, (x) P,,, (x) dr, (x) = ~,,,~, ~ a~P~ (x).(1) n~O For the classical orthogonal polynomials there exist asymptotic formulas inside the orthogonality interval, with the aid of which one proves theorems on the equieonvergence of series (1) with the Fourier series with respect to the trigonometric system for some function associated with the function f(x). If the weight function h{x) is arbitrary, then in the sufficient conditions for the convergence of series (1) to the function f(x) in some isolated point x 0 E (a, b) or in some set Ac (a, b) there usually occurs the boundedness condition of the system {Pn(x)} of polynomials at this point x 0 or on the entire set A, i.e., the validity of the inequalitiesUnder these conditions series (1) possess on the set A some properties of the Fourier trigonometric series.It is clear that condition (2) depends on the properties of the weight function h(x) on the set A, and, investigating this problem, Steklov [219] formulated in 1921 the assumption that in order that inequalities (2) should hold, it is necessary and sufficient that the weight function h(x) be positive on the set A, i.e., we should have h(x)>O, xO. Ac:_ (a, b).Steklov wrote: "Unfortunately, we do not have a method to express the totality of general sufficient conditions which have to be satisfied by the weight h(x) in order that inequality (2) should hold for all polynomials Translated from Itogi Nauki i Tekhniki, Matematieheskii Analiz, Vol. 15, pp. 5-82, 1977. 0096-4104/79/1206-0631 $07.50 9Plenum Publishing Corporation 631 corresponding to the function h(x) and satisfying the indicated conditions. I think that this inequality is a common property of all orthogonal polynomials whose weight does not vanish inside the given interval; however, presently we have not succeeded to find a rigorous proof of this assertion or to discover an example where this inequality does not hold at each point inside this interval."All the classical orthogonal polynomials are uniformly bounded inside the orthogonality interval. In addition, conditions (2) and (3) are satisfied also for more general systems of polynomials and in particular for those systems of orthogonal polynomials...

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“…In the papers of Horup [182] and Golinskii [57,62,63] one gives, under certain conditions on the weight p(0), the asymptotic formula…”

Section: I-vd--a~ (Oo) '-J Bn (Oo) ?=mentioning

confidence: 99%

V. A. Steklov's problem in the theory of orthogonal polynomials

Suetin¹

1979

J Math Sci

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show abstract

“…The well-known Szegr" theory is the framework of some contributions, such as [8] and [141. The well-known Szegr" theory is the framework of some contributions, such as [8] and [141.…”

Section: Introductionmentioning

confidence: 99%