P-variation

In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number p ≥ 1 {displaystyle pgeq 1} . p-variation is a measure of the regularity or smoothness of a function. Specifically, if f : I → ( M , d ) {displaystyle f:I o (M,d)} , where ( M , d ) {displaystyle (M,d)} is a metric space and I a totally ordered set, its p-variation is In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number p ≥ 1 {displaystyle pgeq 1} . p-variation is a measure of the regularity or smoothness of a function. Specifically, if f : I → ( M , d ) {displaystyle f:I o (M,d)} , where ( M , d ) {displaystyle (M,d)} is a metric space and I a totally ordered set, its p-variation is where D ranges over all finite partitions of the interval I. The p variation of a function decreases with p. If f has finite p-variation and g is an α-Hölder continuous function, then g ∘ f {displaystyle gcirc f} has finite p α {displaystyle {frac {p}{alpha }}} -variation. The case when p is one is called total variation, and functions with a finite 1-variation are called bounded variation functions. One can interpret the p-variation as a parameter-independent version of the Hölder norm, which also extends to discontinuous functions. If f is α–Hölder continuous (i.e. its α–Hölder norm is finite) then its 1 α {displaystyle {frac {1}{alpha }}} -variation is finite. Specifically, on an interval , ‖ f ‖ 1 α -var ≤ ‖ f ‖ α ( b − a ) α {displaystyle |f|_{{frac {1}{alpha }}{ ext{-var}}}leq |f|_{alpha }(b-a)^{alpha }} . Conversely, if f is continuous and has finite p-variation, there exists a reparameterisation, τ {displaystyle au } , such that f ∘ τ {displaystyle fcirc au } is 1 / p − {displaystyle 1/p-} Hölder continuous. If p is less than q then the space of functions of finite p-variation on a compact set is continuously embedded with norm 1 into those of finite q-variation. I.e. ‖ f ‖ q -var ≤ ‖ f ‖ p -var {displaystyle |f|_{q{ ext{-var}}}leq |f|_{p{ ext{-var}}}} . However unlike the analogous situation with Hölder spaces the embedding is not compact. For example, consider the real functions on given by f n ( x ) = x n {displaystyle f_{n}(x)=x^{n}} . They are uniformly bounded in 1-variation and converge pointwise to a discontinuous function f but this not only is not a convergence in p-variation for any p but also is not uniform convergence. If f and g are functions from   to ℝ with no common discontinuities and with f having finite p-variation and g having finite q-variation, with 1 p + 1 q > 1 {displaystyle {frac {1}{p}}+{frac {1}{q}}>1} then the Riemann–Stieltjes Integral is well-defined. This integral is known as the Young integral because it comes from Young (1936). The value of this definite integral is bounded by the Young-Loève estimate as follows where C is a constant which only depends on p and q and ξ is any number between a and b.If f and g are continuous, the indefinite integral F ( w ) = ∫ a w f ( x ) d g ( x ) {displaystyle F(w)=int _{a}^{w}f(x),dg(x)} is a continuous function with finite q-variation: If a ≤ s ≤ t ≤ b then ‖ F ‖ q -var ; [ s , t ] {displaystyle |F|_{q{ ext{-var}};}} , its q-variation on , is bounded by C ‖ g ‖ q -var ; [ s , t ] ( ‖ f ‖ p -var ; [ s , t ] + ‖ f ‖ ∞ ; [ s , t ] ) ≤ 2 C ‖ g ‖ q -var ; [ s , t ] ( ‖ f ‖ p -var ; [ a , b ] + f ( a ) ) {displaystyle C|g|_{q{ ext{-var}};}(|f|_{p{ ext{-var}};}+|f|_{infty ;})leq 2C|g|_{q{ ext{-var}};}(|f|_{p{ ext{-var}};}+f(a))} where C is a constant which only depends on p and q.