SavedAdd your journal

Journals / Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics / 2022 / Cilt: 71 - Sayı: 4

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

Ramazan AKGÜN
Pages
1059–1079
Publication Date
2022-12-30
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Estimates of simultaneous approximation by integral functions of finite degree in $L^{p(.)}$ are proved.

📌 3 Key Takeaways

🎯
Objective & Hypothesis: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.
📊
Key Findings & Data:Approximation properties of Bernstein singular integrals in these spaces are obtained.
💡
Conclusion & Impact:Estimates of simultaneous approximation by integral functions of finite degree in $L^{p(.)}$ are proved.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Modulus of smoothness · simultaneous approximation · Bernstein singular integral

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.

🔖

Cite This Article

AKGÜN, R. (2022). Approximation properties of Bernstein's singular integrals in variable exponent Lebesgue spaces on the real axis. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Cilt: 71 - Sayı: 4, 1059–1079. https://ajindex.com/articles/867587

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (15)

  1. 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
  2. Ackhiezer, N. I., Theory of Approximation, Fizmatlit, Moscow, 1965, English translation of 2nd edition, Frederick Ungar, New York, 1956.
  3. Akgün, R., Approximation of functions of weighted Lebesgue and Smirnov spaces, Mathematica (Cluj), 54(77)(Special) (2012), 25-36.
  4. Akgün, R., Sharp Jackson and converse theorems of trigonometric approximation in weighted Lebesgue spaces, Proc. A. Razmadze Math. Inst., 152 (2010), 1-18.
  5. Akgün, R., Inequalities for one sided approximation in Orlicz spaces, Hacet. J. Math. Stat., 40(2) (2011), 231-240.
  6. Akgün, R., Some convolution inequalities in Musielak Orlicz spaces, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 42(2) (2016), 279-291.
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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.
  12. Bernstein, S. N., Collected Works, Mir, Vol. I, Izdat. Akad. Nauk SSSR, Moscow, 1952, 11-104.
  13. 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.
  14. Cruz-Uribe, D., Fiorenza, A., Approximate identities in variable $L_p$ spaces, Math. Nachr., 280(3) (2007), 256-270.
  15. Cruz-Uribe, D., Fiorenza, A., Variable Lebesgue Spaces, Foundations and Harmonic Analysis, Birkhauser, Applied and Numerical Harmonic Analysis, 2013