r,ξ(t)) class by product summability method"> r,ξ(t)) class of functions; (E,1) means; (C,1) means; (E,1)(C,1) product means; Fourier series; Lebesgue integral.">

Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 5 (2010), 113 -- 122

DEGREE OF APPROXIMATION OF FUNCTIONS BELONGING TO LIPα CLASS AND WEIGHTED (Lr,ξ(t)) CLASS BY PRODUCT SUMMABILITY METHOD

Hare Krishna Nigam

Abstract. A good amount of work has been done on degree of approximation of functions belonging to Lipα, Lip(α,r), Lip(ξ(t),r) and W(Lr, ξ(t)) classes using Cesàro and (generalized) Nörlund single summability methods by a number of researchers like Alexits [1], Sahney and Goel [11], Qureshi and Neha [9], Quershi [7, 8], Chandra [2], Khan [4], Leindler [5] and Rhoades [10]. But till now no work seems to have been done so far in the direction of present work. Therefore, in present paper, two quite new results on degree of approximation of functions f∈ Lipα and f∈ W(Lr,ξ(t)) class by (E,1)(C,1) product summability means of Fourier series have been obtained.

2010 Mathematics Subject Classification: 42B05, 42B08.
Keywords: Degree of approximation; Lipα class; W(Lr,ξ(t)) class of functions; (E,1) means; (C,1) means; (E,1)(C,1) product means; Fourier series; Lebesgue integral.

Full text

References

  1. G. Alexits, Convergence problems of orthogonal series. Translated from German by I Földer. International series of Monograms in Pure and Applied Mathematics, vol. 20, Pergamon Press, New York-Oxford-Paris, 1961. MR0218827(36:1911).

  2. P. Chandra, Trigonometric approximation of functions in Lp norm, J. Math. Anal. Appl. 275 (2002), No. 1, 13-26. MR1941769(2003h:42003). Zbl 1011.42001.

  3. G. H. Hardy, Divergent series, first edition, Oxford University Press, 1949, 70.

  4. H. H. Khan, On degree of approximation of functions belonging to the class Lip(α ,p), Indian J. Pure Appl. Math., 5 (1974), no. 2, 132-136. MR0380230(52 :1130). Zbl 0308.41010.

  5. L. Leindler, Trigonometric approximation in Lp norm, J. Math. Anal. Appl., 302 (2005). MR2107350(2005g:42004). Zbl 1057.42004.

  6. L. McFadden, Absolute Nörlund summability, Duke Math. J., 9 (1942), 168-207. MR0006379(3,295g). Zbl 0061.12106.

  7. K. Qureshi, On the degree of approximation of a periodic function f by almost Nörlund means, Tamkang J. Math., 12 (1981), No. 1, 35-38. MR0714492(85h:42002). Zbl 0502.42002.

  8. K. Qureshi, On the degree of approximation of a function belonging to the class Lipα , Indian J. pure Appl. Math., 13 (1982), no. 8, 898-903. MR0670332(84c:41013). Zbl 0498.42002.

  9. K. Qureshi and H. K. Neha, A class of functions and their degree of approximation, Ganita, 41 (1990), no. 1, 37-42. MR1155949. Zbl 0856.42005.

  10. B. E. Rhaodes, On the degree of approximation of functions belonging to Lipschitz class by Hausdorff means of its Fourier series, Tamkang Journal of Mathematics, 34 (2003), no. 3, 245-247. MR2001920(2004g:41023). Zbl 1039.42001.

  11. B. N. Sahney and D. S. Goel, On the degree of continuous functions, Ranchi University Math. Jour., 4 (1973), 50-53. MR0352846(50:5332). Zbl 0296.41008.

  12. E. C. Titchmarsh, The Theory of functions, Oxford University Press, 1939, 402-403.

  13. A. Zygmund, Trigonometric series, 2nd rev. ed., Vol. 1, Cambridge Univ. Press, Cambridge, New York, 1959. MR0107776(21:6498). Zbl 0085.05601.





Hare Krishna Nigam
Department of Mathematics,
Faculty of Engineering and Technology,
Mody Institute of Technology and Science (Deemed University)
Laxmangarh, Sikar-332311,
Rajasthan, India.
e-mail: harekrishnan@yahoo.com


http://www.utgjiu.ro/math/sma