• Kyungpook Mathematical Journal
• /
• v.58 no.4
• /
• pp.733-746
• /
• 2018

The concept of f-statistical convergence which is, in fact, a generalization of statistical convergence, has been introduced recently by Aizpuru et al. (Quaest. Math. 37: 525-530, 2014). The main object of this paper is to prove an f-statistical analog of the classical Korovkin type approximation theorem of Boyanov and Veselinov. It is shown that the f-statistical analog is intermediate between the classical theorem and its statistical analog. As an application, we estimate the rate of f-statistical convergence of the sequence of positive linear operators defined from $C^*[0,{\infty})$ into itself.

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.4
• /
• pp.825-837
• /
• 2010

In this paper, we obtain a Korovkin type approximation theorem for double sequences of positive linear operators of two variables from $H_w$ (K) to C (K) via A-statistical convergence. Also, we construct an example such that our new approximation result works but its classical case does not work. Furthermore, we study the rates of A-statistical convergence by means of the modulus of continuity.

• Kyungpook Mathematical Journal
• /
• v.62 no.3
• /
• pp.467-484
• /
• 2022

In this work, we define bivariate Bernstein-Kantorovich type operators on a triangular domain and obtain some approximation results for these operators. We start off by computing some moment estimates and prove a Korovkin type convergence theorem. Then, we estimate the rate of convergence using the partial and complete modulus of continuity, and derive a Voronovskaya-type asymptotic theorem. Further, we calculate the order of approximation with regard to the Peetre's K-functional and a Lipschitz type class. In addition, we construct the associated GBS type operators and compute the rate of approximation using the mixed modulus of continuity and class of the Lipschitz of Bögel continuous functions for these operators. Finally, we use the two operators to approximate example functions in order to compare their convergence.

• Korean Journal of Mathematics
• /
• v.30 no.3
• /
• pp.539-553
• /
• 2022

In the present paper, we aim to obtain several approximation properties of Stancu form Szász-Mirakjan-Durrmeyer operators based on Bézier basis functions with shape parameter λ ∈ [-1, 1]. We estimate some auxiliary results such as moments and central moments. Then, we obtain the order of convergence in terms of the Lipschitz-type class functions and Peetre's K-functional. Further, we prove weighted approximation theorem and also Voronovskaya-type asymptotic theorem. Finally, to see the accuracy and effectiveness of discussed operators, we present comparison of the convergence of constructed operators to certain functions with some graphical illustrations under certain parameters.

• The Pure and Applied Mathematics
• /
• v.10 no.2
• /
• pp.127-132
• /
• 2003

The object of this paper is to extend and generalize the work of Brosowski ［Fixpunktsatze in der approximationstheorie. Mathematica Cluj 11 (1969), 195-200］, Hicks & Humphries ［A note on fixed point theorems. J. Approx. Theory 34 (1982), 221-225］, Khan & Khan ［An extension of Brosowski-Meinardus theorem on invariant approximation. Approx. Theory Appl. 11 (1995), 1-5］ and Singh ［An application of a fixed point theorem to approximation theory J. Approx. Theory 25 (1979), 89-90; Application of fixed point theorem in approximation theory. In: Applied nonlinear analysis (pp. 389-394). Academic Press, 1979］ in metric spaces having convex structure, and in metric linear spaces having strictly monotone metric.

• The Journal of Korean Institute of Communications and Information Sciences
• /
• v.21 no.1
• /
• pp.251-268
• /
• 1996

In this paper, a higher-order feedforward neural network called ridge polynomial network (RPN) which shows good approximation capability for nonlnear continuous functions defined on compact subsets in multi-dimensional Euclidean spaces, is presented. This network provides more efficient and regular structure as compared to ordinary higher-order feedforward networks based on Gabor-Kolmogrov polynomial expansions, while maintating their fast learning property. the ridge polynomial network is a generalization of the pi-sigma network (PSN) and uses a specialform of ridge polynomials. It is shown that any multivariate polynomial can be exactly represented in this form, and thus realized by a RPN. The approximation capability of the RPNs for arbitrary continuous functions is shown by this representation theorem and the classical weierstrass polynomial approximation theorem. The RPN provides a natural mechanism for incremental function approximation based on learning algorithm of the PSN. Simulation results on several applications such as multivariate function approximation and pattern classification assert nonlinear approximation capability of the RPN.

• Journal of applied mathematics & informatics
• /
• v.41 no.4
• /
• pp.801-810
• /
• 2023

The aim of this paper, investigated of weighted space which contained the unbounded functions which is to be approximated by linear operators in terms some Well-known approximation tools such as the modulus of smoothness and K-functional. The characteristics of the identical theorem between modulus of smoothness and K-functional are consider. In addition to the establish the direct, converse and identical theorem by using some linear operators in terms modulus Ditzian-Totik.

• Journal of applied mathematics & informatics
• /
• v.36 no.1_2
• /
• pp.15-28
• /
• 2018

In this paper, we introduce a Stancu type generalization of genuine LupaŞ-Beta operators of integral type. We establish some moment estimates and the direct results in terms of classical modulus of continuity, Voronovskaja-type asymptotic theorem, weighted approximation, rate of convergence and pointwise estimates using the Lipschitz type maximal function. Lastly, we propose a king type modification of these operators to obtain better estimates.

• Bulletin of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.259-263
• /
• 1995

The following classical Oka-Weil approximation theorem on pseudoconvex domains in $C^n$ is well-known. Suppose that $M \subseteq C^n$ is pseudoconvex and that K is a compact subset of M with K = K, where K is the usual holomorphic hull of K in M. Then any function holomorphic in a neighborhood of K can be approximated uniformly on K by functions holomorphic on M (see [5], [6]).

• Communications of the Korean Mathematical Society
• /
• v.38 no.4
• /
• pp.1127-1139
• /
• 2023

We investigate an extension of Schauder's theorem by studying the relationship between various s-numbers of an operator T and its adjoint T^*. We have three main results. First, we present a new proof that the approximation number of T and T^* are equal for compact operators. Second, for non-compact, bounded linear operators from X to Y, we obtain a relationship between certain s-numbers of T and T^* under natural conditions on X and Y . Lastly, for non-compact operators that are compact with respect to certain approximation schemes, we prove results for comparing the degree of compactness of T with that of its adjoint T^*.