On the Geometry of Müntz Spaces

Abstract EN

Let Λ = { λ k } k = 1 ∞ satisfy 0 < λ 1 < λ 2 < ⋯ , ∑ k = 1 ∞ 1 / λ k < ∞ and i n f k ( λ k + 1 - λ k ) > 0 .

We investigate the Müntz spaces M p Λ = s p a n ¯ { t λ k : k = 1,2 , … } ⊂ L p ( 0,1 ) for 1 ≤ p ≤ ∞ .

We show that, for each p , there is a Müntz space F p which contains isomorphic copies of all Müntz spaces as complemented subspaces.

F p is uniquely determined up to isomorphisms by this maximality property.

We discuss explicit descriptions of F p .

In particular F p is isomorphic to a Müntz space M p ( Λ ^ ) where Λ ^ consists of positive integers.

Finally we show that the Banach spaces ( ∑ n ⊕ F n ) p for 1 ≤ p < ∞ and ( ∑ n ⊕ F n ) 0 for p = ∞ are always isomorphic to suitable Müntz spaces M p ( Λ ) if the F n are the spans of arbitrary finitely many monomials over [ 0,1 ] .