Q2.5
Sequences and Series
Properties and Manipulation of Power Series or Formal Series
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Let $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ be a power series with radius of convergence 1 and sum $f$. Let $S \in \mathbb { C }$. The purpose of this question is to prove that $$\left( \lim _ { \substack { x \rightarrow 1 ^ { - } \\ x \in \mathbb { R } } } f ( x ) = S \text { and } a _ { n } = O \left( \frac { 1 } { n } \right) \right) \Rightarrow \left( \sum _ { n \geqslant 0 } a _ { n } \text { converges and } \sum _ { n = 0 } ^ { + \infty } a _ { n } = S \right) . \quad \text{(Strong Tauberian)}$$
(a) Prove that, without loss of generality, we can assume that $S = 0$.
We now suppose that $\lim _ { \substack { x \rightarrow 1 ^ { - } \\ x \in \mathbb { R } } } f ( x ) = S$ and that $a _ { n } = O \left( \frac { 1 } { n } \right)$, with $S = 0$.
(b) We define $\Theta$ as follows $$\Theta = \left\{ \theta : [ 0,1 ] \rightarrow \mathbb { R } ; \forall x \in \left[ 0,1 \left[ , \sum _ { n \geqslant 0 } a _ { n } \theta \left( x ^ { n } \right) \text { converges and } \lim _ { x \rightarrow 1 ^ { - } } \sum _ { n = 0 } ^ { + \infty } a _ { n } \theta \left( x ^ { n } \right) = 0 \right\} . \right. \right.$$ Prove that $\Theta$ is a vector space over $\mathbb { R }$.
(c) Let $P \in \mathbb { R } [ X ]$ such that $P ( 0 ) = 0$. Prove that $P \in \Theta$.
(d) Prove that $$\forall P \in \mathbb { R } [ X ] , \quad \lim _ { \substack { x \rightarrow 1 ^ { - } \\ x \in \mathbb { R } } } ( 1 - x ) \cdot \sum _ { n = 0 } ^ { + \infty } x ^ { n } P \left( x ^ { n } \right) = \int _ { 0 } ^ { 1 } P ( t ) d t$$
We define the function $g : \mathbb { R } \rightarrow \mathbb { R }$ by $$g ( x ) = \begin{cases} 1 & \text { if } x \in [ 1 / 2,1 ] \\ 0 & \text { otherwise } \end{cases}$$
(e) Prove that to establish (Strong Tauberian), it suffices to prove that $g \in \Theta$.
(f) Let $$h ( x ) = \begin{cases} - 1 & \text { if } x = 0 \\ \frac { g ( x ) - x } { x ( 1 - x ) } & \text { if } x \in ] 0,1 [ \\ 1 & \text { if } x = 1 \end{cases}$$ Given $\varepsilon > 0$, prove that there exist $s _ { 1 } , s _ { 2 } \in \mathcal { C } ^ { 0 } ( [ 0,1 ] )$ satisfying $$s _ { 1 } \leqslant h \leqslant s _ { 2 } \text { and } \int _ { 0 } ^ { 1 } \left( s _ { 2 } ( x ) - s _ { 1 } ( x ) \right) d x \leqslant \varepsilon$$ Represent graphically $h$ and two such functions $s _ { 1 } , s _ { 2 }$.
From now on, $\varepsilon > 0 , s _ { 1 }$ and $s _ { 2 }$ are fixed.
(g) Prove that there exist $T _ { 1 } , T _ { 2 } \in \mathbb { R } [ X ]$ such that $$\sup _ { x \in [ 0,1 ] } \left| T _ { 1 } ( x ) - s _ { 1 } ( x ) \right| \leqslant \varepsilon \quad \text { and } \quad \sup _ { x \in [ 0,1 ] } \left| T _ { 2 } ( x ) - s _ { 2 } ( x ) \right| \leqslant \varepsilon$$
We set, for all $x \in [ 0,1 ]$, $$P _ { 1 } ( x ) = x + x ( 1 - x ) \left( T _ { 1 } ( x ) - \varepsilon \right) , \quad P _ { 2 } ( x ) = x + x ( 1 - x ) \left( T _ { 2 } ( x ) + \varepsilon \right) \quad \text{and} \quad Q ( x ) = \frac { P _ { 2 } ( x ) - P _ { 1 } ( x ) } { x ( 1 - x ) }$$
(h) Prove that $$P _ { 1 } ( 0 ) = P _ { 2 } ( 0 ) = 0 , \quad P _ { 1 } ( 1 ) = P _ { 2 } ( 1 ) = 1 , \quad P _ { 1 } \leqslant g \leqslant P _ { 2 } \quad \text{and} \quad 0 \leqslant \int _ { 0 } ^ { 1 } Q ( x ) d x \leqslant 5 \varepsilon$$
(i) Prove that there exists $M > 0$ such that for all $x \in ] 0,1 [$, $$\left| \sum _ { n = 0 } ^ { + \infty } a _ { n } g \left( x ^ { n } \right) - \sum _ { n = 0 } ^ { + \infty } a _ { n } P _ { 1 } \left( x ^ { n } \right) \right| \leqslant M ( 1 - x ) \sum _ { n = 1 } ^ { + \infty } x ^ { n } Q \left( x ^ { n } \right)$$
(j) Conclude.