Английская Википедия:Borel summation

In mathematics, Borel summation is a summation method for divergent series, introduced by Шаблон:Harvs. It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several variations of this method that are also called Borel summation, and a generalization of it called Mittag-Leffler summation.

Definition

There are (at least) three slightly different methods called Borel summation. They differ in which series they can sum, but are consistent, meaning that if two of the methods sum the same series they give the same answer.

Throughout let Шаблон:Math denote a formal power series

<math>A(z) = \sum_{k = 0}^\infty a_kz^k,</math>

and define the Borel transform of Шаблон:Math to be its equivalent exponential series

<math>\mathcal{B}A(t) \equiv \sum_{k=0}^\infty \frac{a_k}{k!}t^k.</math>

Borel's exponential summation method

Let Шаблон:Math denote the partial sum

A weak form of Borel's summation method defines the Borel sum of Шаблон:Math to be

<math> \lim_{t\rightarrow\infty} e^{-t}\sum_{n=0}^\infty \frac{t^n}{n!}A_n(z). </math>

If this converges at Шаблон:Math to some function Шаблон:Math, we say that the weak Borel sum of Шаблон:Math converges at Шаблон:Math, and write <math> {\textstyle \sum} a_kz^k = a(z) \, (\boldsymbol{wB}) </math>.

Borel's integral summation method

Suppose that the Borel transform converges for all positive real numbers to a function growing sufficiently slowly that the following integral is well defined (as an improper integral), the Borel sum of Шаблон:Math is given by

<math>\int_0^\infty e^{-t} \mathcal{B}A(tz) \, dt. </math>

If the integral converges at Шаблон:Math to some Шаблон:Math, we say that the Borel sum of Шаблон:Math converges at Шаблон:Math, and write <math> {\textstyle \sum} a_kz^k = a(z) \,(\boldsymbol B) </math>.

Borel's integral summation method with analytic continuation

This is similar to Borel's integral summation method, except that the Borel transform need not converge for all Шаблон:Math, but converges to an analytic function of Шаблон:Math near 0 that can be analytically continued along the positive real axis.

Basic properties

Regularity

The methods Шаблон:Math and Шаблон:Math are both regular summation methods, meaning that whenever Шаблон:Math converges (in the standard sense), then the Borel sum and weak Borel sum also converge, and do so to the same value. i.e.

<math> \sum_{k=0}^\infty a_k z^k = A(z) < \infty \quad \Rightarrow \quad {\textstyle \sum} a_kz^k = A(z) \,\, (\boldsymbol{B},\,\boldsymbol{wB}). </math>

Regularity of Шаблон:Math is easily seen by a change in order of integration, which is valid due to absolute convergence: if Шаблон:Math is convergent at Шаблон:Math, then

<math> A(z) = \sum_{k=0}^\infty a_k z^k = \sum_{k=0}^\infty a_k \left( \int_{0}^\infty e^{-t}t^k dt \right) \frac{z^k}{k!} = \int_{0}^\infty e^{-t} \sum_{k=0}^\infty a_k \frac{(tz)^k}{k!}dt, </math>

where the rightmost expression is exactly the Borel sum at Шаблон:Math.

Regularity of Шаблон:Math and Шаблон:Math imply that these methods provide analytic extensions to Шаблон:Math.

Nonequivalence of Borel and weak Borel summation

Any series Шаблон:Math that is weak Borel summable at Шаблон:Math is also Borel summable at Шаблон:Math. However, one can construct examples of series which are divergent under weak Borel summation, but which are Borel summable. The following theorem characterises the equivalence of the two methods.

Theorem (Шаблон:Harv).

Let Шаблон:Math be a formal power series, and fix Шаблон:Math, then:

If <math> {\textstyle \sum} a_kz^k = a(z) \, (\boldsymbol{wB}) </math>, then <math> {\textstyle \sum}a_kz^k = a(z) \, (\boldsymbol{B})</math>.
If <math> {\textstyle \sum} a_kz^k = a(z) \, (\boldsymbol{B}) </math>, and <math> \lim_{t \rightarrow \infty} e^{-t}\mathcal B A(zt) = 0, </math> then <math> {\textstyle \sum} a_kz^k = a(z) \, (\boldsymbol{wB}) </math>.

