# Difference between revisions of "Newman's Tauberian Theorem"

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− | + | '''Newman's Tauberian Theorem''' is a [[tauberian theorem]] | |

+ | first proven by D.J. Newman in 1980, in his short proof of | ||

+ | the [[prime number theorem]]. | ||

+ | |||

==Statement== | ==Statement== | ||

− | Let <math>f:(0,+\infty)\to\mathbb C</math> be a bounded function. Assume that its [[Laplace transform]] <math>F(s)=\int_0^\infty f(t)e^{-st} | + | |

+ | Let <math>f:(0,+\infty)\to\mathbb C</math> be a bounded function. Assume that | ||

+ | its [[Laplace transform]] <math>F(s) = \int_0^\infty f(t)e^{-st}dt</math> | ||

+ | (which is well-defined by this formula for <math>\Re s>0</math>) admits an | ||

+ | analytic extension (which we'll denote by the same letter <math>F</math>) | ||

+ | to some [[open set | open]] domain <math>E</math> containing the closed half-plane | ||

+ | <math>\{s\in\mathbb C : \Re s\ge 0\}</math>. Then, the integral | ||

+ | <math>\int_0^\infty f(t) dt</math> converges and its value equals <math>F(0)</math>. | ||

==Proof== | ==Proof== | ||

− | [[Image:Newmans_Tauberian_Contour.PNG|Contour picture]] | + | For every <math>T>0</math>, let <math>F_T(s) = \int_0^T f(t)e^{-st} dt</math>. The function |

+ | <math>F_T</math> is defined and analytic on the entire complex plane <math>\mathbb C</math>. | ||

+ | The conclusion of the theorem is equivalent to the assertion | ||

+ | <math>\lim_{T\to+\infty} F_T(0) = F(0)</math>. We choose some large <math>R>0</math>, | ||

+ | and some arbitrarily small <math>\delta > 0</math> | ||

+ | such that <math>F</math> is defined on the set | ||

+ | <cmath> \{ z \in \mathbb{C} \mid \Re z \ge - \delta, \lvert z \rvert | ||

+ | \le R \} . </cmath> | ||

+ | Let <math>\Gamma</math> be the counterclockwise contour on the boundary | ||

+ | of this set. Let <math>\Gamma_+</math> be the restriction of this contour | ||

+ | to the half-plane <math>\Re z \ge 0</math>. | ||

+ | Let <math>\Gamma_-^1</math> be the restriction of the contour to the set | ||

+ | <cmath> \{ z \in \mathbb{C} \mid \Re z \in (-\delta,0), \lvert z \rvert | ||

+ | = R \} , </cmath> | ||

+ | and let <math>\Gamma_-^2</math> be the restriction to the set | ||

+ | <cmath> \{ z \in \mathbb{C} \mid \Re z = -\delta, \lvert z \rvert \le R\}.</cmath> | ||

+ | Let <math>\Gamma_- = \Gamma_-^1 + \Gamma_-^2</math>, as shown in the diagram | ||

