The Gibbs Phenomenon and Lebesgue Constants for Regular Sonnenschein Matrices
William T. Sledd · Canadian Journal of Mathematics · 1962
Ifψ(x)is a real-valued function which has a jump discontinuity atx= ε and otherwise satisfies the Dirichlet conditions in a neighbourhood ofx= ε then{sn(x)}the sequence of partial sums of the Fourier series forψ(x)cannot converge uniformly atx =ε. Moreover, it can be shown that given τ in [ — π, π] then there is a sequence {tn} such thattn→ ε and This behaviour of{sn(x)}is called the Gibbs phenomenon. If {σn(x)} is the transform of{sn(x)}by a summability methodT, and if {σn(x)} also has the property described then we say thatTpreserves the Gibbs phenomenon.