Holder space | Wikipedia audio article
Channel: wikipedia tts
Date: 03/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/H%C3%B6lder_condition


00:01:42 1 Hölder spaces
00:01:47 2 Compact embedding of Hölder spaces
00:06:53 3 Examples
00:11:19 4 Properties
00:11:44 5 Notes
00:12:23 6 References
00:18:35 Properties



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SUMMARY
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In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α0, such that

  
    
      
        
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    {\displaystyle |f(x)-f(y)|\leq C\|x-y\|^{\alpha }}
  for all x and y in the domain of f.  More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the exponent of the Hölder condition.  A function on an interval satisfying the condition with α  1 is constant. If α = 1, then the function satisfies a Lipschitz condition. For any α  0, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder.
We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line

Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous ⊂  uniformly continuous  = continuouswhere 0  α ≤ 1.