+ | below. | ||

+ | |||

+ | <!-- This code replaces the image given by | ||

+ | [[Image:Newmans_Tauberian_Contour.PNG|Contour picture]] --> | ||

+ | <asy> | ||

+ | size(200); | ||

+ | defaultpen(.7); | ||

+ | |||

+ | path C=circle((0,0),1); | ||

+ | path D=((-.3,-1.2)--(-.3,1.2)); | ||

+ | pair d1,d2; | ||

+ | |||

+ | d1 = intersectionpoints(C,D)[0]; | ||

+ | d2 = intersectionpoints(C,D)[1]; | ||

+ | |||

+ | draw(d2..(1,0)..d1,MidArrow); | ||

+ | draw(d1--d2); | ||

+ | draw(d1..(-1,0)..d2,dashed); | ||

+ | draw((-1.2,0)--(1.5,0),EndArrow); | ||

+ | draw((0,-1.2)--(0,1.5),EndArrow); | ||

+ | |||

+ | label("\Large{$\Gamma_+$}",(1.1,.5)); | ||

+ | label("\Large{$\Gamma_-$}",(-.45,.4)); | ||

+ | label("\Large{$\tilde\Gamma_-$}",(-1.1,.6)); | ||

+ | </asy> | ||

+ | |||

+ | |||

+ | By the [[Cauchy Integral Formula|Cauchy integral formula]], we have | ||

+ | <cmath> F(0)-F_T(0) = \frac{1}{2\pi i} \int_\Gamma K(z) (F(z)-F_T(z)) | ||

+ | \frac{dz}{z} , </cmath> | ||

+ | where | ||

+ | <cmath> K(z) = \left( 1 + \frac{z^2}{R^2}\right) e^{zT} . </cmath> | ||

+ | We will estimate this integral separately in the left and | ||

+ | right half-planes. In principle, <math>K</math> could be arbitrary, but | ||

+ | we have chosen <math>K</math> to make it easier to estimate this | ||

+ | integral. | ||

+ | |||

+ | We first estimate the difference <math>F(z) - F_T(z)</math> for <math>\Gamma_+</math>. | ||

+ | Let <math>M</math> be an upper bound for <math>\lvert f(x) \rvert</math>. | ||

+ | In the the right half-plane <math>\Re z > 0</math>, we note that | ||

+ | <cmath> \lvert F(z) - F_T(z) \rvert = \biggl\lvert \int_T^{\infty} | ||

+ | f(x)e^{-zt} dt \biggr\rvert \le M \int_T^{\infty} e^{-\Re (z) t} dt | ||

+ | = \frac{M e^{-\Re(z)T}}{\Re z} . </cmath> | ||

+ | |||

+ | Thus, we should kill the denominator <math>\Re z</math> for the integral | ||

+ | to converge. On the other hand, we can afford the kernel <math>K(z)</math> | ||

+ | growth as <math>e^{T\Re z}</math> in the right half-plane, which will allow | ||

+ | us corresponding decay in the left half-plane. Hence our choice | ||

+ | <cmath> K(z) = \left(1+\frac{z^2}{R^2}\right)e^{Tz}.</cmath> | ||

+ | This is convenient because for <math>\lvert z \rvert = R</math>, | ||

+ | <cmath> K(z) = \frac{2 z \Re z}{R^2} e^{Tz} , </cmath> | ||

+ | so that <math>K</math> kills the unpleasant denominator <math>\Re z</math> | ||

+ | on <math>\Gamma_+</math>. | ||

+ | |||

+ | We then have | ||

+ | <cmath> \biggl\lvert \int_{\Gamma_+} K(z)\bigl[ F(z) - F_T(z) \bigr] | ||

+ | \frac{dz}{z} \biggr\rvert | ||

+ | \le\int_{\Gamma_+} \frac{2M}{R^2} \lvert dz \rvert = \frac{2\pi M}{R} . </cmath> | ||

+ | |||

+ | To estimate the integral over <math>\Gamma_-</math>, we note that | ||

+ | <math>K(z)F_T(z)/z</math> is analytic in the left | ||

+ | half-plane, so we may change the integration path to the left semicircle | ||

+ | <math>\tilde\Gamma_-</math> of radius <math>R</math>. Now, on <math>\tilde\Gamma_-</math>, we have | ||

+ | <cmath> \lvert F_T(z) \rvert \le M \int_{0}^T \lvert e^{-zt} \rvert dt | ||

+ | = M \frac{e^{-T\Re z } - 1}{\lvert\Re z\rvert} < M \frac{e^{-T \Re | ||

+ | z}}{\lvert \Re z \rvert} . </cmath> | ||

+ | Then as before, | ||

+ | <cmath> \biggl\lvert \int_{\Gamma_-} F_T(z)K(z) \frac{dz}{z} \biggr\rvert | ||

