Fejér means in variable exponent Lebesgue spaces on the real axis

Abstract

Variable exponent Lebesgue spaces are generalizations of classical Lebesgue spaces and have importance in many branches of Mathematical Analysis. Especially, direct and converse theorems and their improvements are studied by many mathematicians in these spaces. In this article, direct and converse predictions for the rate of convergence of Fejér means of functions belonging to the variable Lebesgue space L^p(⋅) (R) are established by using an appropriate K-functional. In this way, the result of Z. Ditzian on Fejér means in classical Lebesgue spaces L^p (R)(1

Keywords

• Fejer means
• Fourier transform
• variable exponent Lebesgue space

Reel eksende değişken üslü Lebesgue uzaylarda Fejér ortalamalar

Abstract

Değişken üslü Lebesgue uzayları klasik Lebesgue uzaylarının genellemeleridir ve Matematiksel Analizin birçok dalında öneme sahiptir. Özellikle direkt ve ters teoremler ve bunların geliştirilmesi bu uzaylarda birçok matematikçi tarafından incelenmektedir. Bu makalede, değişken üslü Lebesgue uzayı L^p(⋅) (R)'ye ait fonksiyonların Fejér ortalamalarının yakınsaklık hızına ilişkin doğrudan ve ters tahminler, uygun bir K-fonksiyonu kullanılarak oluşturulmuştur. Bu şekilde, Z. Ditzian'ın klasik Lebesgue uzaylarında L^p (R)(1

Keywords

• Fejer ortalamalar
• Fourier dönüşüm
• değişken üslü Lebesgue uzayı

References

1. DeVore, R. A., Lorentz, G. G., Constructive approximation, Springer-Verlag, Berlin, (1993).
2. Timan, A. F., Theory of approximation of functions of a real variable, Pergamon Press, New-York, (1963).
3. Akgun, R., Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent, Ukrainian Mathematical Journal, 63, no. 1, 1–26, (2011).
4. Akgun, R., Ghorbanalizadeh, A., Approximation by integral functions of finite degree in variable exponent Lebesgue spaces on the real axis, Turkish Journal of Mathematics, 42, 1887-1903, (2018).
5. Guven, A., Israfilov, D. M., Trigonometric approximation in generalized Lebesgue spaces L^p(x) , Journal of Mathematical Inequalities, 4, no. 2, 285–299, (2010).
6. Guven, A., Trigonometric approximation by matrix transforms in L^p(x) spaces, Analysis and Applications (Singap.), 10, no.1, 47–65, (2012).
7. Israfilov, D. M., Testici, A., Approximation problems in the Lebesgue spaces with variable exponent, Journal of Mathematical Analysis and Applications, 459, no. 1, 112–123, (2018).
8. Sharapudinov, I. I., Approximation of functions in L_2π^p(x) by trigonometric polynomials. (Russian) Russian Academy of Sciences. Izvestiya Mathematics, 77, no. 2, 197–224, (2013); translation in Izvestiya Mathematics, 77, no. 2, 407–434, (2013).

9. Sharapudinov, I. I., Approximation of function from variable exponent Lebesgue and Sobolev spaces by Vallée Poussin means, (Russian) Sbornik: Mathematics 207, no. 7, 131–158, (2016); translation in Sbornik: Mathematics, 207, no. 7-8, 1010–1036, (2016).
10. Jafarov, S. Z., Approximation by means of Fourier trigonometric series in weighted Lebesgue spaces with variable exponent, The Aligarh Bulletin of Mathematics, 41, 63-80, (2022).
11. Jafarov, S. Z., Approximation by means of Fourier series in Lebesgue spaces with variable exponent, Kazakh Mathematical Journal, (3), 57-68, (2021).
12. Ditzian, Z. On Fejér and Bochner-Riesz means, Journal of Fourier Analysis and Applications, 11, no. 4, 489-496, (2005).
13. Cruz-Uribe, D. V., Fiorenza, A., Variable Lebesgue Spaces, Foundations and harmonic analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, (2013).
14. Hunt, R., Muckenhoupt, B., Wheeden, R., Weighted norm inequalities for the conjugate function and Hilbert transform, Transactions of the American Mathematical Society, 176, 227–251, (1973).
15. Guven, A., Kokilashvili, V., On the means of Fourier integrals and Bernstein inequality in the two-weighted setting, Positivity, 14, no. 1, 165-180, (2010).
16. Weisz, F., Convergence and summability of Fourier transforms and Hardy spaces, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, (2017).

