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Search Results (9)

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Keywords = the matrix summability method

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12 pages, 1758 KB  
Article
Kantorovich Version of Vector-Valued Shepard Operators
by Oktay Duman, Biancamaria Della Vecchia and Esra Erkus-Duman
Axioms 2024, 13(3), 181; https://doi.org/10.3390/axioms13030181 - 9 Mar 2024
Cited by 1 | Viewed by 1373
Abstract
In the present work, in order to approximate integrable vector-valued functions, we study the Kantorovich version of vector-valued Shepard operators. We also display some applications supporting our results by using parametric plots of a surface and a space curve. Finally, we also investigate [...] Read more.
In the present work, in order to approximate integrable vector-valued functions, we study the Kantorovich version of vector-valued Shepard operators. We also display some applications supporting our results by using parametric plots of a surface and a space curve. Finally, we also investigate how nonnegative regular (matrix) summability methods affect the approximation. Full article
(This article belongs to the Special Issue Mathematics, Computer Graphics and Computational Visualizations)
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16 pages, 896 KB  
Article
Vector-Valued Shepard Processes: Approximation with Summability
by Oktay Duman and Biancamaria Della Vecchia
Axioms 2023, 12(12), 1124; https://doi.org/10.3390/axioms12121124 - 15 Dec 2023
Cited by 2 | Viewed by 1424
Abstract
In this work, vector-valued continuous functions are approximated uniformly on the unit hypercube by Shepard operators. If λ denotes the usual parameter of the Shepard operators and m is the dimension of the hypercube, then our results show that it is possible to [...] Read more.
In this work, vector-valued continuous functions are approximated uniformly on the unit hypercube by Shepard operators. If λ denotes the usual parameter of the Shepard operators and m is the dimension of the hypercube, then our results show that it is possible to obtain a uniform approximation of a continuous vector-valued function by these operators when λm+1. By using three-dimensional parametric plots, we illustrate this uniform approximation for some vector-valued functions. Finally, the influence in approximation by regular summability processes is studied, and their motivation is shown. Full article
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8 pages, 247 KB  
Article
Unbounded Versions of Two Old Summability Theorems
by Jeff Connor
Axioms 2023, 12(8), 723; https://doi.org/10.3390/axioms12080723 - 26 Jul 2023
Viewed by 1045
Abstract
In this note, we obtain extensions of a theorem of Meyer-König and Zeller and a theorem of Wilansky in that the given results do not require a summability matrix to be a bounded operator from the convergent sequences into themselves. The culmination of [...] Read more.
In this note, we obtain extensions of a theorem of Meyer-König and Zeller and a theorem of Wilansky in that the given results do not require a summability matrix to be a bounded operator from the convergent sequences into themselves. The culmination of the results in this note is that a triangle matrix method T with null columns maps a bounded divergent sequence to a null sequence if and only if the range of T is not closed in the null sequences. Full article
(This article belongs to the Special Issue Operator Theory and Its Applications II)
6 pages, 255 KB  
Proceeding Paper
On Strong Approximation in Generalized Hölder and Zygmund Spaces
by Birendra Singh and Uaday Singh
Comput. Sci. Math. Forum 2023, 7(1), 9; https://doi.org/10.3390/IOCMA2023-14433 - 29 Apr 2023
Viewed by 1117
Abstract
The strong approximation of a function is a useful tool to analyze the convergence of its Fourier series. It is based on the summability techniques. However, unlike matrix summability methods, it uses non-linear methods to derive an auxiliary sequence using approximation errors generated [...] Read more.
The strong approximation of a function is a useful tool to analyze the convergence of its Fourier series. It is based on the summability techniques. However, unlike matrix summability methods, it uses non-linear methods to derive an auxiliary sequence using approximation errors generated by the series under analysis. In this paper, we give some direct results on the strong means of Fourier series of functions in generalized Hölder and Zygmund spaces. To elaborate its use, we deduce some corollaries. Full article
25 pages, 349 KB  
Article
Matrix Summability of Walsh–Fourier Series
by Ushangi Goginava and Károly Nagy
Mathematics 2022, 10(14), 2458; https://doi.org/10.3390/math10142458 - 14 Jul 2022
Cited by 15 | Viewed by 1632
Abstract
The presented paper discusses the matrix summability of the Walsh–Fourier series. In particular, we discuss the convergence of matrix transforms in L1 space and in CW space in terms of modulus of continuity and matrix transform variation. Moreover, we show the [...] Read more.
