Zygmund class of functions

Let $M$ be a positive real number. The Zygmund class $Z _ {M}$ is the class of continuous $2 \pi$- periodic functions $f$ with the property that for all $x$ and all $h > 0$ the inequality

$$| f ( x + h ) - 2f ( x) + f ( x - h ) | \leq M h$$

holds. The class $Z _ {M}$ was introduced by A. Zygmund [1]. In terms of this class one can obtain a conclusive solution to the Jackson–Bernstein problem on direct and inverse theorems in the theory of approximation of functions (cf. Bernstein theorem; Jackson theorem). For example: A continuous $2 \pi$- periodic function $f$ belongs to the Zygmund class $Z _ {M}$ for some $M > 0$ if and only if its best uniform approximation error $E _ {n} ( f )$ by trigonometric polynomials of degree $\leq n$ satisfies the inequality

$$E _ {n} ( f ) \leq \frac{A}{n} ,$$

where $A > 0$ is a constant. The modulus of continuity $\omega ( \delta , f )$ of any function $f \in Z _ {M}$ admits the estimate

$$\omega ( \delta , f ) \leq \frac{M}{2 \mathop{\rm ln} \sqrt {2 } + 1 } \delta \mathop{\rm ln} \frac \pi \delta + O ( \delta )$$

in which the constant $M / 2 \mathop{\rm ln} ( \sqrt {2 } + 1 )$ cannot be improved on for the entire class $Z _ {M}$[3].

References

[1]	A. Zygmund, "Smooth functions" Duke Math. J. , 12 : 1 (1945) pp. 47–76 ((Also: Selected papers of Antoni Zygmund, Vol. 2, Kluwer, 1989, pp. 184–213.))
[2]	S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[3]	A.V. Efimov, "Estimation of the modules of continuity of functions of class " Izv. Akad. Nauk. SSSR Ser. Mat. , 21 : 2 (1957) pp. 283–288 (In Russian)

Comments

The quantity

$$\omega _ {f} ^ {*} ( h) = \sup _ { x } \sup _ {| \delta | \leq n } \ | f( x+ \delta ) - 2f( x) + f( x- \delta ) | ,$$

for a $2 \pi$- periodic function $f$, is its Zygmund modulus. In terms of this, Zygmund's theorem above can also be stated as: A $2 \pi$- periodic function $f$ satisfies $E _ {n} ( f ) \leq n ^ {-} 1 A$ for some $A$ if and only if $\omega _ {f} ^ {*} ( h ) \leq Bh$( $h > 0$) for some $B$.

References

[a1]	E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff