Asymptotic expansion of thick distributions II

Abstract

In this article we continue our research in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19), about the asymptotic expansion of thick distributions. We compute more examples of asymptotic expansion of integral transforms using the techniques developed in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19). Besides, we derive a new “Laplace Formula” for the situation in which a point singularity is allowed.

Keywords

Distributions asymptotic expansion singular integral Laplace formula thick distributions

1. Introduction

The theory of asymptotic expansions of generalized functions finds its usefulness in many areas in mathematics. The theory itself as well as its applications in number theory, geometry, and mathematical physics has been studied by several research groups in recent years [4,9–11,14].

The theory of thick distributions is a generalization of the classical theory of Schwartz distributions, in which it allows a singular point on the test functions [2,12,13].

In a previous paper [14], we have found some results in asymptotic expansion of thick distributions, mainly on the so called “moment asymptotic expansion”. We also calculated several examples of asymptotic expansion of thick distributions as illustrations of our theorems.

This article will present some follow-up research about asymptotic expansion of thick distributions. It can be viewed as a second part of [14]. This article has two main parts: in Section 3, we will compute more examples of asymptotic expansion of certain integral transforms, using the method developed in [14]. These integral transforms all allow a singular point, which is the main characteristic of thick distributions. To be more precise, we will compute asymptotic expansions of the following integral transforms:

\begin{aligned} I_{1} (λ) & = F . p . \int_{R^{n}} ϕ (x) e^{- λ | x |^{2}} d x . \\ I_{2} (λ) & = F . p . \int_{R^{n}} (λ r)^{t} e^{- (λ r)^{2}} ϕ (x) d x . \\ I_{3} (λ) & = F . p . \int_{R^{n}} ϕ (x) e^{λ i r^{2}} d x \end{aligned}

λ \to + \infty

, and

r = | x |

. Here we assume that

ϕ \in D_{*} (R^{n})

, i.e., it is a thick test function and allows a singularity at the origin.

A second main part of this article is devoted to develop a new “Laplace Formula”. We consider the asymptotic evaluation of multidimensional integrals of the type

\begin{aligned} I (λ) = \int_{R} e^{- λ h (x)} ϕ (x) d x, \end{aligned}

where

h (x)

is a real function and where R is a region of

R^{n}

ϕ \in D_{*} (R^{n})

. Under these conditions we find a new Laplace Formula:

\begin{aligned} e^{- λ | Ψ (x) |^{2}} \sim b_{M - n + 1} \frac{g_{M} δ_{*, x_{0}}^{[M]} (x)}{\sqrt{det (D^{2} h (x_{0}))} λ^{(M + n) / 2}} . \end{aligned}

We will give a detailed explanation and derivation of this formula in Section 4.

2. Some reviews

Let us first quote some theorems and propositions from [14].

Theorem 2.1.
Let $f \in K_{}^{^{'}} (R^{n})$ , with moment functions* $μ_{k} (w)$ , then
$\begin{aligned} f (λ x) \sim \sum_{k = - \infty}^{\infty} \frac{C_{n - 1} μ_{k} (w) δ_{}^{[k]}}{λ^{k + n}}, a s λ \to \infty, \end{aligned}$
(2.1)
where* $C_{n - 1}$ , given by $C_{n - 1} = \frac{2 π^{n / 2}}{Γ (n / 2)}$ , is the surface area of the unit sphere $S^{n - 1}$ . This means that $f (λ x) \sim \sum_{k = m}^{\infty} C_{n - 1} λ^{- k - n} μ_{k} (w) δ_{}^{[k]}$ , for some m.*

This means that for any test function $ϕ \in K_{} (R^{n})$ , $N \in Z$ ,
$\begin{aligned} ⟨ f (λ x), ϕ (x) ⟩ = \sum_{k = m}^{N - 1} \frac{⟨ C_{n - 1} μ_{k} (w) δ_{}^{[k]}, ϕ (x) ⟩}{λ^{k + n}} + O (\frac{1}{λ^{N + n}}), \end{aligned}$
where m is the degree of the starting term of the asymptotic expansion of ϕ with respect to the radius r when $r \to 0^{+}$ . Namely, $ϕ (x) = ϕ (r w) \sim \sum_{j = m}^{\infty} a_{j} (w) r^{j}$ when $r \to 0^{+}$ .

