# Bibliographies on *p*-variation and ϕ-variation

## Abstract

- (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)
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 max_{j}(*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## Preview

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