Abstract
Letf be a real-valued function sequence {f k } that converges to ϕ on a deleted neighborhoodD of α. If there is a subsequence {f k(j) } and a number sequencex such that lim j x j =α and either lim j f k(j) (x j )>lim sup x→α ϕ(x) or lim j f k(j) (x j )<lim inf x→α ϕ(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 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: ifϕ(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.
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Fridy, J.A. Matrix summability and a generalized Gibbs phenomenon. Aeq. Math. 14, 405–412 (1976). https://doi.org/10.1007/BF01835989
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DOI: https://doi.org/10.1007/BF01835989