EAP - Proceedings of the Estonian Academy of Sciences

PUBLISHED
SINCE 1952

proceedings
of the estonian academy of sciences

ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)

Open Access Journal

CiteScore: 2.4

Impact Factor (2022): 0.9

A note on families of generalized Nörlund matrices as bounded operators on l_p; pp. 137–145

PDF | doi: 10.3176/proc.2009.3.01

Authors

Ulrich Stadtmüller, Anne Tali

Abstract

We deal with generalized Nörlund matrices A = (N, p_n, q_n) defined by means of two non-negative sequences (p_n) and (q_n) with p₀, q₀ > 0. We are interested in simple conditions such that the associated non-negative triangular matrix A = (a_nk) is a bounded linear operator on l_p (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 A_α = (N, p^α_n, q_n) studied by different authors (see, e.g., Anal. Math., 2003, 29, 227–242; Math. Z., 1993, 214, 273–286) are bounded linear operators on l_p. These matrices need not satisfy the sufficient conditions given by Borwein in the paper mentioned above. Explicit bounds for the norms ||A_α ||_p are given.

References

1. Borwein, D. Generalized Hausdorff and weighted mean matrices as bounded operators on l^p. Math. Z., 1983, 183, 483–487.
doi:10.1007/BF01173925

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

3. Borwein, D. Simple conditions for matrices to be bounded operators on l^p. Canad. Math. Bull., 1998, 41, 10–14.

4. Borwein, D. Weighted mean operators on l^p. Canad. Math. Bull., 2000, 43, 406–412.

5. Borwein, D. and Cass, F. P. Nörlund matrices as bounded operators on l^p. Arch. Math., 1984, 42, 464–469.
doi:10.1007/BF01190697

6. Borwein, D. and Gao, X. Generalized Hausdorff and weighted mean matrices as operators on l^p. J. Math. Anal. Appl., 1993, 178, 517–528.
doi:10.1006/jmaa.1993.1322

7. Borwein, D. and Gao, X. Matrix operators on l^p to l^q. Canad. Math. Bull., 1994, 37, 448–456.

8. Borwein, D. and Jakimovski, A. Matrix operators on l^p. Rocky Mountain J. Math., 1979, 9, 463–477.

9. Cass, F. P. Convexity theorems for Nörlund and strong Nörlund summability. Math. Z., 1968, 112, 357–363.
doi:10.1007/BF01110230

10. Cass, F. P. and Kratz, W. Nörlund and weighted mean matrices as operators on l^p. Rocky Mountain J. Math., 1990, 20, 59–74.
doi:10.1216/rmjm/1181073159

11. Hardy, G. H. An inequality for Hausdorff means. J. London Math. Soc., 1943, 18, 46–50.
doi:10.1112/jlms/s1-18.1.46

12. Jakimovski, A., Rhoades, B. E., and Tzimbalario, J. Hausdorff matrices as bounded operators over l^p. Math. Z., 1974, 138, 173–181.
doi:10.1007/BF01214233

13. Kiesel, R. General Nörlund transforms and power series methods. Math. Z., 1993, 214, 273–286.
doi:10.1007/BF02572404

14. Kiesel, R. On scales of summability methods. Math. Nachr., 1995, 176, 129–138.
doi:10.1002/mana.19951760110

15. Kiesel, R. The law of the iterated logarithm for certain power series and generalized Nörlund methods. Math. Proc. Camb. Phil. Soc., 1996, 120, 735–753.
doi:10.1017/S0305004100001687

16. Kiesel, R. and Stadtmüller, U. Tauberian and convexity theorems for certain (N, p, q)-means. Can. J. Math., 1994, 46, 982–994.

17. Sinha, R. Convexity theorem for (N, p, q) summability. Kyungpook Math. J., 1973, 13, 37–40.

19. Stadtmüller, U. and Tali, A. On some families of certain Nörlund methods and power series methods. J. Math. Anal. Appl., 1999, 238, 44–66.
doi:10.1006/jmaa.1999.6503

20. Stadtmüller, U. and Tali, A. Comparison of certain summability methods by speeds of convergence. Anal. Math., 2003, 29, 227–242.
doi:10.1023/A:1025419305735

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

Back to Issue