Local properties of fourier series via deferred Riesz mean
DOI:
https://doi.org/10.5269/bspm.62309Abstract
The convergence of Fourier series of a function at a point depends upon the behaviour of the function in the neighborhood of that point, and it leads to the local property of Fourier series. In the proposed work, we introduce and study the absolute convergence of the deferred Riesz summability mean, and accordingly establish a new theorem on the local property of a factored Fourier series. We also suggest a direction for future researches on this subject, which are based upon the local properties of the Fourier series via basic notions of statistical absolute convergence.
References
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2. Braha, N. L. and Loku, V, Estimation of the rate of convergence of Fourier series in the generalized H¨older metric by deferred de la Vallee Poussin mean, J. Inequal. Spec. Funct. 9, 122–128, (2018).
3. Braha, N. L., Loku, V. and Srivastava, H. M., Λ2 -Weighted statistical convergence and Korovkin and Voronovskaya type theorems, Appl. Math. Comput. 266, 675–686, (2015).
4. Bor, H., Local property of |N, pn|k-summability of factored Fourier series, Bull. Inst. Math. Acad. Sinica 17, 165–170, (1989).
5. Bor, H., On the local property of |N¯, pn|k-summability of factored Fourier series, J. Math. Anal. 163, 220–226, (1992).
6. Jena, B. B. and Paikray, S. K., Product of deferred Ces`aro and deferred weighted statistical probability convergence and its applications to Korovkin-type theorems, Univ. Sci. 25, 409–433 (2020).
7. Jena, B. B., Paikray, S. K. and Dutta, H., A new approach to Korovkin-type approximation via deferred Cesaro statistical measurable convergence, Chaos, Solitons & Fractals 148, Article ID 111016, 1–9, (2021).
8. Jena, B. B., Paikray, S. K. and Dutta, H., On various new concepts of statistical convergence for sequences of random variables via deferred Ces`aro mean, J. Math. Anal. Appl. 487, Article ID 123950, 1–18, (2020).
9. Jena, B. B., Vandana, Paikray, S. K. and Misra, U. K. On generalized local property of |A; δ|k-summability of factored Fourier series, Int. J. Anal. Appl. 16, 2019–221, (2018).
10. Jena, B. B., Paikray, S. K. and Misra, U. K., Double absolute indexed matrix summability with its applications, Tbilisi Math. J. 11, 1–18, (2018).
11. Krasniqi, X. Z., Lenski, W. and Szal, B., Approximation of integrable functions by generalized de la Vallee Poussin means of the positive order, J. Appl. Anal. Comput. 12, 106–124 (2022).
12. Matsumoto, K., Local property of the summability |R, pn, 1|, Tohoku Math. J 8, 114–124, (1956).
13. Mazhar, S. M., On the summability factors of infinite series, Publ. Math. Debrecen 13, 229–236, (1966).
14. Mishra, K. N., Multipliers for |N, pn| -summability of Fourier series, Bull.Inst. Math. Acad. Sinica 14, 431–438, (1984).
15. Mohanty, R., On the summability |R, logw, 1| of Fourier series, J. London Math. soc. 25, 67–72, (1950).
16. Paikray, S. K., Jati, R. K., Misra, U. K. and Sahoo, N. C., On degree approximation of Fourier series by product means, Gen. Math. Notes 13, 22–30, (2012).
17. Parida, P., Paikray, S. K. and Jena, B. B., Generalized deferred Ces`aro equi-statistical convergence and analogous approximation theorems, Proyecciones J. Math. 39, 307–331, (2020).
18. Tichmarsh, E. C., The Theory of Functions, Oxford University Press, London, (1961).
19. Zygmund, A., Trigonometric series, vol. I, Cambridge Univ. Press, Cambridge, (1959).
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2022-12-28
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