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$$\alpha$$-variation and transformation into $$C^ n$$ functions. (English) Zbl 0634.26006
Connections are investigated between boundedness of the generalized variation in the sense of L. C. Young, and Lipschitz or differentiability conditions for a transformation of this function. For example, if $$s>0,$$ then a continuous function f on an interval $$[a,b]$$ is of bounded (1/s)- variation if and only if there is a homeomorphism $$\Phi$$ of $$[a,b]$$ onto itself such that $$f\circ \Phi \in Lip s.$$ Here, if $$k<s\leq k+1,$$ k a positive integer, then $$f\in Lip s$$ means that there exists the derivative $$f^{(k)}\in Lip(s-k).$$ The authors also give similar conditions for f to be of strongly bounded $$(1/s)$$-variation.
##### MSC:
 26A45 Functions of bounded variation, generalizations 26A16 Lipschitz (Hölder) classes