Abstract. For various Fourier expansions by orthogonal systems a Hörmander-type multiplier condition implies a Littlewood--Paley theory and from it a sharp Marchaud-type inequality is deduced for the $K$-functional related to the operator generating the orthogonal system and for related moduli of smoothness.

AMS Subject Classification (1991): 26A15, 41A27, 42B25

Keyword(s): Littlewood--Paley type inequality, Hörmander type multiplier theorems, Sharp Marchaud inequality

Received June 16, 2004, and in revised form November 5, 2004. (Registered under 5858/2009.)