Asymptotic expansion of thick distributions

Abstract

We give a theory of asymptotic expansions of thick distributions of rapid decay at infinity. We show that the moment asymptotic expansion of standard distributions of rapid decay follows by projection of our result. We also study in which spaces of thick test functions the asymptotic Taylor approximation is valid.

We employ our formulas to obtain the asymptotic expansion of several multidimensional integrals that are divergent but are regularized by using the Hadamard method.

Keywords

thick points delta functions distributions generalized functions asymptotic expansions

1. Introduction

The theory of asymptotic expansions of generalized functions was probably initiated by the work of Łojasiewicz [28], and has been studied by several authors in the recent years; see [14,31,32,38] and the references therein. These studies have provided a deep understanding of the local properties of distributions and of their behavior at infinity and have helped us to understand many aspects of Tauberian theory; applications in several areas, particularly in number theory, geometry, and mathematical physics have been developed.

The aim of this article is to present a theory of the asymptotic expansion of thick distributions, a new class of distributions introduced in one variable in [11] and in several variables by us [41]. Thick distributions help us to solve several puzzles, apparent paradoxes in the applications of distribution theory in quantum field theory [5], in engineering [29,37], and in the understanding of singularities in mathematical physics as considered in [4] or in [6]. We consider spaces with one thick point in this article, but it is possible to consider spaces with a finite number of such singular points, or even spaces with a countable number of thick points [43]. Thick distributions are the distributional theory corresponding to the theory of finite part regularizations given by Blanchet and Faye [2], who develop such a scheme in the context of finite parts, pseudofunctions and Hadamard regularization, subjects considered earlier by Sellier [33,34]; their analysis is aimed at the study of the dynamics of point particles in high post-Newtonian approximations of general relativity [3]. Thick distributions also appear in problems that seem well understood: The textbook formula for the second order distributional derivatives of $1 / r$ in $R^{3}$ $(r = | x |)$ , was given by Frahm [16]; recently, Franklin [17] gave an alternative formula for certain singular functions,1

¹
It is not clear to which singular test functions the formula of [17] applies.

while we gave the correct generalization that applies in the space of thick distributions [42].

The main result in the approach of [14] is the moment asymptotic expansion, that says that if g is a distribution defined in the whole $R^{n}$ that decays very fast at infinity2

Technically rapid decay means $f \in K^{'} (R^{n})$ .

then

g (λ x)

has the asymptotic expansion

g (λ x) \sim \sum_{j = 0}^{\infty} \sum_{| k | = j} \frac{{(- 1)}^{j} μ_{k} D^{k} δ (x)}{k! λ^{n + j}}, as λ \to \infty,

(1.1) where the constants

μ_{k}

are the moments of g,

μ_{k} = ⟨ g (x), x^{k} ⟩

It should be remarked that if ${ν_{k}}_{k \in N^{n}}$ is any sequence indexed by multi-indices, it is possible to consider a corresponding infinite series of Dirac delta functions, $V = V_{{ν_{k}}_{k \in N^{n}}}$ , $V (x) = \sum_{j = 0}^{\infty} \sum_{| k | = j} \frac{{(- 1)}^{j} ν_{k} D^{k} δ (x)}{k!} .$ (1.2) Such series of deltas have a long story, starting with the study of orthogonal polynomials [22–24], the solution of differential equations [27,35,39], the asymptotics of such solutions [18], and the boundary layer theory [13], to name a few topics; see especially [8] and [14, Chapter 7]. However, it is important to observe that unless only a finite number of the $ν_{k}$ are different from zero, the series is divergent in any space of distributions, as follows immediately from Borel’s theorem [14, Theorem 1.5.2]; actually one even can construct series of deltas that diverge as hyperfunctions [14, Chapter 7]. Therefore, if g is a distribution, whose moments $μ_{k} = ⟨ g (x), x^{k} ⟩$ all exist, one can consider the series of deltas $\sum_{j = 0}^{\infty} \sum_{| k | = j} {(- 1)}^{j} μ_{k} D^{k} δ (x) / k!$ , but this series will not converge, to g or to any other distribution, if the series is infinite. On the other hand, the question of convergence is irrelevant when the extra parameter λ is introduced and the series is considered as an asymptotic series, since only finite sums are needed to understand asymptotic series; interestingly, it is sometimes possible to write certain functions and distributions as finite sums of delta functions plus a remainder term [7].

