| Abstract
| - Abstract. Letf be a real-valued function sequence {fk} that converges to ϕ on a deleted neighborhoodD of α. If there is a subsequence {fk(j)} and a number sequencex such that limjxj=α and either limjfk(j)(xj)>lim supx→αϕ(x) or limjfk(j)(xj)<lim infx→αϕ(x), thenf is said to display theGibbs phenomenon at α. IfA is a (real) summability matrix, thenAf is a function sequence given by(Af)n(x)=∑k=0∞an,kfk(x). IfAf displays the Gibbs phenomenon wheneverf does, thenA is said to beGP-preserving. By replacingfk(x) withfk(xj)≡Fk,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 {an,k} that are necessary and/or sufficient forA to beGP-preserving. For example: ifϕ(x)≡0 thenF displaysGP iff limk,jFk,j≠0, and ifA isGP-preserving then limn,kAn,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.
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