Bibliographies on p-variation and ϕ-variation

Abstract

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

• (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.