In this article we prove that thick distributions f that show rapid decay at infinity, in the sense that $f \in K_{*}^{'} (R^{n})$ , satisfy a generalized moment expansion, namely, $f (λ x) ∽ \sum_{k = - \infty}^{\infty} \frac{2 π^{n / 2} μ_{k} (w) δ_{*}^{[k]}}{Γ (n / 2) λ^{k + n}}, as λ \to \infty,$ (1.3) where the $μ_{k} \in D^{'} (S)$ are the moment functions of f, and where the $δ_{*}^{[k]}$ are thick delta functions as defined in Definition 2.4. Notice that, in general, the formal series of thick delta functions $\sum_{k = - \infty}^{\infty} 2 π^{n / 2} μ_{k} (w) δ_{*}^{[k]} / Γ (n / 2)$ is divergent in $K_{*}^{'} (R^{n})$ .

The plan of the article is as follows. In Section 2 we define the standard thick test functions, $D_{*, a} (R^{n})$ , and distributions, $D_{*, a}^{'} (R^{n})$ . Other spaces of thick distributions, corresponding to the spaces $E$ , $S$ , and especially $K$ , are defined in Section 3. We then study asymptotic Taylor series of thick test functions in Section 4, and this allows us to prove the generalized expansion (1.3) in Section 5; we also show how the ordinary moment asymptotic expansion follows by projection. Section 6 provides some formulas arising in finite part integrals that are needed in Section 7, where we give several illustrations of the generalized moment expansion (1.3).

2. The spaces

D_{*, a} (R^{n})

and

D_{*, a}^{'} (R^{n})

Let us recall the construction of the space of thick test functions $D_{*, a} (R^{n})$ and its dual, $D_{*, a}^{'} (R^{n})$ , the space of thick distributions [41]. Let a be a fixed point of $R^{n}$ .

Definition 2.1.
Let $D_{, a} (R^{n})$ denote the vector space of all smooth functions ϕ defined in $R^{n} ∖ {a}$ , with support of the form $K ∖ {a}$ , where K is compact in $R^{n}$ , that admit a strong asymptotic expansion of the form $ϕ (a + x) = ϕ (a + r w) \sim \underset{j = m}{\sum^{\infty}} a_{j} (w) r^{j}, as x \to 0,$ (2.1) where m is an integer (positive or negative), and where the $a_{j}$ are smooth functions of w, that is, $a_{j} \in D (S)$ . The subspace $D_{, a}^{[m]} (R^{n})$ consists of those test functions ϕ whose expansion (2.1) begins at m. For a fixed compact K whose interior contains a, $D_{, a}^{[m; K]} (R^{n})$ is the subspace formed by those test functions of $D_{, a}^{[m]} (R^{n})$ that vanish in $R^{n} ∖ K$ .

The asymptotic development of $ϕ (x)$ as $x \to a$ has to be “strong”. This means [14, Chapter 1] that for any differentiation operator $D^{p} = \partial^{| p |} / \partial x_{1}^{p_{1}} \dots \partial x_{n}^{p_{n}}$ , $p = (p_{1}, \dots, p_{n}) \in N^{n}$ , the asymptotic development of $D^{p} ϕ (x)$ as $x \to a$ exists and is equal to the term-by-term differentiation of $\sum_{j = m}^{\infty} a_{j} (w) r^{j}$ . Observe that saying that the expansion exists as $x \to 0$ is the same as saying that it exists as $r \to 0^{+}$ , uniformly with respect to w.

