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In this paper, we give some properties of the Dunkl–Gamma‐type operators including Appell polynomials, using into consideration the generalized power summability method. In the first section are given moments of the new defined operators. In second section are given some direct estimation related to the Dunkl–Gamma‐type operators, including Korovkin‐type theorem. In the third section, we give some results related to the weighted spaces of continuous functions, and in the last section, we give some properties in the sense of A$$ A $$‐statistically convergence, including Voronovskaya and Grüss–Voronovskaya‐type theorem. [ABSTRACT FROM AUTHOR]

In this paper, we establish a novel category of sequence spaces ℓ p q λ and ℓ ∞ q λ by utlizing q-analogue Λ q of Λ-matrix. Our investigation outlines several topological characteristics and inclusion results of these newly introduced sequence spaces, specifically identifying them as BK-spaces. Subsequently, we demonstrate that these novel sequence spaces are of nonabsolute type and establish their isometric isomorphism with ℓ p and ℓ ∞ . Moreover, we obtain the α-, β-, and γ-duals of these sequence spaces. We further characterize the class (ℓ p q λ , X) of matrices, where X is any of the spaces ℓ ∞ , c, or c 0 . Lastly, our study delves into the exploration of specific geometric properties exhibited by the space ℓ p q λ . [ABSTRACT FROM AUTHOR]

The main object of this paper is to construct a new class of modified (p , q) -Gamma-type operators. For this new class of operators, in section one, the general moments are find; in section two, the Korovkin-type theorem and some direct results are proved by considering the modulus of continuity and modulus of smoothness and their behavior in Lipschitz-type spaces. In section three, some results in the weighted spaces are given, and in the end, some shape-preserving properties are proven. [ABSTRACT FROM AUTHOR]

In this paper, we have construct a new version of the Bézier‐type Baskakov–Schurer–Szász–Stancu operators. For this new class of operators, uniform convergence is shown in any compact subset or positive real line. We prove Korovkin‐type theorem, Voronovskaya‐type theorem, and Grüss–Voronovskaya‐type theorem. Moreover, at the end, we express the behavior of the operators in the Lipschitz‐type space using the modulus of continuity and smoothness. [ABSTRACT FROM AUTHOR]

In this paper, we define a new kind of the μ -Bernstein–Schurer–Stancu operators. For these operators, we prove uniform convergence and study their behavior using consideration modulus of continuity and smoothness. Moreover, we present the Korovkin type theorem, Voronovskaya type theorem, and Grüss–Voronovskaya type theorems. [ABSTRACT FROM AUTHOR]

In this paper, we have proved certain kinds of Tauberian theorems for Nörlund-Cesáro summability methods in m-normed space X in statistically sense. We give necessary and sufficient Tauberian condition for this method of summability, in statistically sense. Also, we prove that Tauberian theorems under these summability methods are valid under oscillating slow condition of Schmidt-type and Hardy-type conditions. [ABSTRACT FROM AUTHOR]

In this paper, we construct the Chlodowsky-type Szász operators defined via Boas–Buck-type polynomials. We prove some approximation properties and obtain the rate of the convergence for these operators. We also study the Voronovskaya-type theorem and weighted approximation. [ABSTRACT FROM AUTHOR]

In this paper we study a notion called (E, 1)(C, 1)−weighted statistical convergence and prove a Korovkin type approximation theorem. The rate of convergence for weighted statistical convergence is obtained. In the last section we also give Voronovskaya and Grüss‐Voronovskaya type theorems and some illustrative numerical examples [ABSTRACT FROM AUTHOR]

In this paper, we study the Kantorovich variant of Szász operators based on Sheffer polynomials and prove a Korovkin type theorem. Also, we present the Voronovskaya type theorem and Grüss-Voronovskaya type theorem. [ABSTRACT FROM AUTHOR]

In this paper, we will show Tauberian conditions under which ordinary convergence of the sequence (xn) in 2-normed space X, follows from Tp,qn-summability. In fact we give a necessary and sufficient Tauberian condition for this method of summability. Also, we prove that Tauberian Theorems for these summability methods are valid with Schmidt-type slowly oscillating condition as well as with Hardy-type "big O" condition. [ABSTRACT FROM AUTHOR]

