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Bibliographies on p-variation and ϕ-variation

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1703)

Abstract

This is a pair of annotated reference lists, including all items the authors could find, on

  1. (1)

    p-variation of real-valued functions f as defined by Wiener in 1924 and developed by L. C. Young and E. R. Love in the late 1930's and others since then. Usually f is defined on an interval, but some papers give extensions to multidimensional domains;

  2. (2)

    ϕ-variation, namely the supremum of all sums ∑ i φ(|Δ i f), where Δ i f:=f(x i )-f(x i-1), φ is a continuous, increasing function, 0 at 0, and x 0<x 1<...<x n , n=1,2,.... Thus ϕ(y)=y p gives p-variation.

Not included, however, are works on: (a) “quadratic variation” as studied in probability theory and defined as a limit along a sequence of partitions {x j } with mesh maxj(x j −x j−1 )→0, at some rate, or where the sums converge only in probability; (b) the special case p=1 of ordinary bounded variation; or (c) sequence spaces, called James spaces.

Keywords

  • Fourier Series
  • Fourier Coefficient
  • Sample Path
  • Fourier Multiplier
  • Sample Function

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1999 Springer-Verlag

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Dudley, R.M., Norvaiša, R., Qian, J. (1999). Bibliographies on p-variation and ϕ-variation. In: Differentiability of Six Operators on Nonsmooth Functions and p-Variation. Lecture Notes in Mathematics, vol 1703. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0100748

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  • DOI: https://doi.org/10.1007/BFb0100748

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65975-4

  • Online ISBN: 978-3-540-48814-9

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