We call $D_{, a} (R^{n})$ the space of standard* test functions on $R^{n}$ with a thick point located at $x = a$ . We denote $D_{, 0} (R^{n})$ as $D_{} (R^{n})$ . The topology of this space is constructed as follows.
Definition 2.2.
Let m be a fixed integer and K a compact subset of $R^{n}$ whose interior contains a. The topology of $D_{, a}^{[m; K]} (R^{n})$ is given by the seminorms ${∥ \cdot ∥_{q, s, K}}_{q > m, s ⩾ 0}$ defined as $∥ ϕ ∥_{q, s, K} = sup_{x + a \in K} sup_{| p | ⩽ s} | D^{p} ϕ (a + x) - \sum_{j = m - | p |}^{q - 1} a_{j, p} (w) r^{j} | r^{- q},$ (2.2) where $x = r w$ and3
³
Formulas for the coefficients $a_{j, p}$ can be found in [41].

$D^{p} ϕ (a + x) \sim \sum_{j = m - | p |}^{\infty} a_{j, p} (w) r^{j}$ . The topology of $D_{, a}^{[m]} (R^{n})$ is the inductive limit topology of the $D_{, a}^{[m; K]} (R^{n})$ as $K ↗ \infty$ . The topology of $D_{, a} (R^{n})$ is the inductive limit topology of the $D_{, a}^{[m]} (R^{n})$ as $m ↘ - \infty$ .

A sequence ${ϕ_{l}}_{l = 0}^{\infty}$ in $D_{, a} (R^{n})$ converges to ψ if and only there exists $l_{0} ⩾ 0$ , an integer m, and a compact set K with a in its interior, such that $ϕ_{l} \in D_{, a}^{[m; K]} (R^{n})$ for $l ⩾ l_{0}$ and $∥ ψ - ϕ_{l} ∥_{q, s, K} \to 0$ as $l \to \infty$ if $q > m, s ⩾ 0$ . Notice that if ${ϕ_{l}}_{l = 0}^{\infty}$ converges to ψ in $D_{, a} (R^{n})$ then $ϕ_{l}$ and the corresponding derivatives converge uniformly to ψ and its derivatives in any set of the form $R^{n} ∖ B$ , where B is a ball with center at a; in fact, $r^{| p | - m} D^{p} ϕ_{l}$ converges uniformly to $r^{| p | - m} D^{p} ψ$ over all $R^{n}$ . Furthermore, if ${a_{j}^{l}}$ are the coefficients of the expansion of $ϕ_{l}$ and ${b_{j}}$ are those for ψ, then $a_{j}^{l} \to b_{j}$ in the space $D (S)$ for each $j ⩾ m$ .

We can now consider standard distributions in a space with one thick point, the standard thick distributions.
Definition 2.3.
The space of distributions on $R^{n}$ with a thick point at $x = a$ is the dual space of $D_{, a} (R^{n})$ . We denote it $D_{, a}^{'} (R^{n})$ , or just as $D_{}^{'} (R^{n})$ when $a = 0$ .

Observe that $D (R^{n})$ , the space of standard test functions, is a closed subspace of $D_{, a} (R^{n})$ ; we denote by $Π : D_{, a}^{'} (R^{n}) \to D^{'} (R^{n}),$ (2.3) the projection operator, dual of the inclusion $i : D (R^{n}) \to D_{, a} (R^{n})$ .

Let $C = C_{n - 1} = \frac{2 π^{n / 2}}{Γ (n / 2)},$ (2.4) be the surface area of the unit sphere $S$ of $R^{n}$ .
Definition 2.4.
Let $g \in D^{'} (S)$ be a distribution in $S$ . The thick delta function of degree q, denoted as $g δ_{}^{[q]}$ , or as $g (w) δ_{}^{[q]}$ , acts on a thick test function $ϕ (x)$ as ${⟨ g δ_{}^{[q]}, ϕ ⟩}_{D_{}^{'} (R^{n}) \times D_{} (R^{n})} = \frac{1}{C} {⟨ g (w), a_{q} (w) ⟩}_{D^{'} (S) \times D (S)},$ (2.5) where $ϕ (r w) \sim \sum_{j = m}^{\infty} a_{j} (w) r^{j}$ , as $r \to 0^{+}$ .

