Tauberian theorems

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Problem 10.49 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 10.2 "Shauder bases"

Let the series $\sum_{k = 1}^{\infty } x_{k}$ in a Banach space converge in the Cesàro sense, i.e., $\lim_{n \rightarrow \infty } \frac{S_{1}+S_{2}+\ldots +S_{n}}{n}$ exists (where $S_{n} = \sum_{k = 1}^{n} x_{k}$ ). Prove that if $\left\| x_{n}\right\| = o\left(\frac{1}{n}\right)$ , then the series converges in norm.

Problem 2.20.26 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.20.2 "Problems"

Let $a_{n} \geqslant 0(n = 0,1,2, \ldots )$ and let there exist

\lim _{x \rightarrow R-0} \sum _{n = 0}^{\infty } a_{n} x^{n} = S

Prove that $\sum_{n = 0}^{\infty } a_{n} R^{n} = S$ .

Problem 2.20.27 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.20.2 "Problems"

Prove that if the power series $\sum_{n = 0}^{\infty } a_{n} x^{n}$ converges for $x \in (0 ; 1)$ ,

\lim _{n \rightarrow \infty } \frac{a_{1}+2 a_{2}+\ldots +n a_{n}}{n} = 0, \quad \lim _{x \rightarrow 1-0} \sum _{n = 0}^{\infty } a_{n} x^{n} = A,

then

\sum _{n = 0}^{\infty } a_{n} = A.

Problem 2.20.28 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.20.2 "Problems"

(1)(2)

Let the radius of convergence of the power series $\sum_{n = 0}^{\infty } a_{n} x^{n}$ equal 1 and $a_{n} > 0(n = 0,1,2, \ldots )$ . Prove that:

(1)

if the sum $f(x)$ of this series is bounded on the interval $[0 ; 1)$ , then the series $\sum_{n = 0}^{\infty } a_{n}$ converges and $\lim_{x \rightarrow 1-0} f(x) = \sum_{n = 0}^{\infty } a_{n}$ ;

(2)

if the series $\sum_{n = 1}^{\infty } a_{n}$ diverges, then $\lim_{x \rightarrow 1-0} f(x) = +\infty$ .

Problem VI.3.4 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VI.3 "Asymptotics of Sums"

(a)(b)(c)(d)

Let $a_{n} \downarrow 0, p \geqslant 0, \alpha \in \mathbb {R}$ . Prove that

(a)

if $\sum_{1 \leqslant k \leqslant n} k^{\alpha } a_{k}=O\left(n^{p}\right)$ , then $a_{n}=O\left(n^{p-\alpha -1}\right)$ ;

(b)

if $\sum_{1 \leqslant k \leqslant n} k^{\alpha } a_{k}=o\left(n^{p}\right)$ , then $a_{n}=o\left(n^{p-\alpha -1}\right)$ ;

(c)

if $\quad \sum_{k \geqslant n} k^{\alpha } a_{k}=O\left(n^{-p}\right)$ , then $a_{n}=O\left(n^{-p-\alpha -1}\right)$ ;

(d)

if $\quad \sum_{k \geqslant n} k^{\alpha } a_{k}=o\left(n^{-p}\right)$ , then $a_{n}=o\left(n^{-p-\alpha -1}\right)$ .

Problem VI.3.8 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VI.3 "Asymptotics of Sums"

(a)(b)

Let $a_{n} \downarrow 0, S_{n}=a_{1}+\ldots +a_{n} \sim \frac{n^{1-p}}{1-p}$ , where $0<p<1$ .

(a)

Prove that $a_{n} \sim n^{-p}$ .

(b)

Can one claim that $a_{n} \sim \frac{1}{n}$ , if $S_{n} \sim \ln n$ ?

Problem VI.3.9 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VI.3 "Asymptotics of Sums"

Let $a_{n} \downarrow 0$ and $\sum a_{n}<+\infty$ , with $\sum_{k \geqslant n} a_{k} \sim n^{-p}$ , where $p>0$ . Prove that $a_{n} \sim p n^{-1-p}$ .

