40F05 Absolute and strong summability

Overview

40F05 investigates absolute and strong summability as part of absolute and strong summability. The focus is on convergence acceleration, summability equivalences, and inversion principles that extend classical limits and characterize asymptotic behavior of series and transforms.

Related Wikipedia Page

Wikipedia search: Absolute and strong summability

Useful Links

• Wikipedia search: Absolute and strong summability
• Encyclopaedia of Mathematics
• arXiv: Classical Analysis and ODEs (math.CA)
• Proceedings of the London Mathematical Society
• Studia Mathematica
• Mathematical Proceedings of the Cambridge Philosophical Society

Key Ideas

• Core definitions and canonical constructions for absolute and strong summability
• Relations among regular, strong, and absolute summation procedures
• Direct/inverse principles connecting transforms with convergence properties

Typical Uses

Used to recover analytic information from transformed sequences, compare summation frameworks, and prove convergence statements under weaker hypotheses than ordinary limits.

Applications

• Asymptotic expansions and transform-based analysis in pure and applied mathematics
• Stability checks for computational procedures involving slowly convergent series
• Signal/approximation workflows where generalized summation improves interpretability

References

• Hardy, G.H. and Riesz, M. (1915) The General Theory of Dirichlet's Series. Cambridge: Cambridge University Press.
• Bosanquet, L.S. and Chow, P. (1929) "Some properties of strong summability", Proceedings of the London Mathematical Society, s2-31(1), pp. 1-24.
• Flett, T.M. (1957) "On an extension of absolute summability and strong summability", Proceedings of the London Mathematical Society, s3-7(1), pp. 113-141.

Recommended Textbooks

• Hardy, G.H. and Riesz, M. (1915) The General Theory of Dirichlet's Series. Cambridge: Cambridge University Press.
• Hardy, G.H. (1949) Divergent Series. Oxford: Clarendon Press.
• Zeller, K. and Beekmann, W. (1970) Theorie der Limitierungsverfahren. 2nd edn. Berlin: Springer.