Now we present a very useful property of the modulus of continuity.
A function [omega] of a modulus of continuity type on the interval [0,2[pi]] satisfies the condition [mathematical expression not reproducible]
Alternately, the complete modulus of continuity
of fi which we denote by [omega](f; [delta]) is defined as
where [omega]([d.sup.r]f/[dx.sup.r],) is the modulus of continuity
Blasco, "Modulus of continuity
with respect to semigroups of analytic functions and applications," Journal of Mathematical Analysis and Applications, vol.
Now use properties of modulus of continuity
then we have
The modulus of continuity
has the following properties (, p.
Lemma 4.7 Let X have a density [f.sub.X] with modulus of continuity
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and let ([X.sub.n]) be defined by (5).
An abstract modulus of continuity
, defined by the cosine operator function, will play an important role in our paper.
The full modulus of continuity
of f is defined as follows:
For f [member of] C[0, 1], [delta] > 0, we define the modulus of continuity
[omega](f, [delta]) as follows:
MODULUS OF CONTINUITY
, THE BEST APPROXIMATIONS, AND BLACKMAN- AND ROGOSINSKI-TYPE OPERATORS