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 $L_Φ$. It is shown that for any $N$-function $Φ$ the $L_Φ$-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 $S_Φ(M)$-lacunary with a constant $M = K_{Φ,n} \sqrt{n}$, where $K_{Φ,n}$ coincides with the $L_Φ$-norm of a single coordinate function. Moreover, under the additional convexity condition on $Φ(\sqrt{u})$, 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 $Φ(u)=e^{u^2}-1$ and $Φ(u)=\cosh u - 1$, and expressed in terms of hypergeometric functions.