In this paper, we study the Korovkin-type theorem for modified Meyer–König and Zeller operators via A-statistical convergence and power series summability method. The rate of convergence for this new summability methods is also obtained with the help of the modulus of continuity. Further, we establish Voronovskaya-type and Grüss–Voronovskaya-type theorems for A-statistical convergence. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to use a power series summability method to study some approximation properties of Kantorovich type Szász–Mirakyan operators including Sheffer polynomials. We also establish Voronovskaya type result. [ABSTRACT FROM AUTHOR]

In this paper, we study the Kantorovich-Stancu-type generalization of Szász-Mirakyan operators including Brenke-type polynomials and prove a Korovkin-type theorem via the T -statistical convergence and power series summability method. Moreover, we determine the rate of the convergence. Furthermore, we establish the Voronovskaya- and Grüss-Voronovskaya-type theorems for T -statistical convergence. [ABSTRACT FROM AUTHOR]

In this paper we will prove the Korovkin type theorem for modified Szász–Mirakyan operators via A-statistical convergence and the power summability method. Also we give the rate of the convergence related to the above summability methods, and in the last section, we give a kind of Voronovskaya type theorem for A-statistical convergence and Grüss–Voronovskaya type theorem. [ABSTRACT FROM AUTHOR]

In this paper some properties of the generalized Szasz operators by multiple Appell polynomials are given, using into consideration the power summability method. In the first section are given some direct estimation related to the generalized Szasz operators by multiple Appell polynomials, including Korovkin type theorem. In the second section, we give some results related to the weighted spaces of continuous functions and Voronovskaya type theorem. In the third section, we have proved some results related to the statistical convergence of the generalized Szasz operators by multiple Appell polynomials, using into consideration the A− transformation. At the end of the paper are given some illustrative computational examples which make such summability methods (for example, power series method) more useful and fruitful for applications of functional analysis in approximation theory. [ABSTRACT FROM AUTHOR]

Let (p n) {(p_{n})} and (q n) {(q_{n})} be any two non-negative real sequences, with R n := ∑ k = 0 n p k ⁢ q n - k ≠ 0 {R_{n}:=\sum_{k=0}^{n}{p_{k}q_{n-k}}\neq 0} ( n ∈ ℕ {n\in\mathbb{N}}). Let ∑ k = 0 ∞ a k {\sum_{k=0}^{\infty}a_{k}} be a series of real or complex numbers with partial sums (s n) {(s_{n})} , and set t n p , q := 1 R n ⁢ ∑ k = 0 n p k ⁢ q n - k ⁢ s k {t_{n}^{p,q}:=\frac{1}{R_{n}}\sum_{k=0}^{n}{p_{k}q_{n-k}s_{k}}} for n ∈ ℕ {n\in\mathbb{N}}. In this paper, we present the necessary and sufficient conditions under which the existence of the limit lim n → ∞ ⁡ s n = L {\lim_{n\to\infty}{s_{n}}=L} follows from that of lim n → ∞ ⁡ t n p , q = L {\lim_{n\to\infty}t_{n}^{p,q}=L}. These conditions are one-sided or two-sided if (s n) {(s_{n})} is a sequence of real or complex numbers, respectively. [ABSTRACT FROM AUTHOR]

In this paper we will prove the Korovkin type theorem for Baskakov-Schurer-Szász operators via the A-statistical convergence and the power summability method. Also we give the rate of the convergence related to the above summability methods and in the last chapter we give a kind of Voronovskaya type theorem for the A-statistical convergence and Grüss-Voronovskaya type theorem. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new class of operators which will be called the class of k-quasi-class A(s, t) operators. An operator T ∈ B(H) is said to be k-quasi-class A(s, t) if T*k((|T|*t|T|2s|T*|t)1/t+s-|T*|2t)Tk≥ 0, where s > 0, t > 0 and k is a natural number. We show that an algebraically k-quasi-class A(s, t) operator T is polaroid, has Bishop’s property β and we prove that Weyl type theorems for k-quasi-class A(s, t) operators. In particular, we prove that if T* is algebraically k-quasi-class A(s, t), then the generalized a-Weyl’s theorem holds for T. Using these results we show that T* satisfies generalized the Weyl’s theorem if and only if T satisfies the generalized Weyl’s theorem if and only if T satisfies Weyl’s theorem. We also examine the hyperinvariant subspace problem for k-quasi-class A(s, t) operators. [ABSTRACT FROM AUTHOR]