Relationship to other summation methods

Шаблон:Math is the special case of Mittag-Leffler summation with Шаблон:Math.
Шаблон:Math can be seen as the limiting case of generalized Euler summation method Шаблон:Math in the sense that as Шаблон:Math the domain of convergence of the Шаблон:Math method converges up to the domain of convergence for Шаблон:Math.^[1]

Uniqueness theorems

There are always many different functions with any given asymptotic expansion. However, there is sometimes a best possible function, in the sense that the errors in the finite-dimensional approximations are as small as possible in some region. Watson's theorem and Carleman's theorem show that Borel summation produces such a best possible sum of the series.

Watson's theorem

Watson's theorem gives conditions for a function to be the Borel sum of its asymptotic series. Suppose that Шаблон:Math is a function satisfying the following conditions:

Шаблон:Math is holomorphic in some region Шаблон:Math, Шаблон:Math for some positive Шаблон:Math and Шаблон:Math.
In this region Шаблон:Math has an asymptotic series Шаблон:Math with the property that the error

<math>|f(z)-a_0 -a_1z -\cdots -a_{n-1}z^{n-1}|</math>

is bounded by

for all Шаблон:Math in the region (for some positive constant Шаблон:Math).

Then Watson's theorem says that in this region Шаблон:Math is given by the Borel sum of its asymptotic series. More precisely, the series for the Borel transform converges in a neighborhood of the origin, and can be analytically continued to the positive real axis, and the integral defining the Borel sum converges to Шаблон:Math for Шаблон:Math in the region above.

Carleman's theorem

Carleman's theorem shows that a function is uniquely determined by an asymptotic series in a sector provided the errors in the finite order approximations do not grow too fast. More precisely it states that if Шаблон:Math is analytic in the interior of the sector Шаблон:Math, Шаблон:Math and Шаблон:Math in this region for all Шаблон:Math, then Шаблон:Math is zero provided that the series Шаблон:Math diverges.

Carleman's theorem gives a summation method for any asymptotic series whose terms do not grow too fast, as the sum can be defined to be the unique function with this asymptotic series in a suitable sector if it exists. Borel summation is slightly weaker than special case of this when Шаблон:Math for some constant Шаблон:Math. More generally one can define summation methods slightly stronger than Borel's by taking the numbers Шаблон:Math to be slightly larger, for example Шаблон:Math or Шаблон:Math. In practice this generalization is of little use, as there are almost no natural examples of series summable by this method that cannot also be summed by Borel's method.

Example

The function Шаблон:Math has the asymptotic series Шаблон:Math with an error bound of the form above in the region Шаблон:Math for any Шаблон:Math, but is not given by the Borel sum of its asymptotic series. This shows that the number Шаблон:Math in Watson's theorem cannot be replaced by any smaller number (unless the bound on the error is made smaller).

Examples

The geometric series

Consider the geometric series

<math>A(z) = \sum_{k = 0}^\infty z^k,</math>

which converges (in the standard sense) to Шаблон:Math for Шаблон:Math. The Borel transform is

<math>\mathcal{B}A(tz) \equiv \sum_{k=0}^\infty \frac{z^k}{k!}t^k = e^{zt},</math>

from which we obtain the Borel sum

<math>\int_0^\infty e^{-t}\mathcal{B}A(tz) \, dt = \int_0^\infty e^{-t} e^{tz} \, dt =\frac{1}{1-z}</math>

which converges in the larger region Шаблон:Math, giving an analytic continuation of the original series.

Considering instead the weak Borel transform, the partial sums are given by Шаблон:Math, and so the weak Borel sum is

<math> \lim_{t \rightarrow \infty}e^{-t} \sum_{n=0}^\infty \frac{1 -z^{n+1}}{1-z} \frac{t^n}{n!} = \lim_{t \rightarrow \infty} \frac{e^{-t}}{1-z} \big( e^t - z e^{tz} \big) = \frac{1}{1-z}, </math>

where, again, convergence is on Шаблон:Math. Alternatively this can be seen by appealing to part 2 of the equivalence theorem, since for Шаблон:Math,

<math> \lim_{t \rightarrow \infty} e^{-t} (\mathcal{B} A)(zt) = e^{t(z-1)} = 0. </math>

An alternating factorial series

Consider the series

<math>A(z) = \sum_{k = 0}^\infty k!(-1)^k \cdot z^k,</math>

then Шаблон:Math does not converge for any nonzero Шаблон:Math. The Borel transform is

<math>\mathcal{B}A(t) \equiv \sum_{k=0}^\infty \left(-t\right)^k = \frac{1}{1+t} </math>

for Шаблон:Math, which can be analytically continued to allШаблон:Math. So the Borel sum is

<math>\int_0^\infty e^{-t}\mathcal{B}A(tz) \, dt = \int_0^\infty \frac{e^{-t}} {1+tz} \, dt = \frac 1 z \cdot e^{1/z} \cdot \Gamma\left(0,\frac 1 z \right)</math>

(where Шаблон:Math is the incomplete gamma function).

