We continue our studies on Riesz-type families of summability methods for functions and sequences, started in (Proc. Estonian Acad. Sci., 2008, 57, 70–80) and (Math. Model. Anal., 2010, 15, 103–112). Strong summability methods defined on the basis of a given Riesz-type family {Aα} are considered here. Inclusion theorems for these methods are proved. Our inclusion theorems allow us to compare the summability fields and speeds of convergence. The strong summability methods are also compared with ordinary summability methods Aα and with certain methods of statistical convergence. The proved theorems generalize different results that have already been published and are applied, in particular, to Riesz methods, generalized integral Nörlund methods, and Borel-type methods.
1. Borwein, D. On Borel-type methods of summability. Mathematica, 1958, 5, 128–133.
2. Borwein, D. and Shawyer, B. L. R. On Borel-methods. Tôhoku Math. J., 1966, 18, 283–298.
http://dx.doi.org/10.2748/tmj/1178243418
3. Connor, J. On strong matrix summability with respect to a modulus and statistical convergence. Canad. Math. Bull., 1989, 32, 194–198.
http://dx.doi.org/10.4153/CMB-1989-029-3
4. Connor, J. A topological and functional analytic approach to statistical convergence. In Analysis of Divergence: Control and Management of Divergent Processes (Bray, W. O. and Stanojević, Č. V., eds). Birkhäuser Boston, Boston, MA, 1999, 403–413.
5. Hardy, G. H. Divergent Series. Oxford Press, 1949.
6. Kangro, G. On the summability factors of the Bohr-Hardy type for a given rapidity. I. Eesti NSV Tead. Akad. Toim. Füüs. Mat., 1969, 18, 137–146 (in Russian).
7. Kangro, G. Summability factors for the series λ-bounded by the methods of Riesz and Cesàro. Tartu Ülik. Toimetised, 1971, 277, 136–154 (in Russian).
8. Kolk, E. Inclusion relations between the statistical convergence and strong summability. Acta Comment. Univ. Tartu. Math., 1998, 2, 39–54.
9. Móricz, F. Statistical limits of measurable functions. Analysis, 2004, 24, 1–18.
10. Pavlova, V. and Tali, A. On the convexity theorem of M. Riesz. Proc. Estonian Acad. Sci. Phys. Math., 2002, 18, 18–34.
11. Soomer, V. and Tali, A. On strong summability of sequences. Acta Comment. Univ. Tartu. Math., 2007, 11, 57–68.
12. Stadtmüller, U. and Tali, A. Comparison of certain summability methods by speeds of convergence. Analysis Mathematica, 2003, 29, 227–242.
http://dx.doi.org/10.1023/A:1025419305735
13. Stadtmüller, U. and Tali, A. Strong summability in certain families of summability methods. Acta Sci. Math. (Szeged), 2004, 70, 639–657.
14. Šeletski, A. and Tali, A. Comparison of speeds of convergence in Riesz-type families of summability methods. Proc. Estonian Acad. Sci., 2008, 52, 70–80.
http://dx.doi.org/10.3176/proc.2008.2.02
http://dx.doi.org/10.3846/1392-6292.2010.15.103-112
