Bounds For The Operator Norms Of Some Norlund Matrices
Abbreviated Journal Title
Proc. Amer. Math. Soc.
Mathematics, Applied; Mathematics
Suppose (p(n))(n greater than or equal to 0) is a non-increasing sequence of non-negative numbers with p(0) = 1, p(n) = Sigma(j=0)(n) P-j, n = 0, 1..., and A = A(p(n)) = (a(nk)) is the lower triangular matrix defined by a(nk) = p(n-k)/p(n), 0 less than or equal to k less than or equal to n, and a(nk) = 0, n < k. We show that the operator norm of A as a linear operator on l(p) is no greater than p/(p - 1), for 1 < p < infinity; this generalizes, yet again Hardy's inequality for sequences, and simplifies and improves, in this special case;more generally applicable results of D. Borwein, Cass, and Kratz. When the p(n) tend to a positive limit, the operator norm of A on l(p) is exactly p/(p - 1). We also give some cases when the operator norm of A on l(p) is less than p/(p - 1).
Proceedings of the American Mathematical Society
"Bounds For The Operator Norms Of Some Norlund Matrices" (1996). Faculty Bibliography 1990s. 1650.