Abstract this survey paper contains a detailed selfcontained introduction to korovkintype theorems and to some of their applications concerning the approximation of continuous functions as well as of lpfunctions, by means of positive linear operators. In this work, we generalize the concept of statistical convergence and introduce the concept of weighted statistical convergence of order. The main object of this paper is to prove korovkin type approximation theorems using the notion of f statistical convergence. Approximation properties of these operators are obtained by using either korovkintype statistical approximation theorem or hepingtype convergence theorem for bivariate functions. Some applications for falgebras and orthomorphisms are presented. Let a and b be complex calgebras with identities 1, and 1, respectively.
With korovkin approximations one comes a convergence for the whole approximation with examination of the convergence of the process at a finite number of functions. In this paper, using the concept of bstatistical convergence for sequence of infinite matrices bbi with bibnki we investigate various approximation results concerning the classical korovkin theorem. Applications to positive approximation processes on convex compact sets app. Furthermore, they have been the source of a considerable amount of research in several other fields of mathematics cf. Our goal is to study their statistical convergence to the approximated function. The authors have to be congratulated for their colossal. Korovkintype approximation theorem with algebraic test functions for a sequence of real numbers on a banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results in classical and statistical versions. We present a korovkin type approximation theorem for a sequence of positive linear operators defined on the space of all real valued continuous and periodic functions via a statistical approximation, for the rate of the third order ditziantotik modulus of smoothness. Approximation theory has important applications in the theory of polynomial approximation, in various areas of functional analysis 1,3,5,12,15.
Korovkintype approximation theory has been developed in the context of classical function spaces and in more abstract spaces, such as locally convex ordered. Statistical summability and korovkin type approximation. Also, a korovkintype approximation theorem has been studied via statistical convergence in the space c which is the space of all 2. In this paper we consider power series method which is also member of the class of all continuous summability methods.
In this paper, we study the korovkin type approximation theorem for k a. Pdf korovkin type approximation theorem for almost and. This type of convergence can replace classical results provided by bohmankorovkin theorem. Korovkintype approximation has been widely studied in the literature 9. Korovkintype theorems in weighted spaces via summation. Korovkintype theorems and approximation by positive linear. Some of the examples of approximation theory and statistical convergence studies can be seen in 9, 10, 1824. Korovkintype approximation theory and its applications.
A korovkin type approximation theorem and its applications. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Statistically and relatively modular deferredweighted. In, korovkin type approximation theorems were proved by using the idea of statistical convergence. Pdf we present a korovkin type approximation theorem for a sequence of positive linear operators defined on the space of all real valued. Korovkin type approximation theorems in weighted spaces. In this paper, we prove the following riesz spaces version of the korovkin theorem. Korovkin type approximation theorem via aisummability. Korovkintype theorem, positive operator, approximation by positive operators. Even today, the development of korovkintype approximation theory is far from complete. We study a korovkin type approximation theorem for a. Research article a korovkin type approximation theorem and. We apply our new notion of equistatistical convergence to prove a korovkin type approximation theorem and we show that our theorem is a nontrivial extension of some wellknown korovkin type. Namboodiri department of mathematics, indian institute of technology, powai, bombay400076, india communicated by g.
Namboodiri department of mathematics, indian institute of technology, powai, bombay400076, india. S s symmetry article statistically and relatively modular deferredweighted summability and korovkintype approximation theorems hari mohan srivastava 1,2, bidu bhusan jena 3, susanta kumar paikray 3 and umakanta misra 4 1 department of mathematics and statistics, university of victoria, victoria, bc v8w 3r4, canada 2 department of medical research, china medical university hospital, china. Let e and f be two archimedean riesz spaces with f. In this study, we introduce the notions of pointwise statistical convergence of order and strongly. Very recently, kadak 9 have introduced the concept of weighted statistical convergence involving statistical convergence based on p. Korovkin type approximation for double sequences via statistical a. This paper is a contribution to the problem of approximating continuous functions f defined on a compact hausdorff space x, where the value fx is a compact convex set in r for every x in x. Approximation properties of bivariate generalization of. We also study the rate of statistical asummability of positive linear operators. Statistical approximation by kantorovichtype discrete qbeta. Korovkintype theorems and approximation by positive. Corp, delhi, 1960 first established the necessary and. The first one shows that the statistical korovkin type theorem does not work but our approximation theorem. Full text information korovkintype approximation theory and its applications.