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class of operators: (n, fc)-quasi class Q operators, superclass of (n, fc)-quasi paranormal operators. An operator T is said to be (n, fc)-quasi class Q if it satisfies ... for all X H and for some nonnegative integers n and k. We prove the basic structural properties of this class of operators. It will be proved that If T has a no non-trivial invariant subspace, then the nonnegative operator ... is a strongly stable contraction. In section 4, we give some examples which compare our class with other known classes of operators and as a consequence we prove that (n, fc)-quasi class Q does not have SVEP property. In the last section we also characterize the (n, fc)-quasi class Q composition operators on Fock spaces. [ABSTRACT FROM AUTHOR]

In this paper we will prove the Korovkin type theorem for new modified Szász-Mirakyan operators via A-statistical convergence and power summability method. Also we give the rate of the convergence related to the summability methods and in the last section we give a kind of Voronovskaya type theorem for A-statistical convergence and Grüss-Voronovskaya type theorem. [ABSTRACT FROM AUTHOR]

In this paper, which deals essentially with various summability concepts and summability techniques and shows how these concepts and techniques lead to a number of approximation results, we have used the new concept of weighted A -summability proposed by Mohiuddine (2016) and introduced the notions of statistically weighted B -summability and weighted B -statistical convergence with respect to the weighted regular method. We then prove a Korovkin type approximation theorem for functions of two variables and also present an example via generalized Meyer-König and Zeller type operator to show that our proposed method is stronger than its classical and statistical versions. Furthermore, the rate of convergence of approximating positive linear operators are estimated by means of the modulus of continuity and some Voronovskaja type results are investigated. Computational and geometrical approaches to illustrate some of our results are also presented. [ABSTRACT FROM AUTHOR]

Fuzzy set theory has entered into a large variety of disciplines of sciences, technology and humanities having established itself as an extremely versatile interdisciplinary research area. Accordingly different notions of fuzzy structure have been developed such as fuzzy normed linear space, fuzzy topological vector space, fuzzy sequence space etc. While reviewing the literature in fuzzy sequence space, we have seen that the notion of Tauberian theorems for the Euler-Nörlund mean-convergent sequences of fuzzy numbers has not been developed. In the present paper, we introduce some new concepts about statistical convergence of sequences of fuzzy numbers. The main purpose of this paper is to study Tauberian theorems for the Euler-Nörlund mean-convergent sequences of fuzzy numbers and investigate some other kind of convergences named Euler-Nörlund mean-level convergence so as to fill up the existing gaps in the literature. The results which we obtained in this study are much more general than those obtained by others. [ABSTRACT FROM AUTHOR]

In this paper, we will show Tauberian theorems for the generalized de la Vallée-Poussin mean-convergent sequences of fuzzy numbers. [ABSTRACT FROM AUTHOR]

In this paper, we will show a new Tauberian theorems defined by weighted Nörlund-Cesáro summability method. [ABSTRACT FROM AUTHOR]

A real valued function f defined on a subset of R is δ2-ward continuous if limn→∞Δ³ f(Xn) = 0 whenever limn→∞ >Δ³ Xn = 0, where Δ³ zn -- Zn + 3 - 3zn+2 + 3zn+i - Zn for each positive integer n, R denotes the set of real numbers, and a subset E of R is d²-ward compact if any sequence of points in E has a d²-quasi Cauchy subsequence where a sequence (xn) is d²-quasi Cauchy if limn→∞Δ³zn=0. It turns out that the uniform limit process preserves this kind of continuity, and the set of d²-ward continuous functions is a closed subset of the set of continuous functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of almost statistical convergence of generalized difference sequences of fuzzy numbers involving lacunary sequences. We give the relations between the lacunary strongly almost Cesàro type convergence and lacunary almost statistical convergence. Furthermore, we study some of their properties like completeness, solidity, symmetricity etc. We also give some inclusion relations related to these classes. [ABSTRACT FROM AUTHOR]

We will define the second order Cesaro operator spaces and we will show that those spaces are Banach spaces, are separable, fulfill dominant Lebesgue theorem, are not rearrangement invariant, not reflexive and strictly convex. [ABSTRACT FROM AUTHOR]

In this paper, we will show Tauberian conditions under which λstatistical convergence follows from (V,λ)-statistical convergence. Our results generalize the ones given in [3]. [ABSTRACT FROM AUTHOR]

