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Lower Bounds for the L p ‐Norm in Terms of the Mellin Transform
Author(s) -
Friedman Eduardo,
Skoruppa NilsPeter
Publication year - 1993
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/25.6.567
Subject(s) - mathematics , infimum and supremum , norm (philosophy) , convolution (computer science) , combinatorics , product (mathematics) , mellin transform , mathematical analysis , fourier transform , geometry , political science , law , machine learning , artificial neural network , computer science
Given a measurable function f on (0, ∞) with Mellin transform F ( s ), let | f | p denote the L p ‐norm of f with respect to the measure dx/x . We prove that under certain assumptions, for instance if f is real and non‐negative and F (α) converges for α in an open interval and F (α) ≠ 0, then| f | p ⩾ c p| α |1 ‐ 1 / p| F ( α ) exp ( ‐ αF ' F ( α ) )| , where c p ⩾ (2 e ) −1 . We derive similar inequalities for complex‐valued f , for the L p ‐norm of the derivative of f , and for the supremum of real‐valued f and of its derivative. The lower bounds are eminently applicable when f is a convolution product.