Approximation by linear means of Fourier series and realization functionals in weighted Orlicz spaces

Using one-sided Steklov means, we introduce a new modulus of smoothness in weighted Orlicz spaces and state its equivalence with a special $K$-functional. We prove Stechkin-Nikol'skii-type inequality for trigonometric polynomials and direct estimates for the approximation by Riesz-Zygmund, Vallée-Poussin, and Euler means in weighted Orlicz spaces. By these results, several types of realization functionals equivalent to the above cited $K$-functional in points $1/n$, $n\in\mathbb N$, are constructed.