All Questions
4 questions
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Dense subspace of square integrable functions on the complex disc
Denote by $L^{2}(D,(1-|z|^{2})^{a-1}|z|^{2b-2}dx dy)$ the set of square integrable functions on the complex disc $D= \lbrace z \in C, \; |z| <1 \rbrace$ with respect to the measure $(1-|z|^{2})^{a-...
4
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1
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Exchanging series and integrals
I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a ...
- 91
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Does the following sequence $\{g_n\}$ converge?
Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where
\begin{eqnarray}\label{eqn:constraint1}
f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...
- 550
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Does exist a variant of Frullani's theorem valid for $f(x)=\pi(x)/x$ or $f(x)=\psi(x)/x$, where the numerators are prime-counting functions?
Frullani's theorem is a deep theorem in real analysis with applications, see the Wikipedia Frullani integral and other uses and contexts (see [2]). I wrote two imaginative examples of what can be ...
