Approximation by Gamma type operators

In this study, we introduce newly defined Gamma operators which preserve constants and e^2μ·, μ>0 functions. In accordance with this purpose, we focus on their approximation properties such as uniform convergence, rate of convergence, asymptotic formula, and saturation results. Superior properties of introduced operators have been tested both theoretically and numerically in certain senses to highlight the performance of the new constructions of Gamma operators.     

Approximation by some operators of Szasz-Mirakyan type

In this paper we give theorems on the degree of approximation of functions of one and two variables by certain operators of the Szasz-Mirakyan type.     

Approximation by Durrmeyer type Jakimoski–Leviatan operators

In this study, we introduce the Durrmeyer type Jakimoski–Leviatan operators and examine their approximation properties. We study the local approximation properties of these operators. Further, we investigate the convergence of these operators in a weighted space of functions and obtain the approximation properties. Furthermore, we give a Voronovskaja type theorem for the our new operators. Copyright © 2015 John Wiley & Sons, Ltd.     

Approximation by Nonlinear Multivariate Sampling Kantorovich Type Operators and Applications to Image Processing

In this article, we study a nonlinear version of the sampling Kantorovich type operators in a multivariate setting and we show applications to image processing. By means of the above operators, we are able to reconstruct continuous and uniformly continuous signals/images (functions). Moreover, we study the modular convergence of these operators in the setting of Orlicz spaces L ^?(? ⁿ ) that allows us to deal the case of not necessarily continuous signals/images. The convergence theorems in L ^p (? ⁿ )-spaces, L ^αlog ^β L(? ⁿ )-spaces and exponential spaces follow as particular cases. Several graphical representations, for the various examples and image processing applications are included.     

Approximation by discrete operators

A discrete, positive, weighted algebraic polynomial operator which is based on Gaussian quadrature is constructed. The operator is shown to satisfy the Jackson estimate and an optimal version is obtained.     

Approximation by convolution operators

слЕДУь п. к. сИккЕМА, Мы ИсслЕДУЕМ АппРОксИМ АцИОННыЕ сВОИстВА ОпЕРАтОРОВ $$u_\varrho ^\beta (f,x) = \frac{1}{{\beta _\varrho }}\int\limits_{ - \infty }^\infty {f(x - t)\beta ^\varrho (t) dt(\varrho \to \infty ).} $$ жДЕсьΒ — НЕОтРИцАтЕл ьНАь сУММИРУЕМАь ФУН кцИь, \(\beta _\varrho = \int\limits_{ - \infty }^\infty {\beta ^\varrho (t) dt} \) И ВыпОлНЕНы УслОВИь: (i)Β(0)=1 ИΒ НЕпРЕРыВНА В тО ЧкЕt=0, (ii) \(\mathop {\sup }\limits_{\left| t \right| > \delta } \beta (t)< 1\) Дль кАжДОгОδ>0. ДОкАжАНО, ЧтО ЁкспОНЕ НцИАльНыИ пОРьДОк Ап пРОксИМАцИИ МОжЕт Быть ДОстИгНУт тОлькО Дль ФУНкцИИ ВИДА (fx)=ax+b И (fx)=ae ^bx+c. ЁтО — ИсклУЧИтЕльНыЕ слУЧАИ, пОскОлькУ УкАжАННыЕ ФУНкцИИ ьВ льУтсь ЕДИНстВЕННыМ И НЕпОДВИжНыМИ тОЧкАМ И Дль ОпЕРАтОРОВU _β ^? . ДОкАжАНО тАкжЕ, ЧтО пР И УДАЧНОМ ВыБОРЕΒ МО жНО ДОБИтьсь «пОЧтИ Ёксп ОНЕНцИАльНОгО» пОРьДкА АппРОксИМАц ИИ. НАкОНЕц, В пОслЕДНЕИ т ЕОРЕМЕ УтВЕРжДАЕтсь, ЧтО сУЩЕстВУУт тАкИЕΒ Иf, ЧтОU _β ^? (f,x) пРИp→∞ РАсхОДьтсь НА МНОжЕстВЕ пОлОжИтЕл ьНОИ МЕРы.     

Approximation by integral type Meyer-König-Zeller operators on a simplex

Let T={(x₁,x₂):x₁+x₂≤1, x₁,x₂≤0} be a simplex in ?². The integral type Meyer-König-Zeller operators are constructed on the simplex T, and the degree of approximation of these operators for L^p-functions is obtained.     

Approximation by q Baskakov-Beta-Stancu operators

In the present paper we propose the q analogue of the Baskakov-Beta-Stancu operators. We establish some direct results in the polynomial weighted space of continuous functions defined on the interval [0,∞). In the end, we propose an open problem on Srivastava-Gupta operators.     

Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces

In this paper we introduce a nonlinear version of the Kantorovich sampling type series in a nonuniform setting. By means of the above series we are able to reconstruct signals (functions) which are continuous or uniformly continuous. Moreover, we study the problem of the convergence in the setting of Orlicz spaces: this allows us to treat signals which are not necessarily continuous. Our theory applies to L^p-spaces, interpolation spaces, exponential spaces and many others. Several graphical examples are provided.     

Approximation by chaotic operators and by conjugate classes

We show that every bounded linear operator on a separable, infinite-dimensional Hilbert space H is the sum of two operators in the norm-closure of the set of operators on H that are chaotic in the sense of Devaney. We also observe that the density of several classes of cyclic operators, with respect to the strong operator topology, may be derived from a result by Hadwin et al. (Proc Amer. Math. Soc. 76 (1979) 250-252).     

Approximation by semigroup of spherical operators

This paper concerns about the approximation by a class of positive exponential type multiplier operators on the unit sphere Sn of the （n ＋ 1）- dimensional Euclidean space for n ≥2. We prove that such operators form a strongly continuous contraction semigroup of class （l0） and show the equivalence between the approximation errors of these operators and the K-functionals. We also give the saturation order and the saturation class of these operators. As examples, the rth Boolean of the generalized spherical Abel-Poisson operator ＋Vt^γ and the rth Boolean of the generalized spherical Weierstrass operator ＋Wt^k for integer r ≥ 1 and reals γ, k∈ （0, 1] have errors ｜｜＋r Vt^γ- f｜｜X ω^rγ（f, t^1/γ）X and ｜｜＋rWt^kf - f｜｜X ω^2rk（f, t^1/（2k））X for all f ∈ X and 0 ≤t ≤2π, where X is the Banach space of all continuous functions or all L^p integrable functions, 1 ≤p ≤＋∞, on S^n with norm ｜｜·｜｜X, and ω^s（f,t）X is the modulus of smoothness of degree s 〉 0 for f ∈X. Moreover, ＋r^Vt^γ and ＋rWt^k have the same saturation class if γ= 2k.