An application of Banach limits

Authors:
Z. U. Ahmad and Mursaleen

Journal:
Proc. Amer. Math. Soc. **103** (1988), 244-246

MSC:
Primary 40C05; Secondary 46A45

DOI:
https://doi.org/10.1090/S0002-9939-1988-0938676-7

MathSciNet review:
938676

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the Banach space of bounded sequences, an injection of the set of positive integers into itself having no finite orbits, and the operator defined on by . A positive linear functional with , is called a -mean if for all in . A sequence is said to be -convergent, denoted , if takes the same value, called , for all -means . P. Schaefer [**6**] gave necessary and sufficient conditions on a matrix to ensure that , where is the space of convergent sequences, and additional conditions ensuring that for all , denoting the class of matrices satisfying these conditions by and calling them the -regular matrices. In this paper, we use such matrices to find the sum of a sequence of Walsh functions.

**[1]**S. Banach,*Theorie des operations lineaires*, PWN, Warszawa, 1932.**[2]**N. J. Fine,*On Walsh functions*, Trans. Amer. Math. Soc.**65**(1949), 372-414. MR**0032833 (11:352b)****[3]**G. G. Lorentz,*A contribution to the theory of divergent sequences*, Acta Math.**80**(1948), 167-190. MR**0027868 (10:367e)****[4]**Mursaleen,*On some new invariant matrix methods of summability*, Quart. J. Math. Oxford (2)**34**(1983), 77-86. MR**688425 (84h:40004)****[5]**R. A. Raimi,*Invariant means and invariant matrix methods of summability*, Duke Math. J.**30**(1963), 81-94. MR**0154005 (27:3965)****[6]**P. Schaefer,*Infinite matrices and invariant means*, Proc. Amer. Math. Soc.**36**(1972), 104-110. MR**0306763 (46:5885)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0938676-7

Keywords:
Walsh functions,
Banach limits,
infinite matrices

Article copyright:
© Copyright 1988
American Mathematical Society