Thick deltas of order 0 are called just thick deltas, and we shall use the notation $g δ_{}$ instead of $g δ_{}^{[0]}$ . If ${g_{q}}_{q = - \infty}^{p}$ is an arbitrary* sequence of distributions of $D^{'} (S)$ , then the series $\sum_{q = - \infty}^{p} g_{q} δ_{}^{[q]}$ converges in $D_{}^{'} (R^{n})$ , and such series are the most general thick distributions whose support is the set ${0}$ .
3. Other spaces of thick distributions

Let $A (R^{n})$ be a space of test functions in $R^{n}$ and let $A^{'} (R^{n})$ be the corresponding space of distributions.4

⁴
In the sense of Zemanian [44]; we assume that $D (R^{n}) \subset A (R^{n}) \subset E (R^{n})$ densely and continuously and that differentiation is a continuous map of $A (R^{n})$ .

Our aim in this section is to construct the corresponding spaces of thick test functions and distributions,

A_{*, a} (R^{n})

and

A_{*, a}^{'} (R^{n})

Our construction will apply in multiple cases. For instance, $A (R^{n})$ can be $E (R^{n})$ , the space of all smooth functions and thus $A^{'} (R^{n})$ becomes $E^{'} (R^{n})$ , the space of distributions with compact support; or $A (R^{n})$ can be $S (R^{n})$ , so that $A^{'} (R^{n})$ becomes the space of tempered distributions $S^{'} (R^{n})$ . In this study the case of $K (R^{n})$ and $K^{'} (R^{n})$ plays a central role. Other possibilities include the spaces $O_{M} (R^{n})$ , $O_{C} (R^{n})$ , or $P (R^{n})$ . We shall recall the properties of these spaces when needed, but more details can be found in the textbooks [14,20,21,36].

Definition 3.1.

Let $A (R^{n})$ be a space of test functions in $R^{n}$ . The space $A_{*, a} (R^{n})$ consists of those functions ϕ defined in $R^{n} ∖ {a}$ that can be written as $ϕ_{1} + ϕ_{2}$ , where $ϕ_{1} \in D_{*, a} (R^{n})$ and where $ϕ_{2} \in A (R^{n})$ . The topology of $A_{*, a} (R^{n})$ is the final topology induced by the map $A : D_{*, a} (R^{n}) \times A (R^{n}) \to D_{*, a} (R^{n})$ , $A (ϕ_{1}, ϕ_{2}) = ϕ_{1} + ϕ_{2}$ . The space of thick distributions $A_{*, a}^{'} (R^{n})$ is the corresponding dual space.

Similarly one can consider the spaces $A_{*, a}^{[m]} (R^{n})$ for any integer m; the space $A_{*, a} (R^{n})$ is the inductive limit of the $A_{*, a}^{[m]} (R^{n})$ as $m ↘ - \infty$ .

The topology of $A_{*, a} (R^{n})$ can actually be described in several ways. Suppose for instance that $ρ \in D (R^{n})$ is a test function that satisfies that $ρ (x) = 1$ in a neighborhood of $x = a$ . If $∥ \cdot ∥_{1}$ is a continuous seminorm of $D_{*, a} (R^{n})$ while $∥ \cdot ∥_{2}$ is a continuous seminorm of $A (R^{n})$ , then5

⁵

Technically $ϕ_{2} = (1 - ρ) ϕ$ does not belong to $A (R^{n})$ since it is not defined at $x = a,$ but naturally one can extend the domain of $ϕ_{2}$ by asking that $ϕ_{2} (a) = 0$ ; we shall not mention such clear situations any more.

∥ ϕ ∥ = max {∥ ρ ϕ ∥_{1}, {∥ (1 - ρ) ϕ ∥}_{2}},

(3.1) is a continuous seminorm of

A_{*, a} (R^{n})

and the collection of seminorms so constructed form a basis for the continuous seminorms of

A_{*, a} (R^{n})

. We shall consider alternative constructions of basis of seminorms for various spaces

A_{*, a} (R^{n})

below.

