Lipschitz class

Summary

The Lipschitz class $\mathrm{Lip}\,\alpha$, for $0 < \alpha \le 1$, is the set of functions on an interval that satisfy $\abs{f(x) - f(y)} \le M \abs{x - y}^\alpha$ for some constant $M$. For $\alpha = 1$ these are the Lipschitz continuous functions, and for $\alpha < 1$ the Hölder continuous functions of exponent $\alpha$. In approximation theory these classes are the standard scale of smoothness. For $0 < \alpha < 1$, a continuous $2\pi$-periodic function is in $\mathrm{Lip}\,\alpha$ exactly when its best trigonometric approximation of degree $n$ has error of order $n^{-\alpha}$.