A-summation process in the space of locally integrable functions

Nilay Şahin Bayram, Cihan Orhan


In this paper, using the concept of summation process, we give a
Korovkin type approximation theorem for a sequence of positive linear operators
acting from Lp;q (loc) ; the space of locally integrable functions, into itself. We also
study rate of convergence of these operators.


Summation process, positive linear operators, locally integrable functions, Korovkin type theorem, modulus of continuity, rate of convergence.

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F. Altomare and M. Campiti, Korovkin Type Approximation Theory and Its Applications,

de Gruyter, Berlin, (1994).

I. Aslan and O. Duman, Summability on Mellin-Type nonlinear integral operators, Integral

Transforms and Special Function 30 (2019), no. 6, 492-511.

N. ¸S. Bayram, Strong Summation Process in Locally Integrable Function Spaces, Hacettepe

Journal of Mathematics and Statistics 45 (3) (2016), 683-694.

N. ¸S. Bayram and C. Orhan, Abel Convergence of the Sequence of Positive Linear Operators

in Lp;q (loc), Bulletin of the Belgian Mathematical Society-Simon Stevin, 26 (2019), 71-83.

S. J. Bernau, Theorems of Korovkin type for Lp spaces, Paci c J. Math. 53 (1974), 11-19.

O. Costin and G. V. Dunne, Convergence from divergence, J. Phys. A 51 (2018), no. 4,

LT01, 10 pp.

R. A. Devore, The Approximation of Continuous functions by Positive Linear Operators,

Lecture Notes in Mathematics, Springer-Verlag 293 (1972), Berlin.

K. Donner, Korovkin theorems in Lp spaces, J. Functional Analysis 42 (1981), 12-28.

O. Duman and C. Orhan, Statistical approximation in the space of locally integrable func-

tions, Publ. Math. Debrecen, 63 (2003), 134-144.

O. Duman and C. Orhan, Rates of A- statistical convergence of operators in the space of

locally integrable functions, Appl. Math. Letters, (2008), 431-435.

V. K. Dzyadik, On the approximation of functions by linear positive operators and singular

integrals, Mat. Sbornik 70 (112) (1966), 508-517 ( in Russian ).

A. D. Gadjiev, On P. P. Korovkin type theorems, Math. Zametki 20 (1976).

A. D. Gadjiev, R. O. Efendiyev and E. ·Ibikli, On Korovkins type theorem in the space of

locally ·Integrable functions, Czech. Math. J.,53 (128) (2003), 45-53.

P. P. Korovkin, Linear Operators and The Theory of Approximation, Hindustan Publ. Co.

Delhi (1960).

G. G. Lorentz, A contribution to the theory of divergent sequences. Acta. Math. 80 (1948),


C. Orhan and ·I. Sakao¼glu, Rate of convergence in Lp approximation, Periodica Mathematica

Hungarica, 68 (2014),176-184.

·I. Sakao¼glu and C. Orhan, Strong summation process in Lp spaces, Nonlinear Analysis, 86

(2013), 89-94.

A. I. Stepanets, Approximations in spaces of locally integrable functions, Ukrainian Math. J.

(1994), no. 5 (1995), 638-670 .

A. I. Stepanets, Approximations in spaces of locally integrable functions, Akad. Nauk Ukrainy

Inst. Mat. Preprint no. 18,(1993), 47 pp (Russian).

J. J. Swetits and B. Wood, On degree of Lp approximation with positive linear operators,

Journal of Approximation Theory, 87 (1996), 239-241.

R. Taberski, Aprroximation properties of the integral Bernstein operators and their deriv-

atives in some classes of locally integrable functions, Funct. Approx. Comment. Math. 21

(1992), 85-96.

A. Zygmund, Trigonometric Series, Cambridge University Press, 1979.

DOI: http://dx.doi.org/10.24193/subbmath.2020.2.07


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