On Korovkin Type Theorem in the Space of Locally Integrable Functions

Gadjiev, Akif D.; Efendiyev, R. O.; Ibikli, Ertan

doi:10.1023/a:1022967223553

Cited by 38 publications

(17 citation statements)

References 11 publications

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“…; (see [6]) : The next result shows that Korovkin type theorem does not hold in the whole space L p;q (loc) :…”

Section: Introductionmentioning

confidence: 84%

“…We also obtain rate of convergence in L p;q (loc) approximation with positive linear operators by means of modulus of continuity. Now we recall some information of locally integrable functions given in [6]. Let q(x) = 1 + x 2 ; 1 < x < 1 .…”

Section: Introductionmentioning

confidence: 99%

“…It is known [6] that L k p;q (loc) is the subspace of all functions f 2 L p;q (loc) for which there exists a constant k f such that…”

Section: Introductionmentioning

confidence: 99%

“…(see [6]): It is shown that for all j 2 N, kT j f k p;q 4 kf k p;q : Hence fT j g is an uniformly bounded sequence of positive linear operators from L p;q (loc) into itself. Also…”

mentioning

confidence: 96%

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Strong Summation Process in Locally Integrable Function Spaces

Bayram¹

2016

HJMS

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In this paper, using the concept of strong summation process, we give a Korovkin type approximation theorem for a sequence of positive linear operators acting from L p;q (loc) into itself. We also study modulus of continuity for L p;q (loc) approximation and give the rate of convergence of these operators.Keywords: A summability, positive linear operators, locally integrable functions, Korovkin type theorem, modulus of continuity, rate of convergence.2010 AMS Classi…cation: 41A25, 41A36. IntroductionThe classical theorem of Korovkin Approximation theory has important applications in the theory of polynomial approximation, in functional analysis, numerical solutions of di¤erential and integral equations [1], [7].The purpose this paper is to study a Korovkin type approximation theorem of a function f by means of sequence of positive linear operators from the space of locally integrable functions into itself with the use of a matrix summability method which includes both convergence and almost convergence. We also obtain rate of convergence in L p;q (loc) approximation with positive linear operators by means of modulus of continuity. Now we recall some information of locally integrable functions given in [6].

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“…; (see [6]) : The next result shows that Korovkin type theorem does not hold in the whole space L p;q (loc) :…”

Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

“…It is known [6] that L k p;q (loc) is the subspace of all functions f 2 L p;q (loc) for which there exists a constant k f such that…”

Section: Introductionmentioning

confidence: 99%

“…(see [6]): It is shown that for all j 2 N, kT j f k p;q 4 kf k p;q : Hence fT j g is an uniformly bounded sequence of positive linear operators from L p;q (loc) into itself. Also…”

mentioning

confidence: 96%

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Strong Summation Process in Locally Integrable Function Spaces

Bayram¹

2016

HJMS

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“…For details of proofs see [2] and [8]. Now, we give the following theorem to approximation all functions in C x 2 [0; 1) : This type of results is given in Gadjiev et al [7] for locally integrable functions. …”

Section: Weight Approximationmentioning

confidence: 99%

Quantitative estimates for Jain-Kantorovich operators

Deniz¹

2016

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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QUANTITATIVE ESTIMATES FOR JAIN-KANTOROVICH OPERATORS EMRE DEN · IZAbstract. By using given arbitrary sequences, n > 0, n 2 N with the property that limn!1n n = 0 and limn!1 n = 0, we give a Kantorovich type generalization of Jain operator based on the a Poisson disrtibition. Fristly we give the quantitative Voronovskaya type theorem. Then we also obtain the Grüss Voronovskaya type theorem in quantitative form .We show that they have an arbitrary good order of weighted approximation.

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On discrete Picard operators

Aral

İlarslan

Başcanbaz‐Tunca

2016

Math Methods in App Sciences

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The present paper is concerned with the approximation properties of discrete version of Picard operators. We first give exact equalities for the moments of the operators. In calculations of these moments, Eulerian numbers play a crucial role. We discuss convergence of these operators in weighted spaces and give Voronovskaya-type asymptotic formula. The weighted approximation of the operators in quantitative mean in terms of different modulus of continuities is also considered. We emphasize that the rate of convergence of the operators is better than the one obtained in [1].

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On Korovkin Type Theorem in the Space of Locally Integrable Functions

Abstract: It is shown that a Korovkin-type theorem does not hold in the weighted space of Lp−locally integrable functions on the whole real axis.

Cited by 38 publications

References 11 publications

Strong Summation Process in Locally Integrable Function Spaces

Strong Summation Process in Locally Integrable Function Spaces

Quantitative estimates for Jain-Kantorovich operators

On discrete Picard operators

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