Zygmund class of functions

Let be a positive real number. The Zygmund class is the class of continuous -periodic functions with the property that for all and all the inequality

holds. The class 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 -periodic function belongs to the Zygmund class for some if and only if its best uniform approximation error by trigonometric polynomials of degree satisfies the inequality

where is a constant. The modulus of continuity of any function admits the estimate

in which the constant cannot be improved on for the entire class [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

for a -periodic function , is its Zygmund modulus. In terms of this, Zygmund's theorem above can also be stated as: A -periodic function satisfies for some if and only if () for some .

References

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