APPROXIMATION BY CERTAIN LINEAR POSITIVE OPERATORS

Abstract

A linear operator is a function L having the following properties: (a) The domain D of L is a nonempty set of real functions, all having the same real domain T . (b) For every f ED,L(f) is again a real function with domain T . (c) If f and g belong to D , and if a and b are reals, then (af + bg)ED , and gat' + bg)= aL(f) + bL(g). The operator L is called positive if f ED , and f(x)z 0 for every x ET , implies that L(f)(x) k 0 for every x ET . After the work of Bohman [12] and Korovkin [56] linear positive operators have aroused the interest of several researchers in the theory of approximation of functions. Apart from the earlier known examples of linear positive approximation operators, several new sequences and classes of such operators were constructed and studied. Amongst some of well known sequences and classes of such operators, we mention the ones due to Korovkin [56], Meyer-Konig and Zeller [65], Cheney and Sharma [17,18], Me'ir and Sharma [64], Muller [68], Durrmeyer [25], Baskakov [8], King [55], Rathore [72], May [62], Sahai and Prasad [75], Mazhar and Totik [63],' and Gupta et al. [39] etc. 2 We have also defined a new sequence of linear positive operators Mn (1998) called as Integral Baskakov-type operators to approximate unbounded continuous functions on [0, 00) and studied some problems in ordinary and simultaneous approximation by these operators in the present thesis. For a given approximation method {LA. } the investigation of its direct, inverse and saturation theorems forms one of the most-important theoretical and practical aspect of its study. A direct theorem provides the order of approximation for functions of a specified smoothness. On the other hand, an inverse theorem infers the nature of smoothness of a function from its order of approximation. A saturation theorem is a more curious phenomena. It refers to an inherent limitation (if present) of the approximation method: the order of approximation beyond a certain limit 0(0(A)) (0(2) 0, A — co) is possible only for a trivial (finite dimensional) subspace. The function for which the 0(0(2)) approximation is attained, form the saturation (Favard) class and those with the approximation order. o(Ø(2)) come in the trivial class. Thus a saturation theorem consists of a determination of a saturation order 0(2), the saturation class and the trivial class. In this formulation the error in the approximation could be with respect to a certain function norm, seminorm or a metric, pointwise or local. The first case is often referred to as global. In the pointwise case the error term incorporates a function of the point under consideration and indicates a bias of the method towards certain points (usually the end points of the interval). The common characteristic of global and pointwise results is their involvement with the whole domain (the set of definition) of 3 functions, while the local results concern, both in hypothesis and conclusion, with limited subsets of the domain. The study of direct theorems in approximation theory was initiated by the classical work of Jackson [48] on algebraic and trigonometric polynomials of best approximation. The corresponding inverse theorems were obtained by Bernstein (see e.g., Natanson [69]) by an ingeneous application of his famous inequality. In this trigonometric case the results of Jackson-Bernstein had an essential gap for the case a=1. This was filled, much later, by Zygmund [92] through the introduction of the class Z(Lip*1 ). Further generalizations of the Zygmund class have been found to be very useful in recent developments in approximation theory. Alexits [7] initiated the study of saturation of convolution operators by characterizing the saturation class for Fejer operators. Favard [26] gave a general formulation of the phenomenon of saturation. The current interest in the above problems was revived through the original work of Korovkin [56] on linear positive operators. The recognition of the basic role of linear positive operators triggered a virtual chain reaction in approximation theory. Important contributions to direct theorems are due to Freud [28, 29], Sunouchi [84], Shapiro [79], Pych-Taberska [71], Gupta and Pant [43], Wood [91], Gupta and Srivastava [37] and Kasana and Agrawal [51] etc. For operators possessing certain Bernstein-type inequalities an approach towards inverse theorem is given in DeVore [22]. In this direction, much work has been done by Butzer and his associates. Berens and Lorentz [11] gave an elementry proof of global inverse theorem for Bernstein Polynomials for the case 0 < a < 1. An extension of their technique for the case 0 < a s 2 has been made by Becker and Nessel [9]. Significant 4 contributions to inverse results have been made by Butzer and Berens [14], Butzer and Scherer [15] and May [62] etc. Saturation theorems have been studied by Sunouchi [83], Suzuki [85], Suzuki and Watanabe [86], Ikeno and Suzuki [47], Lorentz [60], DeVore [21, 22], Schnabl [78], Becker, Kucharski and Nessel [10], Micchelli [66, 67], May [62], Gupta and Srivastava [40] and Kasana et al. [52] etc. Several investigations (in our context starting with the wotk of Butzer [13] on Bernstein polynomials) indicated that even when a sequence or class of linear positive operators is saturated with a certain order of approximation, some carefully chosen linear combinations of its members may give a better order of approximation for smoother functions. The first attempt at some how improving the order of approximation was made by P.L. Butzer [13] who showed that by taking a linear combination of the Bernstein polynomials, the order of approximation considerably improves for smoother functions. More general combinations have been studied by Rathore [72] and May [62] for other sequences of linear positive operators. Micchelli [66] offered yet another approach for improving the order of approximation by Bernstein polynomials B„ by considering the iterative combinations UnA —BnYc]. Agrawal and Gupta [4] applied the technique of Micchelli to improve the order of approximation by Phillips operators. Another topic of interest in the present thesis is the phenomena of simultaneous approximation (approximation of derivatives of functions by the corresponding order derivatives of operators). The study in this direction began with a remarkable result for the Bernstein polynomials 13, (f) owing to Lorentz [59], who proved that B,,k)(f;x)- f (k) (X), n whenever the latter exists at the particular point 5 x E[0,1], k = 1,2,3,... being arbitrary. His method for this point-wise convergence in simultaneous approximation has since been applied by several workers to other operators. Rathore [73, 74] made a more detailed study of simultaneous approximation and established the existence of Voronvskaja type asymptotic formulae in simultaneous approximation. Several other researchers obtained direct, inverse and saturation theorems in ordinary and simultaneous approximation both locally as well as globally for many other sequences and classes of linear positive operators, in this direction we refer to [1, 2, 3,4, 6, 16, 19, 20, 24, 27, 31, 33, 34, 35, 36, 39, 41, 42, 44, 50, 53, 54, 57, 58, 61, 75, 76, 82, 89]. Several linear positive operators of summation type have been appropriately modified to become L - approximation method. The first such modification was made by Kantorovitch [49] for the case of Bernstein polynomials. Another modification of Bernstein polynomials was defined by Durrmeyer [25] and later studied extensively by Derriennic [19, 20]. Sahai and Prasad [75] and Mazhar and Totik [63] modified Lupas and Szasz operators respectively in an analogous manner to make it possible to approximate functions belonging to Li, [0,00),p z 1. In this thesis we study the Lp- approximation (1 s p < 00) by linear combinations of the operators Mn and obtain local direct and inverse theorems. For significant contributions in this