DOI: 10.33205/cma.1932912 ISSN: 2651-2939

On Best Lacunary System in Orlicz Spaces

Olga Zavarzina
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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.

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