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 ![]() |
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 |
Zygmund class of functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zygmund_class_of_functions&oldid=49251