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Let $\{q_j\}_{j\in\mathbb N}$ be an enumeration of the rational numbers in the unit interval $[0,1]$. Let $\{a_j\}_{j\in\mathbb N}$ be positive real numbers with $\textstyle\sum_j a_j < \infty$. Define for all $x\in [0,1]$ $$g(x):=\sum_{j=0}^\infty \,a_j \sqrt[3]{x-q_j}.$$ It is claimed in this wiki page that $$g^{\prime}(x)=\frac{1}{3}\sum_{j=0}^\infty \frac{a_j}{\sqrt[3]{(x-q_j)^2}}$$ holds at any point $x$ where the sum is finite.

It is not clear to me how to justify this. Any help would be appreciated.

Syang Chen
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1 Answers1

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First, $$ g'(x) = \lim_{h \to 0} \frac{g(x+h) - g(x)}{h} = \lim_{h \to 0}\lim_{n \to \infty} \sum_{j=0}^n \frac{g_j(x+h) - g_j(x)}{h} $$ where $g_j(x) := a_j(x-q_j)^{1/3}$. It should be clear that $$ g_j'(x) = \frac{a_j}{3(x-q_j)^{2/3}}, $$ and so all that is needed is to justify swapping the order of the limits. But consider $\mu$ the counting measure on $\mathbb{N}$: the limit can be written as $$ g'(x) = \lim_{m \to \infty} \int s_m(j) d\mu(j) $$ where $$ s_m(j) := \frac{g_j(x+1/m) - g_j(x)}{1/m}. $$ For a fixed $j$, $\lim_{m \to \infty} s_m(j) = g_j'(x)$, and this is finite since $[0,1]$ is closed. Also, $$ \int g_j'(x) d\mu(j) = \sum_{j=0}^\infty \frac{a_j}{3(x-q_j)^{2/3}} < \infty. $$ Thus the sequence $\{s_m\}$ is eventually dominated by an integrable function, and the dominated convergence theorem can be employed to conclude $$ g'(x) = \int \lim_{m \to \infty} s_m(j) d\mu(j) = \int g'_j(x) d\mu(j) = \sum_{j=0}^\infty \frac{a_j}{3(x-q_j)^{2/3}}. $$

Rookatu
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  • What is the dominating function for $s_m$? Why is it integrable? – Syang Chen Mar 25 '14 at 04:35
  • Since $\lim_{m \to \infty} s_m(j) = g_j'(x) < \infty$, you can let $M$ be large enough so that $m \geq M$ implies $s_m(j) \leq g_j'(x) + 2^{-j}$. This is an integrable function which eventually dominates ${s_m}$ therefore. It is integrable by the hypothesis that the sum is finite. Also, I'd appreciate a comment from whoever downvoted. – Rookatu Mar 25 '14 at 05:45
  • Presumably $M$ will depend on $j$ though. – Syang Chen Mar 26 '14 at 00:23