For the proof of this theorem, please see [14].
Proposition 2.2.
If $ϕ \in D_{} (R^{n})$ , $ψ \in E_{} (R^{n})$ , $ϕ (x) \sim \sum_{j = m}^{\infty} a_{j} (w) r^{j}$ and $ψ (x) \sim \sum_{j = p}^{\infty} b_{j} (w) r^{j}$ , as $x \to 0$ , and $υ > 0$ then
$\begin{aligned} F . p . \int_{R^{n}} ϕ (υ x) ψ (x) d x & = υ^{- n} F . p . \int_{R^{n}} ϕ (x) ψ (x / υ) d x \\ - \sum_{l = p}^{- n - m} υ^{- n - l} \log υ \int_{S} a_{- l - n} (w) b_{l} (w) d σ (w) . \end{aligned}$
(2.2)

We can also consider (2.2) as a transformation formula for the pseudofunction $P f (ψ (x))$ .
Proposition 2.3.
If $ψ \in E_{} (R^{n})$ , with* $ψ (x) \sim \sum_{j = p}^{\infty} b_{j} (w) r^{j}$ , as $x \to 0$ , and $λ > 0$ then
$\begin{aligned} P f (ψ (λ x)) & = P f {ψ (z); λ x} \\ = P f {ψ (λ z); x} + C_{n - 1} \sum_{j = - \infty}^{- n - p} \frac{\log λ}{λ^{n + j}} b_{- n - j} (w) δ_{*}^{[j]} (x) . \end{aligned}$
(2.3)

Proof.
Indeed,
$\begin{aligned} ⟨ P f {ψ (z); λ x}, ϕ (x) ⟩ = λ^{- n} F . p . \int_{R^{n}} ϕ (λ^{- 1} x) ψ (x) d x, \end{aligned}$
(2.4)
while
$\begin{aligned} ⟨ P f {ψ (λ z); x}, ϕ (x) ⟩ = F . p . \int_{R^{n}} ϕ (x) ψ (λ x) d x, \end{aligned}$
(2.5)
and thus (2.3) follows from (2.2) by taking $υ = 1 / λ$ .

The above two propositions are also from [14]. Of course they can be applied to a more broad situation where the finite part integral $F . p . \int_{R^{n}} ϕ (x) ψ (x) d x$ exists. For a more complete explanation of finite part integrals we refer to [14] and [3,4,8].
3. Computation of some examples

In this section, we want to compute some examples of asymptotic expansions of certain integrals with a parameter. In this section, we assume that $ϕ \in D_{*} (R^{n})$ , and $ϕ (r w) \sim \sum_{j = m}^{\infty} a_{j} (w) r^{j}$ as $r \to 0^{+}$ .

