Bibliographies on p-variation and ϕ-variation

  • R. M. Dudley
  • R. Norvaiša
  • Jinghua Qian
Part of the Lecture Notes in Mathematics book series (LNM, volume 1703)


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(xi-1), φ is a continuous, increasing function, 0 at 0, and x0<x1<...<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.


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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Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • R. M. Dudley
  • R. Norvaiša
  • Jinghua Qian

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