Skip to main content
SpringerLink
Log in

Differentiation and integration of complex order of functions represented by trigonometrical series and generalized zeta-functions

  • Published:
Acta Mathematica Academiae Scientiarum Hungarica Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. H. Bohr, Über die Summabilität Dirichletscher Reihen,Gött. Nachr., (1909), pp. 247–262.

  2. L. S. Bosanquet, A solution of the Cesàro summability problem for successively derived Fourier series,Proc. London Math. Soc. (2),46 (1940), pp. 270–289.

    Google Scholar 

  3. L. S. Bosanquet, The Cesàro summability of the successively derived allied series of a Fourier series,Proc. London Math. Soc. (2),49 (1945), pp. 63–76.

    Google Scholar 

  4. H. S. M. Coxeter, The functions of Schläfli and Lobatschefsky,Quart. J. Math., Oxford Ser.,6 (1935), pp. 13–29.

    Google Scholar 

  5. G. Doetsch,Handbuch der Laplace-Transformation, Vol. I–III (Basel, 1950–56).

  6. L. Lejes Tóth, On the volume of a polyhedron in non-Euclidean space,Publ. Math. Debrecen,4 (1956), pp. 256–261.

    Google Scholar 

  7. J. Hadamard, Essai sur l'étude des fonctions données par leur développement de Taylor,Journal de Math. (4),8 (1892), pp. 101–186.

    Google Scholar 

  8. H. Hadwiger, Der Begriff der Ultrafunktion,Vierteljahrsschr. naturf. Ges. Zürich,92 (1947), pp. 31–42.

    Google Scholar 

  9. G. H. Hardy, The application of Abel's method of summation to Dirichlet's series,Quart. J. Math.,47 (1916), pp. 176–192.

    Google Scholar 

  10. G. H. Hardy,Divergent series (Oxford, 1949).

  11. G. H. Hardy andJ. E. Littlewood, Some properties of fractional integrals. I–II,Math. Zeitschrift,27 (1928), pp. 565–606;34 (1932). pp. 403–439.

    Google Scholar 

  12. G. H. Hardy andM. Riesz,The general theory of Dirichlet's series (Cambridge, 1915).

  13. G. H. Hardy andW. W. Rogosinski,Fourier series (Cambridge, 1944).

  14. E. Hille,Functional analysis and semigroups, Amer. Math. Soc. Coll. Publ. XXXI (1948).

  15. K. Knopp,Theorie und Anwendung der unendlichen Reihen,4. edition (Berlin-Heidelberg, 1950).

  16. M. Kuniyeda, Note on Perron's integral and summability-abscissae of Dirichlet's series,Quart. J. Math.,47 (1916), pp. 193–219.

    Google Scholar 

  17. J. Marcinkiewicz andA. Zygmund, On the behaviour of trigonometric series and power series,Trans. Amer. Math. Soc.,50 (1941), pp. 407–453.

    Google Scholar 

  18. M. Mikolás, Farey series and their connection with the prime number problem. I,Acta Sci. Math. Szeged,13 (1950), pp. 93–117.

    Google Scholar 

  19. M. Mikolás, Mellinsche Transformation und Orthogonalität bei ζ(s, u); Verallgemeinerung der Riemannschen Funktionalgleichung von ζ(s),Acta Sci. Math. Szeged,17 (1956), pp. 143–164.

    Google Scholar 

  20. M. Mikolás, Integral formulae of arithmetical characteristics relating to the zeta-function of Hurwitz,Publ. Math. Debrecen,5 (1957), pp. 44–53.

    Google Scholar 

  21. M. Mikolás, A simple proof of the functional equation for the Riemann zeta-function and a formula of Hurwitz,Acta Sci. Math. Szeged,18 (1957), pp. 261–263.

    Google Scholar 

  22. M. Mikolás, Über die Charakterisierung der Hurwitzschen Zetafunktion mittels Funktionalgleichungen,Acta Sci. Math. Szeged. 19 (1958), pp. 247–250.

    Google Scholar 

  23. M. Mikolás, On a problem of Hardy and Littlewood in the theory of diophantine approximations,Publ. Math. Debrecen (under press).

  24. M. L. Misra, The summability (A) of the successively derived series of a Fourier series and its conjugate series,Duke Math. Journal,14 (1947), pp. 167–177.

    Google Scholar 

  25. N. Obreschkoff, Über dieC-Summierbarkeit der derivierten Reihen der Fourierschen Reihen,Abh. Preuss. Akad. Wiss. Math.-Nat. Kl., (1941), No. 15, pp. 1–28.

    Google Scholar 

  26. W. F. Osgood,Lehrbuch der Funktionentheorie, Vol. I, 2. edition (Leipzig-Berlin, 1912).

  27. A. Rényi, On the summability of Cauchy—Fourier series,Publ. Math. Debrecen,1 (1950), pp. 162–164.

    Google Scholar 

  28. M. Riesz, Sur la représentation analytique des fonctions définies par des séries de Dirichlet,Acta Math.,35 (1912), pp. 253–270.

    Google Scholar 

  29. M. Riesz, L'intégrale de Riemann—Liouville et le problème de Cauchy,Acta Math.,81 (1949), pp. 1–223.

    Google Scholar 

  30. E. C. Titchmarsh, Principal value Fourier series,Proc. London Math. Soc. (2),23 (1925), pp. 41–43.

    Google Scholar 

  31. E. C. Titchmarsh,The theory of the Riemann zeta-function (Oxford, 1951).

  32. H. Weyl, Bemerkungen zum Begriff des Differentialkoeffizienten gebrochener Ordnung,Vierteljahrsschr. naturf. Ges. Zürich,62 (1917), pp. 296–302.

    Google Scholar 

  33. E. T. Whittaker andG. N. Watson,A course of modern analysis, 4. edition (Cambridge, 1927).

  34. M Zamansky, Sur la sommation des séries de Fourier dérivées,Comptes Rendus Acad. Sci. Paris,231 (1950), pp. 1118–1120.

    Google Scholar 

  35. A. Zygmund,Trigonometrical series, 2. edition (New York, 1952).

Download references

Authors
  1. M. Mikolás
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

A lecture on the main results of this work has been delivered by the author at theInternational Congress of Math. in Edinburgh (14–21 August 1958).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mikolás, M. Differentiation and integration of complex order of functions represented by trigonometrical series and generalized zeta-functions. Acta Mathematica Academiae Scientiarum Hungaricae 10, 77–124 (1959). https://doi.org/10.1007/BF02063292

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02063292

Keywords

Search

Navigation