Example 3.1.
$f (x) = e^{- | x |^{2}}$ . We want to compute the asymptotic expansion of the integral
$\begin{aligned} I_{1} (λ) = F . p . \int_{R^{n}} ϕ (x) e^{- λ | x |^{2}} d x . \end{aligned}$
Notice that $e^{- | x |^{2}} = e^{- r^{2}} \sim \sum_{j = 0}^{\infty} \frac{(- 1)^{j} r^{2 j}}{j!}$ as $r \to 0^{+}$ , then using Proposition 2.2, we have
$\begin{aligned} I_{1} (λ) & = F . p . \int_{R^{n}} ϕ (x) e^{- | λ^{1 / 2} x |^{2}} d x \\ = λ^{- \frac{n}{2}} F . p . \int_{R^{n}} ϕ (x / λ^{1 / 2}) e^{- | x |^{2}} d x \\ - \frac{1}{2} \sum_{k = 0}^{⌊ \frac{- n - m}{2} ⌋} \frac{λ^{k} (- 1)^{k} \log λ}{k!} \int_{S} a_{- n - 2 k} (w) d σ (w) . \end{aligned}$
On the other hand,
$\begin{aligned} λ^{- \frac{n}{2}} F . p . \int_{R^{n}} ϕ (x / λ^{1 / 2}) e^{- | x |^{2}} d x \\ = λ^{- \frac{n}{2}} \int_{R^{n}} (ϕ (x / λ^{1 / 2}) - \sum_{j = m}^{- n} a_{j} (w) {(\frac{r}{λ^{1 / 2}})}^{j}) e^{- | x |^{2}} d x \\ + \sum_{j = m}^{- n} (λ^{- \frac{n + j}{2}} \int_{S} a_{j} (w) d σ (w)) (F . p . lim_{ε \to 0^{+}} \int_{ε}^{+ \infty} r^{j + n - 1} e^{- r^{2}} d r) . \end{aligned}$
Observe that
$\begin{aligned} F . p . lim_{ε \to 0^{+}} \int_{ε}^{+ \infty} r^{j + n - 1} e^{- r^{2}} d r & = \frac{1}{2} F . p . lim_{ε \to 0} \int_{ε^{2}}^{+ \infty} u^{\frac{j + n - 2}{2}} e^{- u} d u \\ = \frac{1}{2} Γ_{reg} (\frac{j + n}{2}), \end{aligned}$
where
$\begin{aligned} Γ_{reg} (μ + 1) & = F . p . \int_{0}^{+ \infty} x^{μ} e^{- x} dx \\ = {\begin{cases} Γ (μ + 1) & when~ μ \neq - 1, - 2, - 3, \dots \\ (- 1)^{k} γ (k + 1) / k! & when~ k = - μ - 1 = 0, 1, 2, \dots \end{cases}, \end{aligned}$
where $γ (z) = Γ^{'} (z) / Γ (z)$ is the logarithmic derivative of the gamma function [7], so that [4]
$\begin{aligned} γ (k + 1) = \sum_{j = 1}^{k} \frac{1}{j} - γ . \end{aligned}$

For a detailed computation of $Γ_{reg} (μ + 1)$ , please see [14], equations (6.16)–(6.21).

Hence
$\begin{aligned} F . p . lim_{ε \to 0} \int_{ε}^{+ \infty} r^{j + n - 1} e^{- r^{2}} d r = {\begin{cases} \frac{1}{2} Γ (\frac{j + n}{2}) & when~ j + n is odd \\ {(- 1)}^{\frac{j + n}{2}} \frac{γ (- \frac{j + n}{2} + 1)}{2 (- \frac{j + n}{2})!} & when~ j + n is even \end{cases} . \end{aligned}$
On the other hand, it is easy to compute that
$\begin{aligned} λ^{- \frac{n}{2}} \int_{R^{n}} (ϕ (x / λ^{1 / 2}) - \sum_{j = m}^{- n} a_{j} (w) {(\frac{r}{λ^{1 / 2}})}^{j}) e^{- | x |^{2}} d x \\ \sim \frac{1}{2} \sum_{j = - n + 1}^{+ \infty} (λ^{- \frac{n + j}{2}} \int_{S} a_{j} (w) d σ (w)) Γ (\frac{j + n}{2}) . \end{aligned}$
In summary, we have, if $- n - m$ is even,
$\begin{aligned} I_{1} (λ) & = F . p . \int_{R^{n}} ϕ (x) e^{- | λ^{1 / 2} x |^{2}} d x \\ \sim \frac{1}{2} \sum_{k = \frac{- n - m}{2}}^{0} (\frac{λ^{k} (- 1)^{k} (γ (k + 1) - \log λ)}{k!} \int_{S} a_{- n - 2 k} (w) d σ (w) \\ + λ^{k - 1 / 2} Γ (- k + \frac{1}{2}) \int_{S} a_{- n - 2 k + 1} (w) d σ (w)) \\ + \frac{1}{2} \sum_{j = - n + 2}^{+ \infty} (λ^{- \frac{n + j}{2}} \int_{S} a_{j} (w) d σ (w)) Γ (\frac{j + n}{2}) . \end{aligned}$
(3.1)
And the situation when $- n - m$ is odd is similar with just a few index change.

Notice that this expansion was also computed in [1]. Here we summarize the details in [14] and [1].
Example 3.2.

