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Harmonic and Logarithmic Summability of Orthogonal Series are Equivalent up to a Set of Measure Zero
Author(s) -
Móricz F.,
Stadtmüller U.
Publication year - 1999
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/s0024610798006875
Subject(s) - mathematics , logarithm , series (stratigraphy) , measure (data warehouse) , zero (linguistics) , power series , almost everywhere , mathematical analysis , convergence (economics) , harmonic , pure mathematics , abelian and tauberian theorems , set (abstract data type) , paleontology , linguistics , philosophy , physics , computer science , biology , programming language , quantum mechanics , database , economics , economic growth
We prove Tauberian theorems from J p ‐summability methods of power series type to ordinary convergence, respectively M p ‐summability methods of weighted means. Particular cases are the Abel and Cesàro, as well as logarithmic and harmonic summability. Besides numerical series, we also consider orthogonal series with coefficients from L 2 . In the latter case, it turns out that one of our Tauberian conditions is satisfied almost everywhere on the underlying measure space, thereby proving the claim stated in the title.