Exponential approximation in variable exponent Lebesgue spaces on the real line

Abstract

Present work contains a method to obtain Jackson and Stechk in type inequalities of approximation by integral functions of finite degree (IFFD) in some variable exponent Lebesgue space of real functions defined onR:= (-infinity,+infinity). To do this, we employ a transference theorem which produce norm inequalities starting fromnorm inequalities inC(R), the class of bounded uniformly continuous functions defined onR. Let B subset of Rbe ameasurable set,p(x) :B ->[1,infinity)be a measurable function. For the class of functions f belonging to variable exponent Lebesgue spaces Lp(x)(B), we consider difference operator(I-T delta)rf()under the condition thatp(x)satisfies the log-H & ouml;lder continuity condition and1 <= ess infx is an element of Bp(x),ess supx is an element of Bp(x)},delta >= 0and T delta f(x) =1 delta integral delta 0f(x+t)dt,x is an element of R,T0 equivalent to I, is the forward Steklov operator. It is proved that & Vert;(I-T delta)rf & Vert;p() is a suitable measure of smoothness for functions inLp(x)(B), where & Vert;& Vert;p()is Luxemburg norm inLp(x)(B).Weobtain main properties of difference operator & Vert;(I-T delta)rf & Vert;p()inLp(x)(B).We give proof of direct and inversetheorems of approximation by IFFD inLp(x)(R).