$\begin{aligned} f (x) & = | x |^{t} e^{- | x |^{2}} = r^{t} e^{- r^{2}}, where r denotes the (Euclidean) norm of x, and t \in Z . \end{aligned}$

We want to compute the asymptotic expansion of the finite part integral
$\begin{aligned} I_{2} (λ) = F . p . \int_{R^{n}} (λ r)^{t} e^{- (λ r)^{2}} ϕ (x) d x . \end{aligned}$
Obviously, the asymptotic expansion of $r^{t} e^{- r^{2}}$ at the origin is
$\begin{aligned} r^{t} e^{- r^{2}} \sim r^{t} - \frac{r^{t + 2}}{1} + \frac{r^{t + 4}}{2!} - \frac{r^{t + 6}}{3!} + \dots \end{aligned}$
Hence by Proposition 2.2, we have
$\begin{aligned} I_{2} (λ) & = λ^{- n} F . p . \int_{R^{n}} r^{t} e^{- r^{2}} ϕ (\frac{x}{λ}) d x \\ - \sum_{k = 0}^{⌊ (- m - n - t) / 2 ⌋} λ^{2 k + t} \log λ \int_{S} a_{- 2 k - n - t} (w) \frac{{(- 1)}^{k}}{k!} d σ (w) . \end{aligned}$
(3.2)

On the other hand, we compute that
$\begin{aligned} λ^{- n} F . p . \int_{R^{n}} r^{t} e^{- r^{2}} ϕ (\frac{x}{λ}) d x \\ = λ^{- n} \int_{R^{n}} r^{t} e^{- r^{2}} (ϕ (\frac{x}{λ}) - \sum_{j = m}^{- n - t} a_{j} (w) \frac{r^{j}}{λ^{j}}) d x \\ + \sum_{j = m}^{- n - t} (λ^{- n - j} \int_{S} a_{j} (w) d σ (w)) (F . p . lim_{ε \to 0^{+}} \int_{ε}^{+ \infty} r^{j + n - 1 + t} e^{- r^{2}} d r) . \end{aligned}$
And
$\begin{aligned} F . p . lim_{ε \to 0^{+}} \int_{ε}^{+ \infty} r^{j + n - 1 + t} e^{- r^{2}} d r = \frac{1}{2} Γ_{reg} (\frac{j + n + t}{2}), \\ λ^{- n} \int_{R^{n}} r^{t} e^{- r^{2}} (ϕ (\frac{x}{λ}) - \sum_{j = m}^{- n - t} a_{j} (w) \frac{r^{j}}{λ^{j}}) d x \\ \sim \frac{1}{2} \sum_{j = - n - t + 1}^{+ \infty} (λ^{- n - j} \int_{S} a_{j} (w) d σ (w)) Γ_{reg} (\frac{j + n + t}{2}) . \end{aligned}$
In summary, we have, if $m + n + t$ is even,
$\begin{aligned} I_{2} (λ) & = F . p . \int_{R^{n}} {(λ r)}^{t} e^{- {(λ r)}^{2}} ϕ (x) d x \\ \sim \sum_{k = (m + n + t) / 2}^{0} (\frac{λ^{- 2 k + t} {(- 1)}^{k} (γ (- k + 1) - 2 \log λ)}{2 (- k)!} \int_{S} a_{2 k - n - t} (w) d σ (w) \\ + \frac{1}{2} λ^{- 2 k - 1 + t} Γ (k + \frac{1}{2}) \int_{S} a_{2 k - n - t + 1} (w) d σ (w)) \\ + \frac{1}{2} \sum_{j = - n - t + 2}^{+ \infty} (λ^{- n - j} \int_{S} a_{j} (w) d σ (w)) Γ (\frac{j + n + t}{2}) . \end{aligned}$
(3.3)
And when $m + n + t$ is odd the result is similar with just a few index change.
Example 3.3.

