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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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A note on families of generalized Nörlund matrices as bounded operators on lp; pp. 137–145

(Full article in PDF format) doi: 10.3176/proc.2009.3.01


Authors

Ulrich Stadtmüller, Anne Tali

Abstract

We deal with generalized Nörlund matrices A = (N, pn, qn) defined by means of two non-negative sequences (pn) and (qn) with p0, q0 > 0. We are interested in simple conditions such that the associated non-negative triangular matrix A = (ank) is a bounded linear operator on lp (1 < p < ¥). Using results of D. Borwein (Canad. Math. Bull., 1998, 41, 10–14), we provide sufficient conditions and bounds for the norm ||A ||p. Our main question is whether certain families of generalized Nörlund matrices Aα = (N, pαn, qn) studied by different authors (see, e.g., Anal. Math., 2003, 29, 227–242; Math. Z., 1993, 214, 273–286) are bounded linear operators on lp. These matrices need not satisfy the sufficient conditions given by Borwein in the paper mentioned above. Explicit bounds for the norms ||Aα ||p are given.

Keywords

operator theory, Banach space lp, bounded linear operators, generalized Nörlund matrices, Nörlund, Riesz and Euler–Knopp matrices.

References

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  2. Borwein , D. Nörlund operators on lp. Canad. Math. Bull. , 1993 , 36 , 8–14.

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21. Tali , A. Convexity conditions for families of summability methods. Tartu Ülik. Toimetised , 1993 , 960 , 117–138.

 
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Current Issue: Vol. 68, Issue 4, 2019




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No. 1: 20 March
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