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- W4287332523 abstract "We apply ideas from the theory of Uniform Distribution of sequences to Functional Analysis and then study concepts and results in Uniform Distribution itself. Let $E$ be a Banach space. Then: (a) If $F$ is a bounded subset of $E$ and $x in overline{co}(F)$, there is a sequence $(x_n) subseteq F$ which is Ces`{a}ro summable to $x$. (b) If $E$ is separable, $F subseteq E^*$ bounded and $f in overline{co}^{w^*}(F)$, there is a sequence $(f_n) subseteq F$ whose sequence of arithmetic means $frac{f_1+dots+f_N}{N}$, $N ge 1$ weak$^*$-converges to $f$. If we apply (b) to $Omega=B_{M(K)}$, where $K$ is a compact metric space, we get that for every signed Borel measure $mu$ on $K$ with $|mu| le 1$, there are sequences $(x_n) subseteq K$ and $(varepsilon_n) subseteq { pm 1}$ such that the sequence $frac{varepsilon_1 delta_{x_1}+dots+varepsilon_N delta_{x_N}}{N} overset{weak^*}{longrightarrow} mu$. Assuming that $|mu|=1$ we can prove that $(x_n)$ can be chosen to be $|mu|$-u.d., i.e. $ frac{delta_{x_1}+dots+delta_{x_N}}{N} overset{weak^*}{longrightarrow} |mu|$. We generalize a classical theorem of Uniform Distribution which is valid for increasing functions $varphi:I=[0,1] rightarrow Bbb{R}$ with $varphi(0)=0$ and $varphi(1)=1$, for functions $varphi$ of bounded variation on $I$ with $varphi(0)=0$ and total variation $V_0^1 varphi=1$. We show that there are sequences $(x_n) subseteq I$ and $(varepsilon_n) subseteq { pm 1}$ such that for each $x in I$ we have $lim_{N to infty} frac{1}{N} sum_{k=1}^N chi_{[0,x)}(x_k)=upsilon(x) ;, textrm{and} ;, lim_{N to infty} frac{1}{N} sum_{k=1}^N varepsilon_k chi_{[0,x)}(x_k)=varphi(x)$, $upsilon$ is the function of total variation of $varphi$ on $I$." @default.
- W4287332523 created "2022-07-25" @default.
- W4287332523 creator A5051987862 @default.
- W4287332523 creator A5089606467 @default.
- W4287332523 date "2021-02-03" @default.
- W4287332523 modified "2023-09-26" @default.
- W4287332523 title "Uniform Distribution of Sequences and its interplay with Functional Analysis" @default.
- W4287332523 doi "https://doi.org/10.48550/arxiv.2102.02306" @default.
- W4287332523 hasPublicationYear "2021" @default.
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