$\begin{aligned} f (x) = e^{i r^{2}} . \end{aligned}$

Clearly, the asymptotic expansion of $e^{i r^{2}}$ at the origin is
$\begin{aligned} e^{i r^{2}} \sim 1 + \frac{i r^{2}}{1} - \frac{r^{4}}{2!} - \frac{i r^{6}}{3!} + \dots + \frac{(i r^{2})^{k}}{k!} + \dots \end{aligned}$
We can compute that
$\begin{aligned} I_{3} (λ) & = F . p . \int_{R^{n}} ϕ (x) e^{λ i r^{2}} d x \\ = λ^{- \frac{n}{2}} F . p . \int_{R^{n}} ϕ (\frac{x}{λ^{1 / 2}}) e^{i r^{2}} d x \\ - \frac{1}{2} \sum_{k = 0}^{⌊ (- m - n) / 2 ⌋} λ^{k} \log λ \frac{i^{k}}{k!} \int_{S} a_{- 2 k - n} (w) d σ (w) \end{aligned}$
(3.4)
And
$\begin{aligned} λ^{- \frac{n}{2}} F . p . \int_{R^{n}} ϕ (\frac{x}{λ^{1 / 2}}) e^{i r^{2}} d x \\ = λ^{- \frac{n}{2}} \int_{R^{n}} e^{i r^{2}} (ϕ (\frac{x}{λ^{1 / 2}}) - \sum_{j = m}^{- n} a_{j} (w) \frac{r^{j}}{λ^{j / 2}}) d x \\ + \sum_{j = m}^{- n} (λ^{- \frac{n + j}{2}} \int_{S} a_{j} (w) d σ (w)) (F . p . lim_{ε \to 0^{+}} \int_{ε}^{+ \infty} r^{j + n - 1} e^{i r^{2}} d r) . \end{aligned}$
Now let $u = r^{2}$ , we can compute
$\begin{aligned} F . p . lim_{ε \to 0^{+}} \int_{ε}^{+ \infty} r^{j + n - 1} e^{i r^{2}} d r & = \frac{1}{2} F . p . lim_{ε \to 0^{+}} \int_{ε}^{+ \infty} u^{\frac{j + n - 2}{2}} e^{i u} d u \\ = \frac{1}{2} Γ_{reg} (\frac{j + n}{2}) e^{- i π (j + n) / 4}, \end{aligned}$
when we view $F . p . lim_{ε \to 0^{+}} \int_{ε}^{+ \infty} u^{\frac{j + n - 2}{2}} e^{i u} d u$ as the finite part of the Fourier transform of $u_{+}^{\frac{j + n}{2} - 1}$ with the transform variable equal to $- 1$ . And this result can be found in [6], for example.

On the other hand, when $j > - n$ , the Fourier transform of $u_{+}^{\frac{j + n}{2} - 1}$ , with the transform variable equal to $- 1$ , is $\frac{1}{2} Γ (\frac{j + n}{2}) e^{- i π (j + n) / 4}$ . That is, there is no singularity. Hence we have, for $m + n$ even,
$\begin{aligned} I_{3} (λ) & = F . p . \int_{R^{n}} ϕ (x) e^{λ i r^{2}} d x \\ \sim \sum_{k = (m + n) / 2}^{0} (λ^{- k} (\frac{γ (- k + 1) e^{- i π k / 2} - \log λ i^{- k}}{2 (- k)!}) \int_{S} a_{2 k - n} (w) d σ (w) \\ + \frac{1}{2} λ^{- k - 1 / 2} Γ (k + \frac{1}{2}) e^{- i π (2 k + 1) / 4} \int_{S} a_{2 k - n + 1} (w) d σ (w)) \\ + \frac{1}{2} \sum_{j = - n + 2}^{+ \infty} λ^{- \frac{n + j}{2}} Γ (\frac{j + n}{2}) e^{- i π (j + n) / 4} \int_{S} a_{j} (w) d σ (w) . \end{aligned}$
(3.5)
Again, the case when $m + n$ is odd is easy to get.

4. Laplace formula

Let us consider the asymptotic evaluation of multidimensional integrals of the type

\begin{aligned} I (λ) = \int_{R} e^{- λ h (x)} ϕ (x) d x, \end{aligned}

where

h (x)

is a real function and where R is a region of

R^{n}

. We assume that

h (x)

is smooth in a region containing the closure of the region R. It was shown by Focke [5] that the main contribution to

I (λ)

for

λ ≫ 1

comes from the vicinity of the minima of

h (x)

\bar{R}

. The isolated critical points of h where minima occur are classified into three types according to their location: the interior critical points; the critical points of the smooth parts of the boundary; the critical points situated on the non-smooth parts of the boundary. The asymptotic expansion takes a different form for each type of critical points. In this section we want to discuss the asymptotic expansion of

I (λ)

when R contains a first type critical point

x_{0}

, i.e., when it contains an interior critical point.

