On Best Lacunary System in Orlicz Spaces

Stechkin's classical results on the best lacunary system in Lₚ spaces, given by the direction cosines on the unit sphere, are extended to Orlicz spaces LΦ. It is shown that for any N-function Φ the LΦ- norm of a linear combination of the direction cosines is completely determined by the l₂-norm of the coefficient vector. Consequently, the system is SΦ(M)-lacunary with a constant M=K_Φ,n √n, where K_Φ,n coincides with the LΦ- norm of a single coordinate function. Moreover, under the additional convexity condition on Φ(√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)=eu²-1 and Φ(u)=u-1, and expressed in terms of hypergeometric functions.