A function \(f\) belongs to the class \(\operatorname{Lip}\alpha \), for some \(\alpha \in (0,1]\) if \[ \left| f(x)-f(y)\right| =O\left( \left| x-y\right| ^{\alpha }\right) \text{.} \] If instead of \(O\left( \left| x-y\right| ^{\alpha }\right) \) we have \(o\left( \left| x-y\right| ^{\alpha }\right) \) then \(f\) belongs to the class \(\operatorname{lip}\alpha \).
Considering the wavelet approximation of a function \(f\) the authors obtain several estimations for the best wavelet approximation of \(f\) with respect to the supremum norm and with respect to the \(L_{p}\) type norm in the cases when \(f\in \operatorname{Lip}\alpha \), respectively \(f\in \operatorname{lip}\alpha \). An even more general case is discussed, when \(f\in \operatorname{Lip}\left( \xi ,p\right) \) respectively \(f\in\operatorname{lip}\left( \xi ,p\right) \). Here, \(1\leq p<\infty \), \(\xi \) is a monotone increasing function and \(f\in\operatorname{Lip}\left( \xi ,p\right) \) if \[ \left\{ \frac{1}{2\pi }\int\limits_{0}^{2\pi }\left| f(x+t)-f(x)\right| ^{p}dx\right\} ^{1/p}=O(\xi (t)) \] and \(f\in \operatorname{lip}\left( \xi ,p\right) \) if \[ \left\{ \frac{1}{2\pi }\int\limits_{0}^{2\pi }\left| f(x+t)-f(x)\right| ^{p}dx\right\} ^{1/p}=o(\xi (t)). \]