The elements of $A_{*, a} (R^{n})$ can be described as those smooth functions defined in $R^{n} ∖ {a}$ that show the behavior of thick test functions near $x = a$ while at infinity show the behavior of the elements of $A (R^{n})$ . Similar considerations apply to the dual spaces.

3.1. The spaces

E_{*, a} (R^{n})

and

E_{*, a}^{'} (R^{n})

The space $E_{*, a} (R^{n})$ is the space of multipliers of $D_{*, a} (R^{n})$ [41]. Its elements are smooth functions defined in $R^{n} ∖ {a}$ that behave like elements of $D_{*, a} (R^{n})$ near $x = a$ but without any restriction at infinity. No topology for the space $E_{*, a} (R^{n})$ was considered in [41]; with the topology given in the Definition 3.1 then the dual space $E_{*, a}^{'} (R^{n})$ is the subspace of $D_{*, a}^{'} (R^{n})$ formed by those thick distributions whose support, a subset of $R^{n} ∖ {a}$ , is of the form $K ∖ {a}$ where K is compact. For a fixed m the family of seminorms ${∥ \cdot ∥_{q, s, K}}$ defined in (2.2), for $q > m, s ⩾ 0$ , and K a compact set of $R^{n}$ that contains a, form a basis of seminorms for the topology of $E_{*, a}^{[m]} (R^{n})$ .

3.2. The spaces $K_{, a} (R^{n})$ and $K_{, a}^{'} (R^{n})$

Let us employ the notation $ρ_{p} (r) = 1, if 0 ⩽ r ⩽ 1 and r^{- p}, if r > 1 .$ (3.2) The spaces $K$ and $K^{'}$ play a rather important role in the theory of asymptotic expansions of distributions [14]. The space $K (R^{n})$ is the inductive limit of the spaces $K_{q} (R^{n})$ as $q ↗ \infty$ , where $K_{q} (R^{n})$ is the space of all smooth functions in $R^{n}$ that satisfy $D^{p} ϕ (x) = O (| x |^{q - | p |})$ (3.3) for all multi-indices $p \in N^{n}$ ; the topology of $K_{q} (R^{n})$ is defined by the family of seminorms $∥ ϕ ∥_{K_{q} (R^{n}), s} = sup_{x \in R^{n}} max_{| p | ⩽ s} ρ_{q - | p |} (x) | D^{p} ϕ (x) | .$ (3.4)

We can construct a family of seminorms of $K_{q, *} (R^{n})$ as follows; the general case of $K_{*, a} (R^{n})$ is then obtained by a change of variables. If $ϕ \in K_{q, *}^{[m]} (R^{n})$ , put $∥ ϕ ∥_{K_{q, *}^{[m]}, l, s} = max {∥ ϕ ∥_{l, s}^{i}, ∥ ϕ ∥_{K_{q, *}, s}^{o}},$ (3.5) where $\begin{array}{rclr} ∥ ϕ ∥_{l, s}^{i} = sup_{| x | ⩽ 1} sup_{| p | ⩽ s} | D^{p} ϕ (x) - \sum_{j = m - | p |}^{l - 1} a_{j, p} (w) r^{j} | r^{- l}, & (3.6) \\ ∥ ϕ ∥_{K_{q, *}, s}^{o} = sup_{| x | ⩾ 1} max_{| p | ⩽ s} ρ_{q - | p |} (x) | D^{p} ϕ (x) | . & (3.7) \end{array}$

3.3. The spaces

S_{*, a} (R^{n})

and

S_{*, a}^{'} (R^{n})

The family of seminorms $∥ ϕ ∥_{S_{*}^{[m]}, l, k, s} = max {∥ ϕ ∥_{l, s}^{i}, ∥ ϕ ∥_{S_{*}, k, s}^{o}},$ (3.8) where $∥ ϕ ∥_{l, s}^{i}$ is given by (3.6) and where $∥ ϕ ∥_{S_{*}, k, s}^{o} = sup_{| x | ⩾ 1} max_{| p | ⩽ s} ρ_{- k} (x) | D^{p} ϕ (x) |$ (3.9) defines the topology of the space $S_{*}^{[m]} (R^{n})$ .

