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Weighted Weak Behaviour of Fourier‐Jacobi Series
Author(s) -
Guadalupe José J.,
Pérez Mario,
Varona Juan L.
Publication year - 1992
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19921580111
Subject(s) - mathematics , interval (graph theory) , fourier series , bounded function , series (stratigraphy) , type (biology) , fourier analysis , pure mathematics , jacobi operator , fourier transform , combinatorics , mathematical analysis , jacobi polynomials , orthogonal polynomials , paleontology , ecology , biology
Let w ( x ) = (1 ‐ x ) α (1 + x ) β be a Jacobi weight on the interval [‐1, 1] and 1 < p < ∞. If either α > −1/2 or β > −1/2 and p is an endpoint of the interval of mean convergence of the associated Fourier‐Jacobi series, we show that the partial sum operators S n are uniformly bounded from L p ,1 to L p ,∞ , thus extending a previous result for the case that both α, β > −1/2. For α, β > −1/2, we study the weak and restricted weak ( p, p )‐type of the weighted operators f→ uS n ( u −1 f ), where u is also Jacobi weight.