Değişken üslü Lebesgue uzaylarında konvolüsyonların bazı özellikleri

Öz

Bu çalışmada değişken üslü Lebesgue uzaylarında konvolüsyon tanımlandı ve Steklov ortalamalarının sonlu lineer birleşimleri ile yaklaşımının mümkün olduğu kanıtlandı. Ayrıca yaklaşım birimi ile oluşturulan özel konvolüsyonlar dizisinin başlangıç fonksiyonuna yakınsadığı gösterildi.

Anahtar Kelimeler

• Konvolüsyon
• değişken üslü Lebesgue uzayları
• yaklaşım birimi

Convolutions and approximations in the variable exponent spaces

Öz

A convolution in the variable exponent Lebesgue spaces is defined and the possibility its approximation by finite linear combinations of Steklov means is proved. Moreover, the convergence of the special convolutions sequence constructed via approximate identity to the original function is showed.

Anahtar Kelimeler

• convolution
• variable exponent Lebesgue spaces
• approximate identity

Kaynakça

1. Cruz-Uribe, D. V., and Fiorenza A., Variable Lebesgue Spaces Foundation and Harmonic Analysis, Birkhäsuser, (2013).
2. Sharapudinov, I. I., Some questions of approximation theory in the Lebesgue spaces with variable exponent, Itogi Nauki i Techniki Yug Rossii Mathematicheskix Monogaphs., vol. 5, Southern Mathematical Institute of the Vladikavkaz Scientic Center of the Russian Academy of Sciences and Republic of North Ossetia-Alania, Vladikavkaz, (2012).
3. Sharapudinov, I. I., Some aspects of approximation theory in the spaces , Analysis of Mathematical, 33, 2, 135-153, (2007).
4. Sharapudinov, I. I., Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by de la Vall´ee-Poussin means, Sbornik: Mathematics, 207, 7, 1010-1036, (2016).
5. Shakh-Emirov, T. N., On Uniform Boundedness of some Families of Integral Convolution Operators in Weighted Variable Exponent Lebesgue Spaces, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics., 14 4, 1, 422-427, (2014).
6. Volosivets, S. S., Approximation of functions and their conjugates in variable Lebesgue spaces. Sbornik: Mathematics, 208, 1, 48-64, (2017).
7. Jafarov, S. Z., Approximation of the functions in weighted Lebesgue spaces with variable exponent, Complex Variables and Elliptic Equations, 63, 10, 1444-1458, (2018).
8. Guven, A., and Israfilov, D. M., Trigonometric approximation in generalized Lebesgue spaces , Journal of Mathematical Inequalities, 4, 2, 285-299, (2010).

9. Israfilov, D. M., and Testici, A., Approximation in weighted Smirnov spaces, Complex Variable and Elliptic Equations, Vol. 60, 1, 45-58, (2015).
10. Israfilov, D. M., and Testici, A., Approximation in Smirnov classes with variable exponent, Complex Variable and Elliptic Equations, Vol. 60, 9, 1243-1253, (2015).
11. Israfilov, D. M., and Testici, A., Multiplier and Approximation Theorems in Smirnov Classes with Variable Exponent, Turkish Journal of Mathematics, Vol. 42, 1442-1456, (2018).
12. Israfilov, D. M., and Testici, A., Approximation by Faber-Laurent rational functions in Lebesgue spaces with variable exponent, Indagationes Mathematica, Vol. 27, 4, 914-922, (2016).
13. Israfilov, D. M., and Gursel, E., On Some Properties of Convolutions in Variable Exponent Lebesgue Space, Complex Analysis and Operator Theory, Vol. 11, 8, 1817-1824, (2017).
14. Israfilov, D. M., Gursel, E., and Aydin, E., Maximal convergence of Faber Series in Smirnov Classes with Variable Exponent, Bulletin of the Brazilian Mathematical Society, 49, 955-963, New Series (2018).
15. Israfilov, D. M., and Gursel, E., Approximation by -Faber polynomials in the variable Smirnov classes, Mathematical Methods in the Applied Sciences, 44, 7479-7490, (2021).
16. Israfilov, D. M., and Gursel, E., Faber--Laurent series in variable Smirnov classes, Turkish Journal of Mathematics, 44, 2, 389-402, (2020).
17. Jafarov, S. Z., Approximation by means of Fourier series in Lebesgue space with variable exponent, Kazakh Mathematical Journal, 20, 3, 57-63, (2020).
18. Jafarov, S. Z., Linear methods for summing Fourier series and approximation in weihgted Lebesgue spaces with variable exponent, Ukrainian Mathematical Journal, 66, 10, 1509-1518, (2015).
19. Akgun, R., Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent. Ukrains' kyi Matematychnyi Zhurnal, 63, 1, 3-23, (2011).
20. Akgun, R., Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth. Georgian Mathematical Journal, 18, 2, 203-235, (2011).
21. Akgun, R., and Kokilashvili, V., On converse theorems of trigonometric approximation in weighted variable exponent Lebesgue spaces, Banach Journal of Mathematical Analysis, 5, 1, 70-82, (2011).
22. Bilalov, B., and T., Guseynov, Z. G., On the basicity from exponents in Lebesgue spaces with variable exponent, TWMS Journal of Pure and Applied Mathematics, Vol.1, 1, 14-23, (2010).
23. Bilalov, B. T., and Guseynov Z. G., Basicity of a system of exponents with a piece-wise linear phase in variable spaces. Mediterranean Journal Mathematics, 9, 3, 487-498, (2012).
24. Guliyeva, F., and Sadigova, S. R., Bases of the pertrubed system of exponents in generalized weighted Lebesgue space with a general weight, Africa Mathematica, Vol.8, 6, 781-791, (2017).
25. Najafov, T., and Nasibova, N., On the Noetherness of the Riemann problem in generalized weighted Hardy classes, Azerbaijan Journal of Mathematics, vol. 5, 2, 109-124, (2015).
26. Kováčik O. Rákosník J. : On spaces and , Czechoslovak Mathematical Journal, 41, 592—618, (1991).