This paper deals with some classes of bounded linear operators on Hilbert spaces. The main emphasis is put onto the classes A(k*) and A(k*)P; k > 0. Some additional results are given for other classes, like PA(k*), M - A(k*) and spectral properties of operators belonging to A(k*) are considered. We also describe under what conditions a matrix-operator TA;B belongs to A(k*), A(k*)P or PA(k*). [ABSTRACT FROM AUTHOR]

In this paper, we propose to introduce a new Λ 2 -weighted statistical convergence. Based upon this definition, we prove some Korovkin type theorems. We also find the rate of the convergence for this kind of weighted statistical convergence and derive some Voronovskaya type theorems. [ABSTRACT FROM AUTHOR]

First, we will see that if T is a contraction of the k-quasi-∗-class A operator, then the nonnegative operator D = T∗k(∣T²∣ - ∣T∗∣²)Tk is a contraction whose power sequence {Dn}n=1∞ converges strongly to a projection P and TT∗kP = 0. Second, it will be proved that if T is a contraction of the k-quasi-∗-class A operator, then either T has a non-trivial invariant subspace or T is a proper contraction. Finally it will be proved that if T belongs to the k-quasi-∗-class A and is a contraction, then T has a Wold-type decomposition and T has the PF property. [ABSTRACT FROM AUTHOR]

The main object of this article is to give several necessary conditions for periodic functions. These conditions involve the monotonicity properties and the extreme values of the periodic functions. Some illustrative examples and graphical representations are also provided. [ABSTRACT FROM AUTHOR]

In this paper criterion for L1-convergence of a certain cosine sums with quasi hyper-convex coefficients is obtained. [ABSTRACT FROM AUTHOR]

Abstract: The main object of this paper is to prove a Korovkin type theorem for the test functions 1, in the space of all continuous -periodic functions on the real line . Our analysis is based upon the statistical summability involving the idea of the generalized de la Vallée Poussin mean. We also investigate the rate of the de la Vallée Poussin statistical summability of positive linear operators in the space . Finally, we provide an interesting illustrative example in support of our result. [Copyright &y& Elsevier]

In this paper, we introduce a new class of operators, called k-quasi class A n* operators, which is a superclass of hyponormal operators and a subclass of (n, k)-quasi-*-paranormal operators. We will show basic structural properties and some spectral properties of this class of operators. We show that, if T is k-quasi class An* then ... Also, we will prove Browder's theorem and a-Browders theorem for k-quasi class An* operator. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the absolute-(p; r) -*-paranormal which is a further generalization of both absolute-k-*-paranormal operator and p-*-paranormal operator. Also, we introduce the (p; r; q)-*-paranormal operator, which is also a further generalization of both absolute-(p; r) -*-paranormal operator and p -*-paranormal operator. We will show ba- sic structural properties of these two classes of operators. Finally, we will see some properties of the p -*-paranormal operator. Some spectral properties are also presented: we show that, if T is p-*-paranormal then σjp(T) = σp(T), and σja(T) = σa(T). [ABSTRACT FROM AUTHOR]

We introduce the Λ 2-strong convergence of numerical sequences and with it we generalize the concept of Λ-strong convergence of the results published by F. Móricz [2]. [ABSTRACT FROM AUTHOR]

We introduce the spaces of A(λ)-null, A(λ)-convergent, and A(λ)-bounded sequences. We examine some topological properties of the spaces and give some inclusion relations concerning these sequence spaces. Furthermore, we compute α-, β-, and γ-duals of these spaces. Finally, we characterize some classes of matrix transformations fromthe spaces of A(λ)-bounded and A(λ)-convergent sequences to the spaces of bounded, almost convergent, almost null, and convergent sequences and present a Steinhaus type theorem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a new class of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators, which considerably extends the well-known class of the classical Baskakov-Schurer-Szász-Stancu approximation operators. For this new class of approximation operators, we present a Korovkin type theorem and a Grüss-Voronovskaya type theorem, and also study the rate of its convergence. Moreover, we derive several results which are related to the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the weighted spaces. Finally, we prove some shape-preserving properties for the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators and, as a special case, we deduce the corresponding shape-preserving properties for the classical Baskakov-Schurer-Szász-Stancu approximation operators. [ABSTRACT FROM AUTHOR]