Summability of a Tchebysheff system of functions

We consider a special type of tchebysheff systems of functions {u i(·)} i=0 n and {v i(·)} i=0 n defined on the intervals (0,1] and [1,+∞),respectively,such that u i(t) = t α0 ∫ t 1 t α1 1 ∫ t1 1 t α2 2 ⋯ ∫ ti-1 1 t αi idt idt i-1⋯dt 1 and v i(t) = t β0 ∫ 1 t t β1 1 ∫ 1 t1 t β2 2 ⋯ ∫ 1 ti-1 t βi idt idt i-1⋯dt 1. we find necessary and sufficient conditions under which functions from the investigated systems belong to the corresponding lebesgue spaces l p(0,1) and l p(1,+∞). in order to prove the main results we obtain lower and upper estimates of these functions that are of independent interest. copyright © 2010 hindawi publishing corporation.