Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent (Q765109)

Let \(T= (-\pi,\pi]\) and let \(p:T\to(1,+\infty)\) be a \(2\pi\)-periodic measurable function such that \(\text{ess\,sup}_{x\in T}p(x)< +\infty\). Let us consider the Lebesgue space \(L^{p(.)}_{2\pi}(T)\) of the \(2\pi\)-periodic measurable functions defined on \(T\) such that \[ \int_T|f(x)|^{p(x)} dx<\infty, \] with the norm \[ \| f\|_{p,\pi}= \text{inf}\Biggl(\alpha> 0: \int_T |f(x)/\alpha|^{p(x)}\, dx\leq 1\Biggr). \] In general, this Banach space is not \(p\)-continuous and not translation invariant. The author studies the approximation properties of the trigonometric system in this space. He considers the moduli of smoothness of fractional order and obtains direct and inverse approximation theorems and a constructive characterization of a Lipschitz-type class.