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.

arXiv comment13 pages, uses cma.sty