Exponential approximation in variable exponent Lebesgue spaces on the real line

Present work contains a method to obtain Jackson and Stechkin type inequalities of approximation by integral functions of finite degree (IFFD) in some variable exponent Lebesgue space of real functions defined on R := (−∞, +∞). To do this, we employ a transference theorem which produce norm inequalities starting from norm inequalities in C(R), the class of bounded uniformly continuous functions defined on R. Let B ⊆ R be a measurable set, p (x) : B → [1, ∞) be a measurable function. For the class of functions f belonging to variable exponent Lebesgue spaces Lp(x) (B), we consider difference operator (I − Tδ) r f (·) under the condition that p(x) satisfies the log-Hölder continuity condition and 1 ≤ ess infx∈B p(x), ess supx∈B p(x) < ∞, where I is the identity operator, r ∈ N := {1, 2, 3, · · · }, δ ≥ 0 and (∗) Tδf (x) = 1 δ Z δ 0 f (x + t) dt, x ∈ R, T0 ≡ I, is the forward Steklov operator. It is proved that (∗∗) k(I − Tδ) r fkp(·) is a suitable measure of smoothness for functions in Lp(x) (B), where k·kp(·) is Luxemburg norm in Lp(x) (B) . We obtain main properties of difference operator k(I − Tδ) r fkp(·) in Lp(x) (B) . We give proof of direct and inverse theorems of approximation by IFFD in Lp(x) (R) .