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On the generalised jump of a function by its Fourier coefficients

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Summary

Author studies the summability (C,1+α+ρ) of the sequence nBn(x) under weaker conditions than those ofMinakishisundaram [3] and thus generalises his theorems. on the « Jump of a function » and by applying a tauberian theorem obtaing a criteria for the (C, α+ρ) summability of the conjugate series.

References

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    Bosanquet andHyslop,On the absolute summability of the allied Series of a Fourier Series, « Mathematical Zeitschuft », 32 (1937), pp. 489–512.

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    Chow, H. C.,A Further note on a theorem of O. Szaz, « Jonrnal of London Mathematical Society », Vol. 17 (1942), pp. 16–18.

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    Minakshisundaram, S. A note on the Theory of Fourier Series, « Proceedings of the National Institute of Sciences, India », 10 (1944), pp. 235–245.

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    Prasad, B. N.,On the Summability of allied Series of Fourier Series and allied Series, « Journal of London Mathematical Society », (1931), Vol. 6, pp. 274–278.

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    Sulaxana Kumari,On the Generalised Jump of a Function by its Fourier Coefficients, « Proceedings of the National Institute of Sciences of India », 24 (1958), pp. 204–216.

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    Zygmund,Trigonometric Series, Warsaw, 1950.

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    Hardy, G. H. andLittlewood, J. E.,Notes on the theory of Series XVI: Two Tauberian Theorems, « Journ. Lond. Math. Soc. », 6, pp. 281–286.

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    Hobson, E. W.,The Theory of Functions of a Real Variable, (1950), Vol. II, Cambridge Univ. Press.

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Shrivastava, K.C. On the generalised jump of a function by its Fourier coefficients. Annali di Matematica 52, 349–362 (1960). https://doi.org/10.1007/BF02415679

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Keywords

  • Fourier Coefficient
  • Weak Condition
  • Author Study
  • Tauberian Theorem
  • Conjugate Series