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.
Keywords
References
- S. V. Astashkin: A generalized Khintchine inequality in rearrangement invariant spaces, Funct. Anal. Appl., 42 (2008), no. 2, 144–147.
- S. V. Astashkin: Independent functions in rearrangement invariant spaces and the Kruglov property, Sb. Math., 199 (2008), no. 7, 945–963.
- S. V. Astashkin and G. P. Curbera: Local Khintchine inequality in rearrangement invariant spaces, Ann. Mat. Pura Appl., 194 (2015), no. 3, 619–643.
- D. Azagra, E. Le Gruyer and C. Mudarra: Kirszbraun's theorem via an explicit formula, Canad. Math. Bull., 64 (2021), no. 1, 142–153.
- D. P. Bertsekas: Convex Optimization Theory, Athena Scientific, Belmont, MA, 2009.
- J. Bourgain: Bounded orthogonal systems and the $\Lambda(p)$-set problem, Acta Math., 162 (1989), 227–245.
- T. Figiel, P. Hitczenko, W. B. Johnson, G. Schechtman and J. Zinn: Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities, Trans. Amer. Math. Soc., 349 (1997), no. 3, 997–1027.
- V. F. Gaposhkin: On asymptotically best lacunary systems, Math. USSR-Izv., 1 (1967), no. 5, 967–982.
Details
Primary Language
English
Subjects
Operator Algebras and Functional Analysis
Journal Section
Research Article
Authors
Olga Zavarzina
*
0009-0002-3656-6826
Russian Federation
Early Pub Date
October 21, 2026
Publication Date
-
Submission Date
April 17, 2026
Acceptance Date
September 9, 2026
Published in Issue
Year 2026 Number: Advanced Online Publication
