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 Rez∫01(1 − zt)ndg(t) ≧ 0 where
z = 1 − eix.