Problem VI.3.10 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VI.3 "Asymptotics of Sums"

(a)(b)

Let $a_{n} \downarrow 0$ . Prove that

(a)

$\varlimsup_{\varepsilon \cdot \rightarrow +0} \frac{1}{\ln (1 / \varepsilon )} \sum a_{n}^{1+\varepsilon }=\varlimsup_{N \rightarrow \infty } \frac{1}{\ln \ln \left(1 / a_{N}\right)} \sum_{1 \leqslant n \leqslant N} a_{n}$ ;

(b)

$\overline{\lim }_{\varepsilon \rightarrow +0} \varepsilon \sum a_{n}^{1+\varepsilon } \leqslant \overline{\lim }_{N \rightarrow \infty } \frac{1}{\ln \left(1 / a_{N}\right)} \sum_{1 \leqslant n \leqslant N} a_{n} \leqslant e \overline{\lim }_{\varepsilon \rightarrow +0} \varepsilon \sum a_{n}^{1+\varepsilon }$ .

Can the coefficient $e$ in inequality b) be decreased?

Problem VI.3.11 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VI.3 "Asymptotics of Sums"

Let $\alpha_{n}>0, \sum \alpha_{n}<+\infty$ and $f(t)=t \sum \frac{\alpha_{n}}{1+t^{2} \alpha_{n}}$ for $t \geqslant 0$ . Prove that the behavior of the function $f$ as $t \rightarrow +\infty$ is closely related to the behavior of the sequence $\left\{ n \sqrt{\alpha_{n}}\right\}$ :

\frac{\pi }{2} \lim n \sqrt{\alpha _{n}} \leqslant \lim _{t \rightarrow +\infty } f(t) \leqslant \varlimsup _{t \rightarrow +\infty } f(t) \leqslant \frac{\pi }{2} \overline{\lim } n \sqrt{\alpha _{n}}

Moreover,

\sup _{t \geqslant 0} f(t) \leqslant \frac{\pi }{2} \sup n \sqrt{\alpha _{n}}

and the coefficient $\frac{\pi }{2}$ cannot be decreased.

Problem 3.9.1 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.9 "Tauberian Theory"

Prove the theorem of Tauber stated in Section 9.1.

Problem 3.9.8 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.9 "Tauberian Theory"

If $\alpha > 0$ and $\alpha x = 1$ , prove that

1-\alpha < \alpha [x] \leq 1

and deduce from this that $e^{x} K(x)$ is bounded, as asserted in the proof of Theorem 9.12.

Problem 3.9.14 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.9 "Tauberian Theory"

Complete the details in the following outline of Wiener's proof of Littlewood's theorem. Assume $\left|n a_{n}\right| \leq 1, f(r) = \sum_{0}^{\infty } a_{n} r^{n}$ , and $f(r) \rightarrow 0$ as $r \rightarrow 1$ . If $s_{n} = a_{0}+\cdots +a_{n}$ , it is to be proved that $s_{n} \rightarrow 0$ as $n \rightarrow \infty$ .

(a)

$\left|s_{n}-f(1-1 / n)\right| < 2$ . Hence $\left\{ s_{n}\right\}$ is bounded.

(b)

If $\phi (x) = s_{n}$ on $[n, n+1)$ and $0 < x < y$ , then

\left|\phi (y)-\phi (x)\right| \leq \frac{y+1-x}{x}

(c)

$\int_{0}^{\infty } x e^{-x t} \phi (t) d t = f\left(e^{-x}\right) \rightarrow 0$ as $x \rightarrow 0$ . Hence

\lim _{x \rightarrow \infty } \int _{0}^{\infty } K\left(\frac{x}{u}\right) \phi (u) \frac{d u}{u} = 0

K(x) = \left(\frac{1}{x}\right) \exp \left(-\frac{1}{x}\right)

(d)

$\int_{0}^{\infty } K(x) x^{-i t} \frac{d x}{x} = \Gamma (1+i t) \neq 0$ if $t$ is real.

(e)

Put $H(x) = 1 /(\varepsilon x)$ if $(1+\varepsilon )^{-1} < x < 1, H(x) = 0$ otherwise. Conclude that

\lim _{x \rightarrow \infty } \frac{1}{\varepsilon x} \int _{x}^{(1+\varepsilon ) x} \phi (y) d y = 0

(f)

By (b) and (e), $\lim_{x \rightarrow \infty } \phi (x) = 0$ . Note: If $n a_{n} \rightarrow 0$ is assumed to hold, then a modification of step $(a)$ is all that is needed for the proof.

References

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Интегралы. Ряды. 2-е изд., ФИЗМАТЛИТ, 2003.

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.

[Rud91] Rudin W. Functional Analysis. 2nd ed., McGraw-Hill, 1991.