Summability integrals
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- by George Brauer PDF
- Proc. Amer. Math. Soc. 61 (1976), 85-89 Request permission
Abstract:
An integral on the set of natural numbers $N$ is defined. If $E$ is a subset of $N$ such that the characteristic function ${\chi _E}$ of $E$ is evaluated to some number $\sigma$ by a regular nonnegative summation matrix $A$, then ${\mu _A}(E)$ is defined to be $\sigma ;{\mu _A}$ defines a finitely additive measure on $N$. If $s$ is a sequence which can be written as a linear combination of characteristic functions $\Sigma _{i = 1}^n{a_i}{\chi _{{E_i}}}$, where each sequence ${\chi _{{E_i}}}$ is evaluated by $A$, then ${\smallint _N}sd{\mu _A}$ is defined to be $\Sigma {a_i}{\mu _A}({E_i})$. Finally the definition of the integral is naturally extended to $L(A)$, the class of sequences which can be approximated by linear combinations of characteristic functions [2, pp. 85-88]. It is shown that if $A$ and $B$ are two nonnegative regular matrices such that the convergence field of $A$ includes that of $B$, then $L(A)$ includes $L(B)$ provided $B$ is normal. Finally for a nonnegative regular matrix $A = ({a_{nk}})$, the spaces ${L^p}(A)$ spanned by sequences such that $\{ \Sigma _{k = 0}^\infty {a_{nk}}|{s_k}{|^p}\}$ is bounded and $\lim \Sigma _{k = 0}^\infty {a_{nk}}s_k^p$ exists are studied. It is shown that if $p$ is greater than one, then the sequences in ${L^{pā}}(A)$ give rise to a set of bounded linear functionals on ${L^p}(A)$ which are weak star dense in the dual of ${L^p}(A)$.References
- J. R. Edwards and S. G. Wayment, A summability integral, J. Reine Angew. Math. 255 (1972), 85ā93. MR 304919
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 61 (1976), 85-89
- MSC: Primary 40C05; Secondary 28A25
- DOI: https://doi.org/10.1090/S0002-9939-1976-0440242-9
- MathSciNet review: 0440242