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Zygmund class of functions

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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
How to Cite This Entry:
Zygmund class of functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zygmund_class_of_functions&oldid=49251
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article