Altomare, francesco campiti, michele korovkintype approximation theory and its applications. Pdf a korovkin type approximation theorem in statistical. Korovkintype theorem which is one of the fundamental methods in approximation theory to describe uniform convergence of any sequence of positive linear operators is discussed on weighted spaces, for univariate and multivariate functions, respectively. In this paper, we study the korovkin type approximation theorem for ka. Publication date 1994 isbn 3110141787 9783110141788.
Korovkintype approximation theory in riesz spaces springerlink. The concept of statistical convergence for sequence. In this work, we introduce the korovkin type approximation theorems on. One of the main problems in this theory is to state sufficient and necessary conditions under which a given subset is a korovkin subset in for with respect to. The study of the korovkintype approximation theory is a wellestablished area of research, which concern with the problem of approximation a function f by means of a sequence a n of positive linear operators. In practice a continuous function can be approximated by polynomials. Finally, we obtain an interleave between rieszs representation theory and lebesguestieltjes integral i, for rieszs. Classical approximation theory has started with the proof of weierstrass approximation theorem and after that korovkin linear operators and approximation theory, hindustan publ. A korovkintype approximation theorem and power series.
We present a korovkin type approximation theorem for a sequence of positive linear operators defined on the space of all real valued continuous and periodic. We present two general sequences of positive linear operators. Devore, the approximation of continuous functions by positive linear operators, lecture notes in mathematics, vol. Much of approximation theory involves polynomials and approximation underneath the supremum norm. The book is an excellent reference text and may be used as a textbook for a graduate level course.
Title korovkin type approximation closures for vectorvalued. In this article, we prove a korovkin type approximation theorem for a function of two variables by using the notion of statistical asummability. On the korovkin approximation theorem and volkovtype. Responsibility francesco altomare, michele campiti. Astatistical extension of the korovkin type approximation. Statistical convergence is quite effective in the approximation theory. Until the study of gadjiev and orhan, there was no study related to statistical convergence and approximation theory. Korovkintype approximation theory on commutative banach algebras m. A korovkin type approximation theorem for setvalued functions klaus keimel and walter roth communicated by john b.
Appendix d a subject classification of korovkintype approximation theory with a subject index. Numerous and frequentlyupdated resource results are available from this search. In this short paper, we give a generalization of the classical korovkin approximation theorem korovkin in linear operators and approximation theory, 1960, volkovtype theorems volkov in dokl. This theory has fruitful connections with real analysis, functional analysis, harmonic analysis, measure theory and probability theory, summability theory and partial di. A good understanding of the weierstrass theorem from the 1800s is a good place to start. In this paper, using the concept of bstatistical convergence for sequence of infinite matrices b b i with b i b n k i we investigate various approximation results concerning the classical korovkin theorem. We also prove a similar korovkintype theorem for nets of operators. Journal of approximation theory 34, 237246 1982 korovkintype approximation on calgebras b. This survey paper contains a detailed selfcontained introduction to korovkintype theorems and to some of their applications concerning the approximation of continuous functions as well as of lpfunctions, by means of positive linear operators. Similar problems have been settled for nets of operators that are not necessarily equicontinuous or linear.
Korovkintype approximation theory and its applications in. The korovkin theorems are simple yet powerful tools for deciding whether a given sequence of positive linear operators on or is an approximation process. Korovkintype approximation theorems provide basic and e ective tools to determine whether a given sequence of positive linear operators, acting on some function spaces is an approximation process or, equivalently, converges strongly to the identity operator. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Korovkin type theorems for weighted approximation 1277 we need to know the limits of the function. Equistatistical relative convergence and korovkintype. The series is devoted to the publication of monographs and highlevel textbooks in mathematics, mathematical methods and their applications. In this paper, using the concept of astatistical convergence for double sequences, we provide a. A relationship between the newly established korovkin type approximation theorems via f statistical convergence, the classical korovkin theorems and their statistical analogs has been studied. The first is introduced by using a class of dependent random variables, and the second is a mixture between two linear operators of discrete type. In this vein there is also the stoneweierstrass theorem.
The study of the korovkin type approximation theory is a wellestablished area of research, which deals with the problem of approximating a function f by means of a sequence ln f of positive lin. Korovkin type approximation theorem for functions of two. In mathematics the korovkin approximation is a convergence statement in which the approximation of a function is given by a certain sequence of functions. In this study, using the concept of astatistical convergence we investigate a korovkin type approximation result for a sequence of positive linear operators defined on the space of all continuous. Abstract korovkin theory in modular spaces in the sense of power series method. Furthermore, we obtain these types of approximation theorems by means of summability which is a stronger convergence method than. Then we present two examples of sequences of positive linear operators. Korovkintype theorems and approximation by positive linear operators. Korovkin type approximation theorems in bstatistical. Fibonacci statistical convergence and korovkin type.