The presented paper discusses the matrix summability of the Walsh–Fourier series. In particular, we discuss the convergence of matrix transforms in L1 space and in CW space in terms of modulus of continuity and matrix transform variation. Moreover, we show the sharpness of our result. We also discuss some properties of the maximal operator t(f) of the matrix transform of the Walsh–Fourier series. As a consequence, we obtain the sufficient condition so that the matrix transforms tn(f) of the Walsh–Fourier series are convergent almost everywhere to the function f. The problems listed above are related to the corresponding Lebesgue constant of the matrix transformations. The paper sets out two-sides estimates for Lebesgue constants. The proven theorems can be used in the case of a variety of summability methods. Specifically, the proven theorems are used in the case of Cesàro means with varying parameters. Full article
(This article belongs to the Section E6: Functional Interpolation)
11 pages, 280 KB  
Article
Quasi-Density of Sets, Quasi-Statistical Convergence and the Matrix Summability Method
by Renata Masarova, Tomas Visnyai and Robert Vrabel
Axioms 2022, 11(3), 88; https://doi.org/10.3390/axioms11030088 - 23 Feb 2022
Cited by 1 | Viewed by 2325
Abstract
In this paper, we define the quasi-density of subsets of the set of natural numbers and show several of the properties of this density. The quasi-density dp(A) of the set AN is dependent on the sequence [...] Read more.
In this paper, we define the quasi-density of subsets of the set of natural numbers and show several of the properties of this density. The quasi-density dp(A) of the set AN is dependent on the sequence p=(pn). Different sequences (pn), for the same set A, will yield new and distinct densities. If the sequence (pn) does not differ from the sequence (n) in its order of magnitude, i.e., limnpnn=1, then the resulting quasi-density is very close to the asymptotic density. The results for sequences that do not satisfy this condition are more interesting. In the next part, we deal with the necessary and sufficient conditions so that the quasi-statistical convergence will be equivalent to the matrix summability method for a special class of triangular matrices with real coefficients. Full article
(This article belongs to the Special Issue Approximation Theory and Related Applications)
16 pages, 334 KB  
Review
Ideals, Nonnegative Summability Matrices and Corresponding Convergence Notions: A Short Survey of Recent Advancements
by Pratulananda Das
Axioms 2022, 11(1), 1; https://doi.org/10.3390/axioms11010001 - 21 Dec 2021
Viewed by 2441
Abstract
In this survey article, we look into some recent results concerning summability matrices, both regular as well as those which are not regular (called semi-regular) and generated matrix ideals as the overall view of the inter relationship between the notions of ideal convergence [...] Read more.
In this survey article, we look into some recent results concerning summability matrices, both regular as well as those which are not regular (called semi-regular) and generated matrix ideals as the overall view of the inter relationship between the notions of ideal convergence and summability methods by regular summability matrices. Full article
(This article belongs to the Special Issue Operator Theory and Its Applications)
5 pages, 229 KB  
Article
Orlicz–Pettis Theorem through Summability Methods
by Fernando León-Saavedra, María del Pilar Romero de la Rosa and Antonio Sala
Mathematics 2019, 7(10), 895; https://doi.org/10.3390/math7100895 - 25 Sep 2019
Cited by 6 | Viewed by 2664
Abstract
This paper unifies several versions of the Orlicz–Pettis theorem that incorporate summability methods. We show that a series is unconditionally convergent if and only if the series is weakly subseries convergent with respect to a regular linear summability method. This includes results using [...] Read more.
This paper unifies several versions of the Orlicz–Pettis theorem that incorporate summability methods. We show that a series is unconditionally convergent if and only if the series is weakly subseries convergent with respect to a regular linear summability method. This includes results using matrix summability, statistical convergence with respect to an ideal, and other variations of summability methods. Full article
12 pages, 302 KB  
Article
New Definitions about A I -Statistical Convergence with Respect to a Sequence of Modulus Functions and Lacunary Sequences
by Ömer Kişi, Hafize Gümüş and Ekrem Savas
Axioms 2018, 7(2), 24; https://doi.org/10.3390/axioms7020024 - 13 Apr 2018
Cited by 2 | Viewed by 4777
Abstract
In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of I -statistical convergence, which is a recently introduced summability method. The names of our new methods are A I -lacunary statistical [...] Read more.
In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of I -statistical convergence, which is a recently introduced summability method. The names of our new methods are A I -lacunary statistical convergence and strongly A I -lacunary convergence with respect to a sequence of modulus functions. These spaces are denoted by S θ A I , F and N θ A I , F , respectively. We give some inclusion relations between S A I , F , S θ A I , F and N θ A I , F . We also investigate Cesáro summability for A I and we obtain some basic results between A I -Cesáro summability, strongly A I -Cesáro summability and the spaces mentioned above. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications)
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