Summability integrals

Brauer, George

doi:10.1090/S0002-9939-1976-0440242-9

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 ${L^{p’}}(A)$ 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), 85–93. 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