Abstract:
We consider the approximation of a special class of complex functions
f 
that is analytic on the disk
\textU = \left\ z \in \rmC:\dfrac12 < \left| z \right| < 1 \right\ 
, whose origin is its essential singularity. We obtain the exact Jackson inequality between the best approximation
E_n - 1\left( f \right)_2 
of functions
f 
and
m-order continuous modules of functions
z^rf^\left( r \right) 
. We also obtain the exact Jackson inequality between the best approximation
E_n - 1\left( f \right)_2 
of function
f 
and
\calK 
-functional. Then we obtain the exact Jackson inequality between the best approximation
E_n - 1\left( f \right)_2 
of functions
f 
and weighted integral of
m-order continuous modules of functions
z^rf^\left( r \right) 
. Finally, we obtain the best approximation and n-widths in the functions classes of
m-order continuous modules of functions
z^rf^\left( r \right) 
, in the functions classes of weighted integral of
m-order continuous modules of functions
z^rf^\left( r \right) 
and in the functions classes of
\calK 
-functional respectively.