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The Full Müntz Theorem in C [0, 1] and L 1 [0, 1]
Author(s) -
Borwein Peter,
Erdélyi Tamás
Publication year - 1996
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/54.1.102
Subject(s) - mathematics , combinatorics , span (engineering) , sequence (biology) , discrete mathematics , chemistry , biochemistry , civil engineering , engineering
The main result of this paper is the establishment of the ‘full Müntz Theorem’ in C [0, l]. This characterizes the sequences{ λ i }i = 1 ∞of distinct, positive real numbers for which span{ 1 , x λ 1, x λ 2, … }is dense in C [0, 1]. The novelty of this result is the treatment of the most difficult case when inf i λ i = 0 while sup i λ i = ∞. The paper settles the L ∞ and L 1 cases of the following. THEOREM (Full Müntz Theorem in L p [0,1]). Let p ∈ [l, ∞]. Suppose that{ λ i } i = 0 ∞is a sequence of distinct real numbers greater than −1/ p . Then span{ x λ 0, x λ 1, … }is dense in L p [0, 1] if and only ifΣ i = 0 ∞λ i + 1 / p( λ i + 1 / p ) 2 + 1 = ∞ .