Matrix summability and a generalized Gibbs phenomenon

Letf be a real-valued function sequence {f _k } that converges to on a deleted neighborhoodD of . If there is a subsequence {f _k(j) } and a number sequencex such that lim _j x _j = and either lim _j f _k(j) (x _j )>lim sup _x (x) or lim _j f _k(j) (x _j ) x (x), thenf is said to display theGibbs phenomenon at . IfA is a (real) summability matrix, thenAf is a function sequence given by(Af) _n (x)= _k=0 a _n,k f _k (x). IfAf displays the Gibbs phenomenon wheneverf does, thenA is said to beGP-preserving. By replacingf _k (x) withf _k (x _j )F _k,j , the Gibbs phenomenon is viewed as a property of the matrixF, andGP-preserving matrices are determined by properties of the matrix productAF. The general results give explicit conditions on the entries {a _n,k } that are necessary and/or sufficient forA to beGP-preserving. For example: if(x)0 thenF displaysGP iff lim _k,j F _k,j 0, and ifA isGP-preserving then lim _n,k A _n,k 0. IfA is a triangular matrix that is stronger than convergence, thenA is notGP-preserving. The general results are used to study the preservation of the Gibbs phenomenon by matrix methods of Nörlund, Hausdorff, and others.