We also assume that $ϕ (x)$ is smooth except for at the point $x_{0}$ . In fact, we assume that $ϕ (x) \in K_{*, x_{0}} (R^{n})$ . That is, it is tempered when $| x | \to \infty$ and, it admits an asymptotic expansion on $r = | x |$ when $x \to x_{0}$ .

By assumption, $x_{0}$ is an interior minimum of h, non-degenerate. Namely, ${\frac{\partial h}{\partial x_{i}} |}_{x_{0}} = 0$ and its Hessian matrix

\begin{aligned} A = {(\frac{\partial^{2} h}{\partial x_{i} \partial x_{j}}) |}_{x_{0}} \end{aligned}

is positive definite. Thus, we can find

y = Ψ (x) = (ψ_{i} (x))

, with

Ψ (x_{0}) = 0

, with positive Jacobian near

x_{0}

, and such that,

\begin{aligned} h (x) = h (x_{0}) + | Ψ (x) |^{2} . \end{aligned}

Thus

\begin{aligned} I (λ) = e^{- λ h (x_{0})} \int_{R} e^{- λ | Ψ (x) |^{2}} ϕ (x) d x . \end{aligned}

By the computation we did in Example 3.1 above,

\begin{aligned} I_{1} (λ) & = F . p . \int_{R^{n}} ϕ (x) e^{- | λ^{1 / 2} x |^{2}} d x \\ \sim \frac{1}{2} \sum_{k = \frac{- n - m}{2}}^{0} (\frac{λ^{k} {(- 1)}^{k} (γ (k + 1) - \log λ)}{k!} \int_{S} a_{- n - 2 k} (w) d σ (w) \\ + λ^{k - 1 / 2} Γ (- k + \frac{1}{2}) \int_{S} a_{- n - 2 k + 1} (w) d σ (w)) \\ + \frac{1}{2} \sum_{j = - n + 2}^{+ \infty} (λ^{- \frac{n + j}{2}} \int_{S} a_{j} (w) d σ (w)) Γ (\frac{j + n}{2}) . \\ e^{- λ | Ψ (x) |^{2}} & \sim \sum_{m = - \infty}^{- 1} C_{n - 1} (\frac{\sqrt{π} {(- 2)}^{m + 1} δ_{*}^{[2 m - n + 1]} (Ψ (x))}{2 (2 m + 1)!! λ^{m + 1 / 2}} - \frac{γ δ_{*}^{[2 m - n + 2]} (Ψ (x))}{k! λ^{m + 1}}) \\ + \sum_{m = 0}^{\infty} C_{n - 1} (\frac{\sqrt{π} (2 m)! δ_{*}^{[2 m - n + 1]} (Ψ (x))}{(m! 2^{2 m + 1}) λ^{m + 1 / 2}} + \frac{2 m!! δ_{*}^{[2 m - n + 2]} (Ψ (x))}{2^{m + 1} λ^{m + 1}}) . \end{aligned}

Now, let us consider the leading term of this expansion. Let us pick up a test function $ϕ (x) \in K_{*, x_{0}} (R^{n})$ . We know that it admits an asymptotic expansion

\begin{aligned} ϕ (x) \sim \sum_{k = m}^{\infty} a_{k} (w) | x |^{k} \end{aligned}

with some integer m. In the following context let us use

b_{k - n + 1}

to denote the constant coefficients of the expansion of

e^{- λ | Ψ (x) |^{2}}

for simplicity. i.e.

\begin{aligned} e^{- λ | Ψ (x) |^{2}} \sim \sum_{k = - \infty}^{\infty} b_{k - n + 1} \frac{δ_{*}^{[k]} (Ψ (x))}{λ^{(k + n) / 2}} . \end{aligned}

Before discussing the expansion, let us present the following fact:

Proposition 4.1.