4. Taylor series in spaces of test functions

If ϕ is a smooth function in $R^{n}$ then we can consider its Taylor series $S_{ϕ} (x) = \sum_{j = 0}^{\infty} \sum_{| k | = j} a_{k} x^{k}, where a_{k} = \frac{D^{k} ϕ (0)}{k!} .$ (4.1) Naturally $S_{ϕ} (x)$ does not have to converge for any $x \neq 0$ and even if convergent it may not converge to $ϕ (x)$ . However, $S_{ϕ} (x)$ is an asymptotic series6

⁶

Asymptotic series do not have to be divergent, but many such series used in applied problems are [14, Chapter 1].

x \to 0

ϕ (x) = \sum_{j = 0}^{M} \sum_{| k | = j} a_{k} x^{k} + O (| x |^{M + 1})

(4.2) for any

M \in N

Similarly, if ϕ is a thick test function of a space $A_{q, *}^{[m]} (R^{n})$ then the series $S_{ϕ}^{*, m} (x) = \sum_{j = m}^{\infty} a_{j} (w) r^{k},$ (4.3) provides an asymptotic approximation of $ϕ (x)$ as $x \to 0$ , and, likewise, the series is usually divergent or if convergent it does not have to converge to $ϕ (x)$ .

If $ϕ \in A (R^{n})$ then, in general, the Taylor series $S_{ϕ} (x)$ cannot provide an asymptotic series in the topology of $A (R^{n})$ since actually, in many cases, the individual terms, the monomials $x^{k}$ , do not even belong to $A (R^{n})$ ; consider the spaces $D (R^{n})$ or $S (R^{n})$ , for instance. It follows from the results of [9] (see [14, Chapter 6]) that in the space $K (R^{n})$ the series $S_{ϕ} (ε x)$ gives an asymptotic expansion of $ϕ (ε x)$ as $ε \to 0$ , and actually $K$ is the smallest space of test functions with this property. Our aim is to show that a similar result holds for thick test functions of the space $K_{*} (R^{n})$ , but before we do so we would like to give an example.

Example 4.1.

Let $Q$ be the space of polynomials in n variables, with the LF topology as an inductive limit of finite dimensional spaces [36]. Consider the space of test functions $A (R^{n}) = D (R^{n}) \oplus Q$ , with the direct sum topology. Then the series $S_{ϕ} (x)$ makes sense as a formal series in $A (R^{n})$ since $\sum_{j = 0}^{M} \sum_{| k | = j} a_{k} x^{k}$ belongs to $A (R^{n})$ for any M. However, in general, $S_{ϕ} (ε x)$ does not give an asymptotic expansion of $ϕ (ε x)$ as $ε \to 0$ (just take ϕ as any non-zero element of $D (R^{n})$ ).

Our aim is to prove an asymptotic Taylor formula in the space $K_{*} (R^{n})$ . In order to do so we first need an auxiliary result.

Lemma 4.2.

Let $ϕ \in K_{q, *}^{[k]} (R^{n})$ . Suppose that $q ⩾ k ⩾ m$ . Then for each continuous seminorm $∥ \cdot ∥$ of $K_{q, *}^{[m]} (R^{n})$ there exists a constant M such that $∥ ϕ (ε x) ∥ ⩽ M ε^{k}, 0 < ε < 1 .$ (4.4)

Proof.

Indeed, it is enough to prove the result for the norms $∥ ϕ ∥_{K_{q, *}^{[m]}, l, s}$ defined in (3.5).

In fact, observe that if $ϕ \in K_{q, *}^{[k]} (R^{n})$ then for each multi-index p there exists a constant $M_{p}$ such that $\begin{array}{rclr} | D^{p} ϕ (x) | ⩽ M_{p} | x |^{k - | p |}, | x | ⩽ 1, & (4.5) \\ | D^{p} ϕ (x) | ⩽ M_{p} | x |^{q - | p |}, | x | ⩾ 1 . & (4.6) \end{array}$ Hence if $0 < ε < 1$ , ${∥ ϕ (ε x) ∥}_{K_{q, *}^{[m]}, l, s} ⩽ M_{s} ε^{k}$ (4.7) for some constant $M_{s}$ . □

We therefore obtain the ensuing Taylor formula.

