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Holder spaces. https://math.ucdenver.edu/~jmandel/classes/7760f05/spaces.pdf https://math.ucdenver.edu/~jmandel/classes/7760f05/spaces.pd... """ Let Ω be an

by rrmm 5y ago

Holder spaces. https://math.ucdenver.edu/~jmandel/classes/7760f05/spaces.pdf https://math.ucdenver.edu/~jmandel/classes/7760f05/spaces.pd... """ Let Ω be an open set in Rn, 0<α≤1, and k a nonnegative integer. The (uniform) Holder spaces Ck,α(Ω) consist of functions whose k−th order derivatives are uniformly Holder continuous with exponent α in Ω. The(local) Holder spaces Ck,α(Ω) consist of functions whose k−th order derivatives are locally Holder continuous with exponent α """ The Holder continuity condition looks sorta like the limit definition of a derivative with an exponent on the denominator. Also: https://en.wikipedia.org/wiki/H%C3%B6lder_condition https://en.wikipedia.org/wiki/H%C3%B6lder_condition

vitus 5y ago
> The Holder continuity condition looks sorta like the limit definition of a derivative with an exponent on the denominator. This is actually stronger than the epsilon-delta definition of continuity, as noted in the wiki article -- "For any α > 0, the condition implies the function is uniformly continuous." Normally, you'd have something like For all ϵ > 0, there exists δ(x, ϵ) > 0 such that if |x - x₀| < δ, then |f(x) - f(x₀)| < ϵ. (namely, at any given point, there's a ball of nearby points such that all their corresponding outputs are close together.) But, with absolute continuity, δ is purely a function of ϵ, namely, that ball doesn't change size even as we move further away. For instance, f(x)=x is absolutely continuous, while f(x)=x^2 is not. (for the former case, we can use δ = ϵ/2; for the latter case, we'd need a δ that looks something like 2ϵ·x) In this case, since the Holder condition depends only on the distance between x and y, it automatically implies absolute continuity.
- jbrot 5y ago
  Small correction: the definition you shared is that of “uniform continuity” [1] not “absolute continuity” [2]. Absolute continuity is a slightly stronger condition where instead of providing a single (x1, x2) pair with |x1 - x2| < delta, you can now provide a finite list of pairs. [1] https://en.m.wikipedia.org/wiki/Uniform_continuity https://en.m.wikipedia.org/wiki/Uniform_continuity [2] https://en.m.wikipedia.org/wiki/Absolute_continuity https://en.m.wikipedia.org/wiki/Absolute_continuity
  - vitus 5y ago
    Ah, good catch! The wiki quote that I cited does reference uniform continuity, so the rest still holds. (left for posterity so your comment makes sense)
Ericson2314 5y ago
Thank you!! I guess in hindsight they did talk about the exponents and the iterated differentiabilty, but I didn't make the connection. I think this was at the end of the analysis textbook too, in one of those "looking ahead future topics" sections.

by rrmm 5y ago

vitus 5y ago

> The Holder continuity condition looks sorta like the limit definition of a derivative with an exponent on the denominator. This is actually stronger than the epsilon-delta definition of continuity, as noted in the wiki article -- "For any α > 0, the condition implies the function is uniformly continuous." Normally, you'd have something like For all ϵ > 0, there exists δ(x, ϵ) > 0 such that if |x - x₀| < δ, then |f(x) - f(x₀)| < ϵ. (namely, at any given point, there's a ball of nearby points such that all their corresponding outputs are close together.) But, with absolute continuity, δ is purely a function of ϵ, namely, that ball doesn't change size even as we move further away. For instance, f(x)=x is absolutely continuous, while f(x)=x^2 is not. (for the former case, we can use δ = ϵ/2; for the latter case, we'd need a δ that looks something like 2ϵ·x) In this case, since the Holder condition depends only on the distance between x and y, it automatically implies absolute continuity.

jbrot 5y ago
Small correction: the definition you shared is that of “uniform continuity” [1] not “absolute continuity” [2]. Absolute continuity is a slightly stronger condition where instead of providing a single (x1, x2) pair with |x1 - x2| < delta, you can now provide a finite list of pairs. [1] https://en.m.wikipedia.org/wiki/Uniform_continuity https://en.m.wikipedia.org/wiki/Uniform_continuity [2] https://en.m.wikipedia.org/wiki/Absolute_continuity https://en.m.wikipedia.org/wiki/Absolute_continuity
- vitus 5y ago
  Ah, good catch! The wiki quote that I cited does reference uniform continuity, so the rest still holds. (left for posterity so your comment makes sense)

Ericson2314 5y ago

Thank you!! I guess in hindsight they did talk about the exponents and the iterated differentiabilty, but I didn't make the connection. I think this was at the end of the analysis textbook too, in one of those "looking ahead future topics" sections.