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Approximation of functions and their derivatives by analytic maps on certain Banach spaces
Author(s) -
Azagra D.,
Fry R.,
Keener L.
Publication year - 2011
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdr032
Subject(s) - mathematics , banach space , separable space , lipschitz continuity , bounded function , hilbert space , polynomial , pure mathematics , analytic function , space (punctuation) , discrete mathematics , mathematical analysis , philosophy , linguistics
Let X be a separable Banach space that admits a separating polynomial; in particular, let X be a separable Hilbert space. Let f : X → ℝ be bounded and Lipschitz, with uniformly continuous derivative. Then, for each ε > 0, there exists an analytic function g : X → ℝ with | g − f | < ε and ‖ g ′ − f ′ ‖ < ε.