This integral converges for all Шаблон:Math, so the original divergent series is Borel summable for all suchШаблон:Math. This function has an asymptotic expansion as Шаблон:Math tends to 0 that is given by the original divergent series. This is a typical example of the fact that Borel summation will sometimes "correctly" sum divergent asymptotic expansions.

Again, since

<math> \lim_{t \rightarrow \infty} e^{-t} (\mathcal B A)(zt) = \lim_{t \rightarrow \infty} \frac{e^{-t}}{1 + zt} = 0, </math>

for all Шаблон:Math, the equivalence theorem ensures that weak Borel summation has the same domain of convergence, Шаблон:Math.

An example in which equivalence fails

The following example extends on that given in Шаблон:Harv. Consider

<math>A(z) = \sum_{k = 0}^\infty \left( \sum_{\ell=0}^\infty \frac{(-1)^\ell(2\ell + 2)^k}{(2\ell+1)!} \right) z^k. </math>

After changing the order of summation, the Borel transform is given by

<math>

\begin{align} \mathcal B A(t)&= \sum_{\ell = 0}^\infty \left( \sum_{k=0}^\infty \frac{\big((2\ell+2) t\big)^k}{k!} \right) \frac{(-1)^\ell}{(2\ell+1)!} \\ &= \sum_{\ell=0}^\infty e^{(2\ell+2)t}\frac{(-1)^\ell}{(2\ell+1)!} \\ &= e^t \sum_{\ell=0}^\infty \big(e^t\big)^{2\ell+1} \frac{(-1)^\ell}{(2\ell+1)!} \\ & = e^t \sin(e^t). \end{align}

</math>

At Шаблон:Math the Borel sum is given by

<math> \int_0^\infty e^t \sin(e^{2t}) \, dt = \int_1^\infty \sin(u^2) \, du = \sqrt{\frac{\pi}{8}} - S(1) < \infty,

</math>

where Шаблон:Math is the Fresnel integral. Via the convergence theorem along chords, the Borel integral converges for all Шаблон:Math (the integral diverges for Шаблон:Math).

For the weak Borel sum we note that

<math> \lim_{t \rightarrow \infty} e^{(z-1)t}\sin(e^{zt}) = 0 </math>

holds only for Шаблон:Math, and so the weak Borel sum converges on this smaller domain.

Existence results and the domain of convergence

Summability on chords

If a formal series Шаблон:Math is Borel summable at Шаблон:Math, then it is also Borel summable at all points on the chord Шаблон:Math connecting Шаблон:Math to the origin. Moreover, there exists a function Шаблон:Math analytic throughout the disk with radius Шаблон:Math such that

<math> {\textstyle \sum} a_kz^k = a(z) \, (\boldsymbol B), </math>

for all Шаблон:Math.

An immediate consequence is that the domain of convergence of the Borel sum is a star domain in Шаблон:Math. More can be said about the domain of convergence of the Borel sum, than that it is a star domain, which is referred to as the Borel polygon, and is determined by the singularities of the series Шаблон:Math.

The Borel polygon

Suppose that Шаблон:Math has strictly positive radius of convergence, so that it is analytic in a non-trivial region containing the origin, and let Шаблон:Math denote the set of singularities of Шаблон:Math. This means that Шаблон:Math if and only if Шаблон:Math can be continued analytically along the open chord from 0 to Шаблон:Math, but not to Шаблон:Math itself. For Шаблон:Math, let Шаблон:Math denote the line passing through Шаблон:Math which is perpendicular to the chord Шаблон:Math. Define the sets

<math> \Pi_P = \{z \in \mathbb{C} \, \colon \, Oz \cap L_P = \varnothing \}, </math>

the set of points which lie on the same side of Шаблон:Math as the origin. The Borel polygon of Шаблон:Math is the set

<math> \Pi_A = \operatorname{cl}\left( \bigcap_{P \in S_A} \Pi_P \right). </math>

An alternative definition was used by Borel and Phragmén Шаблон:Harv. Let <math> S \subset \mathbb{C} </math> denote the largest star domain on which there is an analytic extension of Шаблон:Math, then <math> \Pi_A </math> is the largest subset of <math> S </math> such that for all <math> P \in \Pi_A </math> the interior of the circle with diameter OP is contained in <math>S </math>. Referring to the set <math> \Pi_A </math> as a polygon is somewhat of a misnomer, since the set need not be polygonal at all; if, however, Шаблон:Math has only finitely many singularities then <math> \Pi_A </math> will in fact be a polygon.