+ | \le \frac{2\pi M}{R}. </cmath> | ||

+ | |||

+ | |||

+ | Now, let <math>N(R)</math> be an upper bound for the quantity | ||

+ | <cmath> \left\lvert F(z) \left(1 + \frac{z^2}{R^2} \right) \frac{1}{z} | ||

+ | \right\rvert </cmath> | ||

+ | on <math>\Gamma_-</math>. Then for <math>\delta < R</math>, | ||

+ | <cmath> \biggl\lvert \int_{\Gamma_-^1} F(z)K(z) \frac{dz}{z} \biggr\rvert | ||

+ | \le N(R) \biggl\lvert \int_{\Gamma_-^1} e^{zT} dz \biggr\rvert | ||

+ | < N(R) \cdot 4 \delta , </cmath> | ||

+ | and | ||

+ | <cmath> \biggl\lvert \int_{\Gamma_-^2} F(z)K(z) \frac{dz}{z} \biggr\rvert | ||

+ | \le N(R) \int_{\Gamma_-^2} e^{-\delta T} \lvert dz \rvert | ||

+ | < N(R) \cdot 2R e^{-\delta T} .</cmath> | ||

+ | Therefore | ||

+ | <cmath> \lvert F(0) - F_T(0) \rvert \le \frac{4\pi M}{R} + N(R) \cdot | ||

+ | 4\delta + N(R) \cdot 2Re^{-\delta T} . </cmath> | ||

+ | But as <math>T</math> becomes arbitrarily large, the last term vanishes, so | ||

+ | that | ||

+ | <cmath> \limsup_{T \to \infty} \lvert F(0) - F_T(0) \rvert | ||

+ | \le \frac{4\pi M}{R} + N(R)\cdot 4 \delta . </cmath> | ||

+ | We can make <math>\delta</math> arbitrarily small, so that the second term | ||

+ | vanishes. Then we pick an arbitrarily large <math>R</math>, so that the | ||

+ | first term vanishes, and the theorem follows. | ||

+ | <math>\blacksquare</math> | ||

+ | |||

+ | ==See also== | ||

+ | |||

+ | * [[Tauberian theorem]] | ||

+ | * [[Prime Number Theorem|Prime number theorem]] | ||

+ | |||

+ | [[Category:Complex analysis]] |

## Latest revision as of 06:55, 12 August 2019

**Newman's Tauberian Theorem** is a tauberian theorem
first proven by D.J. Newman in 1980, in his short proof of
the prime number theorem.

## Statement

Let be a bounded function. Assume that its Laplace transform (which is well-defined by this formula for ) admits an analytic extension (which we'll denote by the same letter ) to some open domain containing the closed half-plane . Then, the integral converges and its value equals .

## Proof

For every , let . The function is defined and analytic on the entire complex plane . The conclusion of the theorem is equivalent to the assertion . We choose some large , and some arbitrarily small such that is defined on the set Let be the counterclockwise contour on the boundary of this set. Let be the restriction of this contour to the half-plane . Let be the restriction of the contour to the set and let be the restriction to the set Let , as shown in the diagram below.

By the Cauchy integral formula, we have
where
We will estimate this integral separately in the left and
right half-planes. In principle, could be arbitrary, but
we have chosen to make it easier to estimate this
integral.

We first estimate the difference for . Let be an upper bound for . In the the right half-plane , we note that

Thus, we should kill the denominator for the integral to converge. On the other hand, we can afford the kernel growth as in the right half-plane, which will allow us corresponding decay in the left half-plane. Hence our choice This is convenient because for , so that kills the unpleasant denominator on .

We then have

To estimate the integral over , we note that is analytic in the left half-plane, so we may change the integration path to the left semicircle of radius . Now, on , we have Then as before,

Now, let be an upper bound for the quantity
on . Then for ,
and
Therefore
But as becomes arbitrarily large, the last term vanishes, so
that
We can make arbitrarily small, so that the second term
vanishes. Then we pick an arbitrarily large , so that the
first term vanishes, and the theorem follows.