Vol. 15, No. 2, 1965

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ISSN: 0030-8730
The generalized Gibbs phenomenon for regular Hausdorff means

Jonah Mann and Donald J. Newman

Vol. 15 (1965), No. 2, 551–555

One says that the means σn(x), of the Fourier series of a function f(x), exhibit the (generalized) Gibbs phenomenon at the point x = x0 if the interval between the upper and lower limit of σn(x), as n →∞ and x xo independently, contains points outside the interval between the upper and lower limits of f(x) as x x0. Theorem. In order that the Hausdorff summability method given by g(t) not display the Gibbs phenomenon for any Lebesgue integrable function, it is necessary and sufficient that 1 g(t) be positive definite. A new inequality which must be satisfied by g(t), whenever 1 g(t) is positive definite, is Re z 01(1 zt)n dg(t) 0 where z = 1 eix.

Mathematical Subject Classification
Primary: 42.20
Received: 24 February 1964
Published: 1 June 1965
Jonah Mann
Donald J. Newman