Details

Primary Language

English

Subjects

Approximation Theory and Asymptotic Methods

Journal Section

Research Article

Authors

Ebru Altıparmak ^*
0000-0001-6722-0807
Türkiye

Ali Güven
0000-0001-8878-250X
Türkiye

Early Pub Date

January 6, 2024

Publication Date

January 19, 2024

Submission Date

September 7, 2023

Acceptance Date

October 25, 2023

Published in Issue

Year 2024 Volume: 26 Number: 1

DOI

https://doi.org/10.25092/baunfbed.1356259

IZ

https://izlik.org/JA36BB32NJ

Cite

RIS / Bibtex

APA

Altıparmak, E., & Güven, A. (2024). Fejér means in variable exponent Lebesgue spaces on the real axis. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 26(1), 188-195. https://doi.org/10.25092/baunfbed.1356259

AMA

1.Altıparmak E, Güven A. Fejér means in variable exponent Lebesgue spaces on the real axis. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2024;26(1):188-195. doi:10.25092/baunfbed.1356259

Chicago

Altıparmak, Ebru, and Ali Güven. 2024. “Fejér Means in Variable Exponent Lebesgue Spaces on the Real Axis”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 26 (1): 188-95. https://doi.org/10.25092/baunfbed.1356259.

EndNote

Altıparmak E, Güven A (January 1, 2024) Fejér means in variable exponent Lebesgue spaces on the real axis. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 26 1 188–195.

IEEE

[1]E. Altıparmak and A. Güven, “Fejér means in variable exponent Lebesgue spaces on the real axis”, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 26, no. 1, pp. 188–195, Jan. 2024, doi: 10.25092/baunfbed.1356259.

ISNAD

Altıparmak, Ebru - Güven, Ali. “Fejér Means in Variable Exponent Lebesgue Spaces on the Real Axis”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 26/1 (January 1, 2024): 188-195. https://doi.org/10.25092/baunfbed.1356259.

JAMA

1.Altıparmak E, Güven A. Fejér means in variable exponent Lebesgue spaces on the real axis. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2024;26:188–195.

MLA

Altıparmak, Ebru, and Ali Güven. “Fejér Means in Variable Exponent Lebesgue Spaces on the Real Axis”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 26, no. 1, Jan. 2024, pp. 188-95, doi:10.25092/baunfbed.1356259.

Vancouver

1.Ebru Altıparmak, Ali Güven. Fejér means in variable exponent Lebesgue spaces on the real axis. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2024 Jan. 1;26(1):188-95. doi:10.25092/baunfbed.1356259

Reel eksende değişken üslü Lebesgue uzaylarda Fejér ortalamalar

Abstract

Değişken üslü Lebesgue uzayları klasik Lebesgue uzaylarının genellemeleridir ve Matematiksel Analizin birçok dalında öneme sahiptir. Özellikle direkt ve ters teoremler ve bunların geliştirilmesi bu uzaylarda birçok matematikçi tarafından incelenmektedir. Bu makalede, değişken üslü Lebesgue uzayı L^p(⋅) (R)'ye ait fonksiyonların Fejér ortalamalarının yakınsaklık hızına ilişkin doğrudan ve ters tahminler, uygun bir K-fonksiyonu kullanılarak oluşturulmuştur. Bu şekilde, Z. Ditzian'ın klasik Lebesgue uzaylarında L^p (R)(1

