Approximation in Weighted Orlicz Spaces with a generating Young function that might be non-convex

Year 2021, Volume: 23 Issue: 2, 850 - 866, 04.07.2021

Ali Doğu , Yunus Emre Yıldırır

https://doi.org/10.25092/baunfbed.869567

Abstract

The aim of this paper is to investigate the order of approximation by some linear summation methods of trigonometric Fourier series in weighted Orlicz spaces which have generating Young functions not necessary to be convex. Obtained estimations base on the fractional modulus of smoothness and the best approximation. Furthermore, a convolution type operator is defined and its estimation by the best approximation is obtained.

Keywords

Linear summation processes , Fourier series , trigonometric approximation , weighted Orlicz spaces , Muckenhoupt weight

Konveks olması gerekmeyen genelleştirilmiş Young fonksiyonu ile üretilen Ağırlıklı Orlicz Uzaylarında yaklaşım

Abstract

Bu çalışmada, konveks olması gerekmeyen Young fonksiyonları ile üretilen ağırlıklı Orlicz uzaylarında trigonometrik Fourier serilerinin bazı lineer toplam metodları ile yaklaşım problemleri incelenmiştir. Elde edilen sonuçlar kesirli düzgünlük modülüne ve en iyi yaklaşım sayısına dayanmaktadır. Ayrıca, konvolüsyon tipli dönüşüm tanımlayıp, bu dönüşüm ile en iyi yaklaşım sayısı arasındaki ilişki değerlendirilmiştir.

Keywords

Lineer toplam metodları , Fourier serileri , , trigonometrik yaklaşım , ağırlıklı Orlicz uzayları , Muckenhoupt ağırlık fonksiyonu

References

• Akgün, R., Some inequalities of trigonometric approximation in weighted Orlicz spaces, Mathematica Slovaca, 66, 1, 217-234, (2016).
• Akgün, R. and Koç, H., Simultaneous approximation of functions in Orlicz spaces with Muckenhoupt weights, Complex Variables and Elliptic Equations, 61, 8, 1107-1115, (2016).
• Chen, Y.M., On two-functional spaces, Studia Mathematica, 24, 61-88, (1964).
• Dogu, A., Avsar, A.H. and Yildirir, Y.E., Some inequalities about convolution and trigonometric approximation in weighted Orlicz spaces. Proceeding of the Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 44, 1, 107-115, (2018).
• Gavriljuk, V.G., Linear summation methods for the Fourier series and best approximation, Ukrainian Mathematical Journal, 15, 4, 412-418, (1963).
• Israfilov, D.M. and Yirtici, E., Convolutions and best approximations in variable exponent Lebesgue spaces, Mathematical Reports (Bucuresti), 18(68), 4, 497-508, (2016).
• Jafarov, S.Z., Linear methotds of summing Fourier series and approximation in weight Orlicz spaces, Turkish Journal of Mathematics, 42, 6, 2916-2925, (2018).
• Jafarov, S.Z., Approximation by linear means of Fourier series in weighted Orlicz spaces Proceeding of the Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 43, 2, 175-187, (2017).
• Jafarov, S.Z., Approximation by linear means of Fourier series in weighted Lebesque spaces with variable exponents, Ukrainian Mathematical Journal, 66, 10, 1509-1518, (2015).
• Khabazi, M., The mean convergence of trigonometric Fourier series in weighted Orlicz classes, Proceedings of A. Razmadze Mathematical Institute, 129, 65-75, (2002).
• Krasnoselíski, M.A. and Ruticki, Y.B., Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron, P. Noordho Ltd., Groningen, (1961).
• Koç, H., Simultaneous approximation by polynomials in Orlicz spaces generated by quasiconvex Young functions. Kuwait J. Science, 43, 4, 18-31, (2016).
• Kokilashvili, V. and Tsanava, T., "On the norm estimate of deviation by linear summability means and an extension of the Bernstein inequality, Proceedings of A. Razmadze Mathematical Institute, 154, 144-146 (2010).
• Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function, Transactions of the American Mathematical Society, 165, 207-226, (1972).
• Ponomarenko, V.G. and Timan, M.F., The properties of convolution type transforms in the Orlicz spaces, theory of approximation of functions, Proceedings of the Institute of Mathematics and Mechanics, 3, Donetsk, (1998).
• Rao, M.M. and Ren, Z.D., Applications of Orlicz spaces, Marcel Dekker Inc., New York, (2002).
• Rao, M.M. and Ren, Z.D., Theory of Orlicz spaces, Marcel Dekker Inc., New York, (1991).
• Stechkin, S.B., Approximation of periodic functions by Fejíer sums, Trudy Matematicheskogo Instituta imeni V. A. Steklova, 2, 48-60, (1961).
• Timan, M.F., Best approximation of a function and linear methods for the summation of Fourier series, Izvestiya Rossiiskoi Akademii Nauk Seriya Matematicheskaya, 29, 587-604, (1965).
• Timan, M.F., Approximation of continuous periodic functions by linear operators constructed on the basis of their Fourier series, Doklayd Akademii Nauk. SSSR, 181, 1339-1342, (1968).
• Timan, M.F., Some linear summation processes for the Fourier series and the best approximation, Doklayd Akademii Nauk. SSSR, 145, 741-743, (1962).
• Yildirir, Y.E. and Cetintas, R., Trigonometric approximation in weighted Orlicz spaces. Proceeding of the Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 42, 1, 25-37, (2016).
• Yildirir, Y.E. and Dogu, A., Approximation in weighted Lorentz spaces, Journal Mathematical Science: Advences and Applications, 54, 1-9, (2018).
• Yildirir, Y.E. and Israfilov, D.M., The properties of convolution type transforms in weighted Orlicz spaces, Glasnik Matematicki, 65, 461-474, (2010).