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A Hardy-Davies-Petersen Inequality for a Class of Matrices

  • P. D. Johnson (a1) and R. N. Mohapatra (a2)

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Let ω be the set of all real sequences a = ﹛an﹜ n ≧0. Unless otherwise indicated operations on sequences will be coordinatewise. If any component of a has the entry oo the corresponding component of a-1 has entry zero. The convolution of two sequences s and q is given by s * q . The Toeplitz martix associated with sequence s is the lower triangular matrix defined by tnk = sn-k (n ≧ k), tnk = 0 (n < k). It can be seen that Ts(q) = s * q for each sequence q and that Ts is invertible if and only if s0 ≠ 0. We shall denote a diagonal matrix with diagonal sequence s by Ds.

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References

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1. Borwein, D., On products of sequences, J. London Math. Soc. 33 (1958), 212220.
2. Davies, G. S. and Petersen, G. M., On an inequality of Hardy's (II, Quart. J. Math. (Oxford) (2) 15 (1964), 3540.
3. Hardy, G. H., Littlewood, J. E. and Pólya, G. , Inequalities (Cambridge, 1934).
4. Johnson, P. D., Jr. and Mohapatra, R. N., The maximal normal subspace of the inverse image of a normal space of sequences by a non-negative matrix transformation, to appear.
5. Johnson, P. D., Jr. and Mohapatra, R. N., Inequalities involving lower-triangular matrices, to appear.
6. Leibowitz, G. M., A note on Cesdro sequence spaces, Tamkang, J. Math. 2 (1971), 151157.
7. Petersen, G. M., An equality of Hardy's, Quart. J. Math. (Oxford) (2) 15 (1964), 3540.
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A Hardy-Davies-Petersen Inequality for a Class of Matrices

  • P. D. Johnson (a1) and R. N. Mohapatra (a2)

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