Modulus of smoothness

In mathematics, moduli of smoothness are used to quantitatively measure smoothness of functions. Moduli of smoothness generalise modulus of continuity and are used in approximation theory and numerical analysis to estimate errors of approximation by polynomials and splines.

The modulus of smoothness of order ${\displaystyle n}$ of a function ${\displaystyle f\in C[a,b]}$ is the function ${\displaystyle \omega _{n}:[0,\infty )\to \mathbb {R} }$ defined by

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