Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the degree of approximation of a function by the partial sums of its Fourier series

Author: Elaine Cohen
Journal: Trans. Amer. Math. Soc. 235 (1978), 35-74
MSC: Primary 42A08
MathSciNet review: 0461004
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: When f is a $2\pi$ periodic function with rth order fractional derivative, $r \geqslant 0$, of p-bounded variation, Golubov has obtained estimates of the degree of approximation of f, in the ${L^q}$ norm, $q > p$, by the partial sums of its Fourier series. Here we consider the analogous problem for functions whose fractional derivatives are of $\Phi$-bounded variation and obtain estimates of the degree of approximation in an Orlicz space norm. In a similar manner we shall extend various results that he obtained on degree of approximation in the sup norm.

References [Enhancements On Off] (What's this?)

  • N. K. Bari, Trigonometricheskie ryady, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961 (Russian). With the editorial collaboration of P. L. Ul′janov. MR 0126115
  • B. I. Golubov, Continuous functions of bounded $p$-variation, Mat. Zametki 1 (1967), 305–312 (Russian). MR 211181
  • B. I. Golubov, Functions of bounded $p$-variation, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 837–858 (Russian). MR 0235080
  • M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961. Translated from the first Russian edition by Leo F. Boron. MR 0126722
  • E. R. Love, A generalization of absolute continuity, J. London Math. Soc. 26 (1951), 1–13. MR 39791, DOI
  • J. Musielak and W. Orlicz, On generalized variations. I, Studia Math. 18 (1959), 11–41. MR 104771, DOI
  • B. Sz.-Nagy, Über gewisse Extremalfragen bei transformierfen trigonometrischen Entwicklungen. I. Periodisches Fall, Berichten Math.-Phys. Kl. Akad. Wiss. Leipzig 90 (1938), 103-134.
  • S. M. Nikol′skiĭ, Fourier series of functions having a derivative of bounded variation, Izvestiya Akad. Nauk SSSR. Ser. Mat. 13 (1949), 513–532 (Russian). MR 0032803
  • W. Pinkewitch, Sur l’ordre du reste de la série de Fourier des fonctions dérivables au sens de Weyl, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4 (1940), 521–528 (Russian, with French summary). MR 0003835
  • Robert Ryan, Conjugate functions in Orlicz spaces, Pacific J. Math. 13 (1963), 1371–1377. MR 157183
  • R. Salem, Essais sur les séries trigonométriques, Thèse présentée à la Faculté des Sciences de l’Université de Paris, Hermann, Paris, 1940. MR 2, 93, 419. N. Wiener, The quadratic variation of a function and its Fourier coefficients, J. Math. and Phys. 3 (1924), 72-94.
  • L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), no. 1, 251–282. MR 1555421, DOI
  • A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42A08

Retrieve articles in all journals with MSC: 42A08

Additional Information

Keywords: Orlicz class, Orlicz space, <!– MATH ${\Delta _2}$ –> <IMG WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img3.gif" ALT="${\Delta _2}$">-condition, <IMG WIDTH="28" HEIGHT="22" ALIGN="BOTTOM" BORDER="0" SRC="images/img9.gif" ALT="$\Delta ’$">-condition, <IMG WIDTH="20" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\Phi$">-modulus of continuity, <IMG WIDTH="20" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$\Phi$">-variation, fractional integral, fractional derivative, degree of approximation
Article copyright: © Copyright 1978 American Mathematical Society