On Best Lacunary System in Orlicz Spaces

Abstract

Stechkin's classical results on the best lacunary system in $L_p$ spaces, given by the direction cosines on the unit sphere, are extended to Orlicz spaces $L_{\Phi}$. It is shown that for any $N$-function $\Phi$ the $L_{\Phi}$-norm of a linear combination of the direction cosines is completely determined by the $l_2$-norm of the coefficient vector. Consequently, the system is $S_{\Phi}(M)$-lacunary with a constant $M = K_{\Phi,n} \sqrt{n}$, where $K_{\Phi,n}$ coincides with the $L_{\Phi}$-norm of a single coordinate function. Moreover, under the additional convexity condition on $\Phi(\sqrt{u})$, this constant is proved to be optimal, so the direction cosines form the best lacunary system in Orlicz spaces. Explicit formulas for the constant are derived for $N$-functions $\Phi(u)=e^{u^2}-1$ and $\Phi(u)=\cosh u - 1$, and expressed in terms of hypergeometric functions.