Keywords

• Fejer ortalamalar
• Fourier dönüşüm
• değişken üslü Lebesgue uzayı

References

1. DeVore, R. A., Lorentz, G. G., Constructive approximation, Springer-Verlag, Berlin, (1993).
2. Timan, A. F., Theory of approximation of functions of a real variable, Pergamon Press, New-York, (1963).
3. Akgun, R., Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent, Ukrainian Mathematical Journal, 63, no. 1, 1–26, (2011).
4. Akgun, R., Ghorbanalizadeh, A., Approximation by integral functions of finite degree in variable exponent Lebesgue spaces on the real axis, Turkish Journal of Mathematics, 42, 1887-1903, (2018).
5. Guven, A., Israfilov, D. M., Trigonometric approximation in generalized Lebesgue spaces L^p(x) , Journal of Mathematical Inequalities, 4, no. 2, 285–299, (2010).
6. Guven, A., Trigonometric approximation by matrix transforms in L^p(x) spaces, Analysis and Applications (Singap.), 10, no.1, 47–65, (2012).
7. Israfilov, D. M., Testici, A., Approximation problems in the Lebesgue spaces with variable exponent, Journal of Mathematical Analysis and Applications, 459, no. 1, 112–123, (2018).
8. Sharapudinov, I. I., Approximation of functions in L_2π^p(x) by trigonometric polynomials. (Russian) Russian Academy of Sciences. Izvestiya Mathematics, 77, no. 2, 197–224, (2013); translation in Izvestiya Mathematics, 77, no. 2, 407–434, (2013).

9. Sharapudinov, I. I., Approximation of function from variable exponent Lebesgue and Sobolev spaces by Vallée Poussin means, (Russian) Sbornik: Mathematics 207, no. 7, 131–158, (2016); translation in Sbornik: Mathematics, 207, no. 7-8, 1010–1036, (2016).
10. Jafarov, S. Z., Approximation by means of Fourier trigonometric series in weighted Lebesgue spaces with variable exponent, The Aligarh Bulletin of Mathematics, 41, 63-80, (2022).
11. Jafarov, S. Z., Approximation by means of Fourier series in Lebesgue spaces with variable exponent, Kazakh Mathematical Journal, (3), 57-68, (2021).
12. Ditzian, Z. On Fejér and Bochner-Riesz means, Journal of Fourier Analysis and Applications, 11, no. 4, 489-496, (2005).
13. Cruz-Uribe, D. V., Fiorenza, A., Variable Lebesgue Spaces, Foundations and harmonic analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, (2013).
14. Hunt, R., Muckenhoupt, B., Wheeden, R., Weighted norm inequalities for the conjugate function and Hilbert transform, Transactions of the American Mathematical Society, 176, 227–251, (1973).
15. Guven, A., Kokilashvili, V., On the means of Fourier integrals and Bernstein inequality in the two-weighted setting, Positivity, 14, no. 1, 165-180, (2010).
16. Weisz, F., Convergence and summability of Fourier transforms and Hardy spaces, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, (2017).

Details

Primary Language

English

Subjects

Approximation Theory and Asymptotic Methods

Journal Section

Research Article

Authors

Ebru Altıparmak ^*
0000-0001-6722-0807
Türkiye

Ali Güven
0000-0001-8878-250X
Türkiye

Early Pub Date

January 6, 2024

Publication Date

January 19, 2024

Submission Date

September 7, 2023

Acceptance Date

October 25, 2023

Published in Issue

Year 2024 Volume: 26 Number: 1

DOI

https://doi.org/10.25092/baunfbed.1356259

IZ

https://izlik.org/JA36BB32NJ

Cite

RIS / Bibtex

APA

Altıparmak, E., & Güven, A. (2024). Fejér means in variable exponent Lebesgue spaces on the real axis. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 26(1), 188-195. https://doi.org/10.25092/baunfbed.1356259

AMA

1.Altıparmak E, Güven A. Fejér means in variable exponent Lebesgue spaces on the real axis. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2024;26(1):188-195. doi:10.25092/baunfbed.1356259

Chicago

Altıparmak, Ebru, and Ali Güven. 2024. “Fejér Means in Variable Exponent Lebesgue Spaces on the Real Axis”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 26 (1): 188-95. https://doi.org/10.25092/baunfbed.1356259.

EndNote

Altıparmak E, Güven A (January 1, 2024) Fejér means in variable exponent Lebesgue spaces on the real axis. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 26 1 188–195.

