Approximation of Conjugate Functions by General Linear Operators of Their Fourier Series at the Lebesgue Points

Włodzimierz Łenski 1  and Bogdan Szal 1
  • 1 UNIVERSITY OF ZIELONA GÓRA FACULTY OF MATHEMATICS, COMPUTER SCIENCE AND ECONOMETRICS ul. Szafrana 4a 65-516 ZIELONA GÓRA, POLAND

Abstract

The pointwise estimates of the deviations r T͂n,A,Bf (·) - f͂͂ (·) and T͂n,A,Bf (·) - f͂͂ (·,ε) in terms of moduli of continuity ω̃f and r ω̃f are proved. Analogical results on norm approximation with remarks and corollary are also given. These results generalized a theorem of Mittal [3, Theorem 1, p. 437].

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  • [1] S. Aljancic, R. Bojanic, M. Tomic, On the degree of convergence of Fejér-Lebesgue sums, Enseign. Math., Geneva, 15 (1969), 21-28.

  • [2] W. Łenski, B. Szal, Approximation of integrable functions by general linear operators of their Fourier series at the Lebesgue points, Acta Math. Hungar. 131(4) (2011), 380-394.

  • [3] M. L. Mittal, A sufficient condition for pF1q-effectiveness of the C1T-method, J. Math. Anal. Appl. 220 (1998), 434-450. Article no. AY975781

  • [4] M. L. Mittal, B. E. Rhoades, V. N. Mishra, Approximation of signals (functions) belonging to the weighted W(Lp, ζ(t)); (p ≥ 1) -class by linear operators, Int. J. Math. Math. Sci., Vol. 2006, (2006), Article ID: 53538.

  • [5] A. Zygmund, Trigonometric Series, Cambridge, 2002.

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