The Best Approximation Problems Of A Specific Class Of Complex Functions Under L₂ Norm

We consider the best approximation of a special class of complex functions that are analytic on the annulus U={zz?C?1/2<|z|<1?,where the origin is their essential singularity.We obtain the exact Jackson inequality between the best approximation En+s-1(f(s))2 and m-order continuous modules of functions f(r)and the similar inequality between the best approximation En+s-1(f(s))2 and K-functional.The best approximation of classes of functions restricted by the continuous modules and by K-functional respectively and the exact Jackson inequality between the best approximation En-1(f)2 and m-order continuous modules of zrf(r)are studied.Subsequently,the exact Jackson inequality between the best approximation En-1(f)2 and K-functional and exact Jackson inequality between the best approximation En-1(f)2 and E-functional have been got.With a weight functions q(t),we research the exact Jackson inequality between the best approximation En-1(f)2 and weighted integral of m-order continuous modules of zrf(r)Further,we have got the best approximation and n-widths in the classes of complex functions restricted by m-order continuous modules of zrf(r),by weighted integral of m-order continuous of zrf(r),by K-functional and by E-functional respectively.Finally,we do research in the exact Jackson inequality between the best approximation R(n-1,n-1)(f)2 and K-functional or between E(n-1,n-1)(f)2 and E-functional.The approximation problems in the class of complex functions restricted by K-functional and by E-functional respectively are solved partly as follow.The research in this paper is inspired by the results of M.Sh.Shabozov's research on the exact Jackson inequality in the best approximation of the analytic complex function defined on the unit disc.We change the domain of the complex function from a disk to an annulus.This change makes the expansion of the analytic functions at the origin from a Taylor series without negative powers to a Laurent series.Firstly,we consider the case where the Laurent series has only negative power term expansions,adjust the translation operator,define the corresponding continuous modules,K-functional,etc.,and obtains some exact constants of inequalities and approximation results of the class of complex functions.Then we put our hands to the general situation.Because the translation operator in the negative power terms and the positive power terms encountered certain difficulties in the problem of unification,only partial results are obtained.