Cesàro and Borel-type summability

Authors:
David Borwein and Tom Markovich

Journal:
Proc. Amer. Math. Soc. **103** (1988), 1108-1112

MSC:
Primary 40G05; Secondary 40E05, 40G10

DOI:
https://doi.org/10.1090/S0002-9939-1988-0954991-5

MathSciNet review:
954991

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Though summability of a series by the Cesàro method does not in general imply its summability by the Borel-type method , it is shown that the implication holds under an additional condition.

**[1]**D. Borwein,*A Tauberian theorem for Borel-type methods of summability*, Canad. J. Math.**21**(1969), 740-747. MR**0243233 (39:4556)****[2]**D. Borwein and T. Markovich,*A Tauberian theorem concerning Borel-type and Cesàro methods of summability*, (submitted).**[3]**G. Faulhaber,*Äquivalenzsätze für die Kreisverfahren der Limitierungstheorie*, Math. Z.**66**(1956), 34-52. MR**0082572 (18:573a)****[4]**G. H. Hardy and J. E. Littlewood,*Theorems concerning the summability of series by Borel's exponential method*, Rend. Circ. Mat. Palermo**41**(1916), 36-53.**[5]**G. H. Hardy,*Divergent series*, Oxford, 1949. MR**0030620 (11:25a)****[6]**J. M. Hyslop,*On the summability of series by a method of Valiron*, Proc. Edinburgh Math. Soc. (2)**4**(1936), 218-223. MR**0002638 (2:89g)****[7]**-,*The generalisation of a theorem on Borel summability*, Proc. London Math. Soc. (2)**41**(1936), 243-256.**[8]**B. Kwee,*An improvement on a theorem of Hardy and Littlewood*, J. London Math. Soc. (2)**28**(1983), 93-102. MR**703468 (84m:40011)****[9]**A. Meir,*Tauberian constants for a family of transformations*, Ann. of Math. (2)**78**(1963), 594-599. MR**0166519 (29:3794)****[10]**V. Swaminathan,*A note on the family**of summability methods*, Math. Z.**138**(1974), 119-122. MR**0348330 (50:828)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
40G05,
40E05,
40G10

Retrieve articles in all journals with MSC: 40G05, 40E05, 40G10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0954991-5

Article copyright:
© Copyright 1988
American Mathematical Society