Approximation properties of Bernstein's singular integrals in variable exponent Lebesgue spaces on the real axis

Year 2022, Volume: 71 Issue: 4, 1059 - 1079, 30.12.2022

Ramazan Akgün

https://doi.org/10.31801/cfsuasmas.1056890

Cited By: 1

Abstract

In generalized Lebesgue spaces $L^{p(.)}$ with variable exponent $p(.)$ defined on the real axis, we obtain several inequalities of approximation by integral functions of finite degree. Approximation properties of Bernstein singular integrals in these spaces are obtained. Estimates of simultaneous approximation by integral functions of finite degree in $L^{p(.)}$ are proved.

Keywords

Modulus of smoothness, simultaneous approximation, Bernstein singular integral, forward Steklov mean, mollifiers, Jackson inequality, entire integral functions of finite degree

References

• Abdullaev, F., Chaichenko, S., Imashqızı, M., Shidlich, A., Direct and inverse approximation theorems in the weighted Orlicz-type spaces with a variable exponent, Turk. J. Math., 44(1) (2020), 284-299. https://doi.org/10.3906/mat-1911-3
• Ackhiezer, N. I., Theory of Approximation, Fizmatlit, Moscow, 1965, English translation of 2nd edition, Frederick Ungar, New York, 1956.
• Akgün, R., Approximation of functions of weighted Lebesgue and Smirnov spaces, Mathematica (Cluj), 54(77)(Special) (2012), 25-36.
• Akgün, R., Sharp Jackson and converse theorems of trigonometric approximation in weighted Lebesgue spaces, Proc. A. Razmadze Math. Inst., 152 (2010), 1-18.
• Akgün, R., Inequalities for one sided approximation in Orlicz spaces, Hacet. J. Math. Stat., 40(2) (2011), 231-240.
• Akgün, R., Some convolution inequalities in Musielak Orlicz spaces, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 42(2) (2016), 279-291.
• Akgün, R., Ghorbanalizadeh, A., Approximation by integral functions of finite degree in variable exponent Lebesgue spaces on the real axis, Turk. J. Math., 42(4) (2018), 1887-1903. https://doi.org/10.3906/mat-1605-26
• Avşar, A. H., Koç, H., Jackson and Stechkin type inequalities of trigonometric approximation in $A^{w,\theta}_{p,q(.)}$, Turk. J. Math., 42(6) (2018), 2979-2993. https://doi.org/10.3906/mat-1712-84
• Avşar, A. H., Yildirir, Y. E., On the trigonometric approximation of functions in weighted Lorentz spaces using Cesaro submethod, Novi Sad J. Math., 48(2) (2018), 41-54. https://doi.org/10.30755/NSJOM.06335
• Bardaro, C., Butzer, P. L., Stens, R. L., Vinti, G., Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals, J. Math. Anal. Appl., 316(1) (2006), 269-306. https://doi.org/10.1016/j.jmaa.2005.04.042
• Bernstein, S. N., Sur la meilleure approximation sur tout laxe reel des fonctions continues par des fonctions entieres de degre n I, Dokl. Acad. Sci. URSS (N.S.), 51 (1946), 331-334.
• Bernstein, S. N., Collected Works, Mir, Vol. I, Izdat. Akad. Nauk SSSR, Moscow, 1952, 11-104.
• Butler, R., On the evaluation of $\int_{0}^{\infty}\frac{\sin^{m}t}{t^{m}}dt$ by the trapezoidal rule, Amer. Math. Monthly, 67(6) (1960), 566-569.
• Cruz-Uribe, D., Fiorenza, A., Approximate identities in variable $L_p$ spaces, Math. Nachr., 280(3) (2007), 256-270.
• Cruz-Uribe, D., Fiorenza, A., Variable Lebesgue Spaces, Foundations and Harmonic Analysis, Birkhauser, Applied and Numerical Harmonic Analysis, 2013.
• Diening, L., Maximal function on generalized Lebesgue spaces $L^{p(x)}$, Math. Ineq. & Appl., 7(2) (2004); 245-253. https://doi.org/10.7153/mia-07-27
• Diening, L., Harjulehto, P., Hastö, P., Ruzicka, M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 2011.
• Diening, L., Ruzicka, M., Calderon-Zymund Operators on Generalized Lebesgue Spaces $L^{P(X)}$ and Problems Related to Fluid Dynamics, Preprint, Mathematische Fakültat, Albert- Ludwings-Universitat Freiburg, 2002, 1-20.
• Dogu, A., Avsar, A. H., Yildirir, Y. E., Some inequalities about convolution and trigonometric approximation in weighted Orlicz spaces, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 44 (1) (2018), 107-115.
• Fan, X., Zhao, D., On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263(2) (2001), 424-446. doi:10.1006/jmaa.2000.7617
• Guven, A., Israfilov, D. M., Trigonometric approximation in generalized Lebesgue spaces $L^{p(x)}$, J. Math. Ineq. 4 (2) (2010), 285-299.
• Jafarov, S. Z., Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents, Ukr. Math. J., 66(10) (2015), 1509-1518. https://doi.org/10.1007/s11253-015-1027-y
• Jafarov, S. Z., Approximation by trigonometric polynomials in subspace of variable exponent grand Lebesgue spaces, Global J. Math. Sci., 8(2) (2016), 836-843.