Let $g (w) δ_{*}^{[k]} (x) \in K_{*}^{^{'}} (R^{n})$ , then

\begin{aligned} g (w) δ_{*}^{[k]} (x - x_{0}) = g (w) δ_{*, x_{0}}^{[k]} (x) \in K_{*, x_{0}}^{^{'}} (R^{n}) . \end{aligned}

Recall, if $ϕ (x) \in K_{*, x_{0}} (R^{n})$ , it admits an asymptotic expansion when $| x - x_{0} | \to 0$ as $\sum_{j = M}^{\infty} a_{j} (w) | x - x_{0} |^{j}$ , and $g (w) δ_{*, x_{0}}^{[k]} (x)$ acts on $ϕ (x)$ as

\begin{aligned} {⟨ g (w) δ_{*, x_{0}}^{[k]} (x), ϕ (x) ⟩}_{K_{*, x_{0}}^{^{'}} (R^{n}) \times K_{*, x_{0}} (R^{n})} = {⟨ g (w), a_{k} (w) ⟩}_{D^{'} (S) \times D (S)} . \end{aligned}

Proof.

\begin{aligned} ⟨ g (w) δ_{*}^{[k]} (x - x_{0}), ϕ (x) ⟩ = ⟨ g (w) δ_{*}^{[k]} (x), ϕ (x + x_{0}) ⟩ \end{aligned}

(4.1)

The statement follows if one observes that $x + x_{0}$ is to move the coordinate system towards $x_{0}$ .

Now, in order to discuss the leading term of the expansion, let us consider the following

\begin{aligned} ⟨ δ_{*}^{[k]} (Ψ (x)), ϕ (x) ⟩ \\ = ⟨ \frac{δ_{*}^{[k]} (y)}{det (D Ψ (x_{0}))}, ϕ (Ψ^{- 1} (y)) ⟩ = ⟨ \frac{δ_{*}^{[k]} (y)}{\sqrt{det (D^{2} h (x_{0}))}}, ϕ (Ψ^{- 1} (y)) ⟩, \end{aligned}

where

D^{2} h (x_{0}) = A

denotes the Hessian matrix at

x_{0}

. Note

Ψ^{- 1} (0) = x_{0}

, if we write

Ψ^{- 1} (y) = x_{0} + {\tilde{Ψ}}^{- 1} (y)

, then

{\tilde{Ψ}}^{- 1} (0) = 0

. We know that

ϕ (x)

admits an asymptotic expansion as

\sum_{j = m}^{\infty} a_{j} (w) | x - x_{0} |^{j}

when

| x - x_{0} | \to 0

. One can see that

\begin{aligned} ϕ (Ψ^{- 1} (y)) \sim \sum_{j = m}^{\infty} a_{j} (\frac{{\tilde{Ψ}}^{- 1} (y)}{| {\tilde{Ψ}}^{- 1} (y) |}) {| {\tilde{Ψ}}^{- 1} (y) |}^{j} as | {\tilde{Ψ}}^{- 1} (y) | \to 0. \end{aligned}

Since

{\tilde{Ψ}}^{- 1} (0) = 0

, there is an

n \times n

matrix B such that

\begin{aligned} {\tilde{Ψ}}^{- 1} (y) = B y + R (y) \end{aligned}

With

\begin{aligned} | R (y) | = O (| y^{2} |) . \end{aligned}

In fact,

B = D (Ψ^{- 1} (x_{0}))

, it is just the derivative of

Ψ^{- 1} (x)

x_{0}

Hence

\begin{aligned} ϕ (Ψ^{- 1} (y)) \sim a_{M} (\frac{B y}{| B y |}) | B y |^{M} as y \to 0 \end{aligned}

Namely, the leading term of the expansion can be viewed as a linear change of variable.

Observe, if we denote $y = r w$ ,

\begin{aligned} a_{M} (\frac{B y}{| B y |}) | B y |^{M} = a_{M} (\frac{B w}{| B w |}) | B w |^{M} r^{M} . \end{aligned}

Thus in conclusion,

\begin{aligned} e^{- λ | Ψ (x) |^{2}} \sim b_{M - n + 1} \frac{g_{M} δ_{*, x_{0}}^{[M]} (x)}{\sqrt{det (D^{2} h (x_{0}))} λ^{(M + n) / 2}}, \end{aligned}

where

g_{M} \in D^{'} (S^{n - 1})

and

\begin{aligned} ⟨ g_{M} (w), a (w) ⟩ = ⟨ 1, a (\frac{B w}{| B w |}) | B w |^{M} ⟩ . \end{aligned}

Footnotes

Acknowledgements

Thanks to National Natural Science Foundation of China to support this research. The grant number is 12001150.

Thanks to Hefei University of Technology to support this research. The grant number from Hefei University of Technology is 407-0371000086.

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