The following theorem, due to Borel and Phragmén provides convergence criteria for Borel summation.

Theorem Шаблон:Harv.

The series Шаблон:Math is Шаблон:Math summable at all <math>z \in \operatorname{int}(\Pi_A)</math>, and is Шаблон:Math divergent at all <math>z \in \mathbb{C}\setminus \Pi_A</math>.

Note that Шаблон:Math summability for <math> z \in \partial \Pi_A</math> depends on the nature of the point.

Example 1

Let Шаблон:Math denote the Шаблон:Math-th roots of unity, Шаблон:Math, and consider

<math>\begin{align}

A(z) & = \sum_{k=0}^\infty (\omega_1^k + \cdots + \omega_m^k)z^k \\

& = \sum_{i=1}^m \frac{1}{1-\omega_iz}, \end{align} </math>

which converges on Шаблон:Math. Seen as a function on Шаблон:Math, Шаблон:Math has singularities at Шаблон:Math, and consequently the Borel polygon <math> \Pi_A</math> is given by the regular [[Regular polygon|Шаблон:Math-gon]] centred at the origin, and such that Шаблон:Math is a midpoint of an edge.

Example 2

The formal series

<math> A(z) = \sum_{k=0}^\infty z^{2^k}, </math>

converges for all <math> |z| < 1 </math> (for instance, by the comparison test with the geometric series). It can however be shown^[2] that Шаблон:Math does not converge for any point Шаблон:Math such that Шаблон:Math for some Шаблон:Math. Since the set of such Шаблон:Math is dense in the unit circle, there can be no analytic extension of Шаблон:Math outside of Шаблон:Math. Subsequently the largest star domain to which Шаблон:Math can be analytically extended is Шаблон:Math from which (via the second definition) one obtains <math> \Pi_A = B(0,1) </math>. In particular one sees that the Borel polygon is not polygonal.

A Tauberian theorem

A Tauberian theorem provides conditions under which convergence of one summation method implies convergence under another method. The principal Tauberian theorem^[1] for Borel summation provides conditions under which the weak Borel method implies convergence of the series.

Theorem Шаблон:Harv. If Шаблон:Math is Шаблон:Math summable at Шаблон:Math, <math>{\textstyle \sum}a_kz_0^k = a(z_0) \, (\boldsymbol{wB}) </math>, and

<math> a_kz_0^k = O(k^{-1/2}), \qquad \forall k \geq 0, </math>

then <math> \sum_{k=0}^\infty a_kz_0^k = a(z_0) </math>, and the series converges for all Шаблон:Math.

Applications

Borel summation finds application in perturbation expansions in quantum field theory. In particular in 2-dimensional Euclidean field theory the Schwinger functions can often be recovered from their perturbation series using Borel summation Шаблон:Harv. Some of the singularities of the Borel transform are related to instantons and renormalons in quantum field theory Шаблон:Harv.

Generalizations

Borel summation requires that the coefficients do not grow too fast: more precisely, Шаблон:Math has to be bounded by Шаблон:Math for some Шаблон:Math. There is a variation of Borel summation that replaces factorials Шаблон:Math with Шаблон:Math for some positive integer Шаблон:Math, which allows the summation of some series with Шаблон:Math bounded by Шаблон:Math for some Шаблон:Math. This generalization is given by Mittag-Leffler summation.

In the most general case, Borel summation is generalized by Nachbin resummation, which can be used when the bounding function is of some general type (psi-type), instead of being exponential type.

Notes

↑ ^1,0 ^1,1 Hardy, G. H. (1992). Divergent Series. AMS Chelsea, Rhode Island.
↑ Шаблон:Cite web

References

[Hardy1992-1] 1,0 ^1,1 Hardy, G. H. (1992). Divergent Series. AMS Chelsea, Rhode Island.

[2] Шаблон:Cite web

[1]

[2]

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