Global Smoothness of Approximating Functions

Anastassiou, George A.; Cottin, Claudia; Gonska, Heiner

doi:10.1524/anly.1991.11.1.43

Cited by 41 publications

(27 citation statements)

References 8 publications

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“…For instance, Anastassiou, Cottin, and Gonska [5] provide the best absolute constant in the case of Bernstein polynomials on the standard m-simplex, while for the classical Sza sz operator, we refer to [4].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Best Constants in Preservation Inequalities Concerning the First Modulus and Lipschitz Classes for Bernstein-Type Operators

Adell

Pérez-Palomares

1998

Journal of Approximation Theory

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We consider families (L t , t # T) of positive linear operators such that each L t is representable in terms of a stochastic process starting at the origin and having nondecreasing paths and integrable stationary increments. For these families, we give probabilistic characterizations of the best possible constants both in preservation inequalities concerning the first modulus and in preservation of Lipschitz classes of first order. As an application, we compute such constants for the Bernstein, Sza sz, Gamma, Baskakov, and Beta operators.1998 Academic Press

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Best Constants in Preservation Inequalities Concerning the First Modulus and Lipschitz Classes for Bernstein-Type Operators

Adell

Pérez-Palomares

1998

Journal of Approximation Theory

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“…Assuming that / is continuous and bounded on I(f G Co(-O) and applying the ii-functional method we can get for certain operators better estimates. In [1], Anastassiou, Cottin, Gonska obtained the best estimate of the modulus of continuity of operators (1.1) in the case where I is a compact interval.…”

Section: The Methods Of If-functionalsmentioning

confidence: 99%

On Smoothness and Approximation Properties of the Kantorovich Type Operators

Powierska¹

2002

Demonstratio Mathematica

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Abstract. In this paper we present inequalities concerning the weighted moduli of continuity of Kantorovich type operators L^f-Moreover, we give some estimates of the degree of approximation of / by Lnf in the Holder type norms. PreliminariesLet I be a finite or infinite interval and let M(I) be the class of all measurable complex-valued functions bounded on I. In the case when I is an infinite interval, denote by M\ oc (I) the class of all functions bounded on every compact subinterval of I. Given any n G N :-{1,2,...}, let J n be a set of indices contained in Z := {0, ±1, ±2,...} and let I be the union of non-overlapping intervals Ij !n (j £ Jn) with increasing left (right) end points. Introduce, formally, for functions / belonging to M{I) or M\ oc {I), the discrete operators L n defined byjeJ n where € ij, n and pj >n are non-negative functions continuous on I. Denote by L* n the Kantorovich type modification of operators (1.1) given by We adopt the following notation. Given any non-negative function w defined on the interval ICR and any x,y £ I we writeFor an arbitrary function g defined on I we introduce the quantityWe denote by A (I) the set of all continuous functions w on I, positive in the interior of I, with values not greater than 1, which satisfy the inequality w(x,y) < w(s) for any three points x, s,y G I such that x < s < y.(Obviously, this inequality holds if, for example, w is non-decreasing or nonincreasing or concave on I). Given two weights w,p G A(J) we define the general weighted modulus of continuity of g on I byIf p(x) = 1 on /, we will write fi w (g; 6) instead of £l w ,p(9', Further, in the case w(x) = 1 on I the weighted modulus il w (g~,6) becomes the ordinary modulus of continuity u(g\ 6) of g on I. In this case, w = 1, the symbol ||(?|| will be used instead of HffH^. Let C W (I) denote the class of all functions g continuous on I, such that ||<7||u, < 00.Taking into account a positive non-decreasing function tp on the interval (0,1], such that ip( 1) < 1, we write iuil(v) . luii• --f 19{x) ~ 9(y)\w{x,y)p{x,y)x,y G I, 0 < \x-y\ < lj.If this quantity is finite we call it the weighted Holder type norm of g on I.In case p = 1 and w = 1 on /, we denote this expression by if p # 1, w = 1, then we will write ||<7||p^ instead of ||.

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“…It should be said that, since B n (W) … W (n \ 1) (cf. [11]), and each w ¥ W coincides with its usual modulus of continuity, such a result can be viewed as a consequence of [4,Theorem 9 and the subsequent Remark (ii)].…”

Section: Introductionmentioning

confidence: 91%