• Jafarov, S. Z., Ulyanov type inequalities for moduli of smoothness, Appl. Math. E-Notes, 12 (2012), 221-227.
• Jafarov, S. Z., S. M. Nikolskii type inequality and estimation between the best approximations of a function in norms of different spaces, Math. Balkanica, 21(1-2) (2007), 173-182.
• Ibragimov, I. I., Teoriya Priblizheniya Tselymi Funktsiyami (In Russian), The Theory of Approximation by Entire Functions, Elm, Baku, 1979, 468 pp.
• Israfilov, D. M., Testici, A., Approximation problems in the Lebesgue spaces with variable exponent, J. Math. Anal. Appl., 459(1) (2018), 112-123. https://doi.org/10.1016/j.jmaa.2017.10.067
• Israfilov, D.M., Testici, A., Approximation by Faber-Laurent rational functions in Lebesgue spaces with variable exponent, Indag. Math., 27(4) (2016), 914-922. https://doi.org/10.1016/j.indag.2016.06.001
• Israfilov, D. M., Yirtici, E., Convolutions and best approximations in variable exponent Lebesgue spaces, Math. Reports, 18(4) (2016), 497-508.
• Koc, H., Simultaneous approximation by polynomials in Orlicz spaces generated by quasiconvex Young functions, Kuwait J. Sci., 43(4) (2016), 18-31.
• Kokilashvili, V., Nanobashvili, I., Boundedness criteria for the majorants of Fourier integrals summation means in weighted variable exponent Lebesgue spaces and application, Georgian Math. J., 20(4) (2013), 721-727. https://doi.org/10.1515/gmj-2013-0038
• Kovacik, Z. O., Rakosnik, J., On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czech. Math. J., 41(116)(4) (1991), 592-618.
• Musielak, J., Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Springer, 1983.
• Nikolski, S. M., Inequalities for entire functions of finite degree and their application to the theory of differentiable functions of several variables, AMS Translation Series Two, 80 (1969), 1-38 Trud. Steklov Math. Inst., 38 (1951), 211-278.
• Orlicz, W., Uber konjugierte exponentenfolgen, Studia Math. 3 (1931), 200-212.
• Paley, R., Wiener, N., Fourier Transforms in the Complex Domain, American Mathematical Society, 1934.
• Rajagopal, K. R., Ruzicka, M., On the modeling elektroreological materials, Mech. Res. Comm., 23(4) (1996), 401-407. https://doi.org/10.1016/0093-6413(96)00038-9
• Ruzicka, M., Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000.
• Samko, S., Differentiation and Integration of Variable Order and the Spaces $L^{p(x)}$, In: Operator Theory for Complex and Hypercomplex Analysis (Mexico City, 1994), 203-219, Contemporary Math. 212, American Mathematical Society, Providence, RI, 1998.
• Sharapudinov, I. I., Some Questions in the Theory of Approximation in Lebesgue Spaces with Variable Exponent, (In Russian), Itogi Nauki Yug. Rossii Matematicheskii Monografiya, 5, Southern Institute of Mathematics of the Vladikavkaz Sceince Centre of the Russian Academy of Sciences and the Government of the Republic of North Ossetia-Alania, Vladikavkaz, 2012, 267 pp.
• Sharapudinov, I. I., Approximation of functions in $L^{p(x)}_{2\pi}$ by trigonometric polynomials, Izv. Math., 77(2) 2013, 407-434. https://doi.org/10.4213/im7808
• Sharapudinov, I. I., On direct and inverse theorems of approximation theory in variable Lebesgue and Sobolev spaces, Azerbaijan J. Math., 4(1) (2014), 55-72.
• Sharapudinov, I. I., Approximation of functions in Lebesgue and Sobolev spaces with variable exponent by Fourier-Haar sums, Sbornik Math., 205(1-2) (2014), 291-306. https://doi.org/10.4213/sm8274
• Sharapudinov, I. I., Some problems in approximation theory in the spaces $L^{p(x)}(E)$, (In Russian), Analysis Math., 33(2) (2007), 135-153. https://doi.org/10.1007/s10476-007-0204-0
• Sharapudinov, I. I., The basis property of the Haar system in the space $L^{p(t)}([0, 1])$ and the principle of localization in the mean, (In Russian), Mat. Sbornik, 130(172)(2) (1986), 275-283. https://doi.org/10.1070/SM1987v058n01ABEH003104
• Taberski, R., Approximation by entire functions of exponential type, Demonstratio Math., 14 (1981), 151-181.
• Taberski, R., On exponential approximation of locally integrable functions, Annales Sociatatis Mathematicae Polonae, Series I: Commentationes Mathematicae, 32 (1992), 159-174.
• Volosivets, S. S., Approximation of functions and their conjugates in variable Lebesgue spaces, Sbornik Math., 208(1) (2017), 44-59. https://doi.org/10.1070/SM8636
• Yeh, J., Real Analysis: Theory of Measure and Integration, 2nd edition, World Scientific, 2006.
• Zhikov, V. V., Averaging of functionals of the calculus of variations and elasticity theory, (In Russian), Math. USSR-Izvestiya, 50(4) (1986), 675-710. https://doi.org/10.1070/IM1987v029n01ABEH000958