IEEE

[1]E. Altıparmak and A. Güven, “Fejér means in variable exponent Lebesgue spaces on the real axis”, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 26, no. 1, pp. 188–195, Jan. 2024, doi: 10.25092/baunfbed.1356259.

ISNAD

Altıparmak, Ebru - Güven, Ali. “Fejér Means in Variable Exponent Lebesgue Spaces on the Real Axis”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 26/1 (January 1, 2024): 188-195. https://doi.org/10.25092/baunfbed.1356259.

JAMA

1.Altıparmak E, Güven A. Fejér means in variable exponent Lebesgue spaces on the real axis. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2024;26:188–195.

MLA

Altıparmak, Ebru, and Ali Güven. “Fejér Means in Variable Exponent Lebesgue Spaces on the Real Axis”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 26, no. 1, Jan. 2024, pp. 188-95, doi:10.25092/baunfbed.1356259.

Vancouver

1.Ebru Altıparmak, Ali Güven. Fejér means in variable exponent Lebesgue spaces on the real axis. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2024 Jan. 1;26(1):188-95. doi:10.25092/baunfbed.1356259

References

1. DeVore, R. A., Lorentz, G. G., Constructive approximation, Springer-Verlag, Berlin, (1993).
2. Timan, A. F., Theory of approximation of functions of a real variable, Pergamon Press, New-York, (1963).
3. Akgun, R., Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent, Ukrainian Mathematical Journal, 63, no. 1, 1–26, (2011).
4. Akgun, R., Ghorbanalizadeh, A., Approximation by integral functions of finite degree in variable exponent Lebesgue spaces on the real axis, Turkish Journal of Mathematics, 42, 1887-1903, (2018).
5. Guven, A., Israfilov, D. M., Trigonometric approximation in generalized Lebesgue spaces L^p(x) , Journal of Mathematical Inequalities, 4, no. 2, 285–299, (2010).
6. Guven, A., Trigonometric approximation by matrix transforms in L^p(x) spaces, Analysis and Applications (Singap.), 10, no.1, 47–65, (2012).
7. Israfilov, D. M., Testici, A., Approximation problems in the Lebesgue spaces with variable exponent, Journal of Mathematical Analysis and Applications, 459, no. 1, 112–123, (2018).
8. Sharapudinov, I. I., Approximation of functions in L_2π^p(x) by trigonometric polynomials. (Russian) Russian Academy of Sciences. Izvestiya Mathematics, 77, no. 2, 197–224, (2013); translation in Izvestiya Mathematics, 77, no. 2, 407–434, (2013).

9. Sharapudinov, I. I., Approximation of function from variable exponent Lebesgue and Sobolev spaces by Vallée Poussin means, (Russian) Sbornik: Mathematics 207, no. 7, 131–158, (2016); translation in Sbornik: Mathematics, 207, no. 7-8, 1010–1036, (2016).
10. Jafarov, S. Z., Approximation by means of Fourier trigonometric series in weighted Lebesgue spaces with variable exponent, The Aligarh Bulletin of Mathematics, 41, 63-80, (2022).
11. Jafarov, S. Z., Approximation by means of Fourier series in Lebesgue spaces with variable exponent, Kazakh Mathematical Journal, (3), 57-68, (2021).
12. Ditzian, Z. On Fejér and Bochner-Riesz means, Journal of Fourier Analysis and Applications, 11, no. 4, 489-496, (2005).
13. Cruz-Uribe, D. V., Fiorenza, A., Variable Lebesgue Spaces, Foundations and harmonic analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, (2013).
14. Hunt, R., Muckenhoupt, B., Wheeden, R., Weighted norm inequalities for the conjugate function and Hilbert transform, Transactions of the American Mathematical Society, 176, 227–251, (1973).
15. Guven, A., Kokilashvili, V., On the means of Fourier integrals and Bernstein inequality in the two-weighted setting, Positivity, 14, no. 1, 165-180, (2010).
16. Weisz, F., Convergence and summability of Fourier transforms and Hardy spaces, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, (2017).