Theorem 4.3.

Let $ϕ \in K_{*}^{[m]} (R^{n})$ with expansion $ϕ (x) \sim S_{ϕ}^{*, m} (x) = \sum_{j = m}^{\infty} a_{j} (w) r^{k}$ , as $x \to 0$ . Then $ϕ (ε x) \sim S_{ϕ}^{*, m} (ε x) = \sum_{j = m}^{\infty} a_{j} (w) r^{k} ε^{k}, ε \to 0,$ (4.8) in the topology of the space $K_{*}^{[m]} (R^{n})$ .

Proof.

The proof of the theorem follows from the Lemma 4.2 by observing that if $ϕ \in K_{q, *}^{[m]} (R^{n})$ and $q ⩾ k ⩾ m$ , then $ϕ (x) - \sum_{j = m}^{k - 1} a_{j} (w) | x |^{j}$ belongs to $K_{q, *}^{[k]} (R^{n})$ . □

It is worth pointing out that we can express the result of the theorem by saying that if $ϕ \in K_{*} (R^{n})$ then $ϕ (ε x) \sim \sum_{j = - \infty}^{\infty} a_{j} (w) r^{k} ε^{k},$ (4.9) as $ε \to 0$ in the topology of $K_{*} (R^{n})$ . Naturally the asymptotic character of the series is as $j \to \infty$ since the terms $a_{j} (w)$ vanish if $j < m$ for some m.

5. The moment asymptotic expansion

If the distribution f belongs to $K^{'} (R^{n})$ then its moments are the numbers $μ_{k} = ⟨ f (x), x^{k} ⟩$ (5.1) for $k \in N^{n}$ . For thick distributions we employ the following definition.

Definition 5.1.

Let $f \in K_{*}^{'} (R^{n})$ . Then its moments are the distributions $μ_{k} \in D^{'} (S)$ defined as ${⟨ μ_{k} (w), φ (w) ⟩}_{D^{'} (S) \times D (S)} = {⟨ f (x), | x |^{k} φ (x / | x |) ⟩}_{K_{*}^{'} (R^{n}) \times K_{*} (R^{n})}$ (5.2) for $k \in Z$ .

Observe that if $f \in K_{*}^{'} (R^{n})$ is a thick distribution and $g = Π (f) \in K^{'} (R^{n})$ is the usual distribution obtained by projection, then the moments $μ_{k}$ of g are related to the moment functions $μ_{k} (w)$ of f by the formula $μ_{k} = ⟨ μ_{k} (w), w^{k} ⟩,$ (5.3) where $k = | k |$ .

Any distribution of $K^{'} (R^{n})$ satisfies the moment asymptotic expansion [14]. We now prove that the thick distributions of $K_{*}^{'} (R^{n})$ also satisfy a generalized moment asymptotic expansion.

Theorem 5.2.

Let $f \in K_{*}^{'} (R^{n})$ , with moment functions $μ_{k} (w)$ , then $f (λ x) ∽ \sum_{k = - \infty}^{\infty} \frac{C_{n - 1} μ_{k} (w) δ_{*}^{[k]}}{λ^{k + n}}, as λ \to \infty,$ (5.4) where $C_{n - 1}$ , given by ( 2.4 ), is the surface area of the unit sphere. This means that $f (λ x) ∽ \sum_{k = m}^{\infty} C_{n - 1} λ^{- k - n} μ_{k} (w) δ_{*}^{[k]}$ , for some m.

Proof.

We shall prove that (5.4) holds in the strong topology of $K_{*}^{'} (R^{n})$ , but to do so it is enough to show that it holds in the weak sense, since weak and strong convergence of sequences are equivalent in spaces of distributions [20]. Thus, we need to show that if $ϕ \in K_{*} (R^{n})$ , say $ϕ \in K_{*}^{[m]} (R^{n})$ , with expansion $ϕ (r w) \sim \sum_{j = m}^{\infty} a_{j} (w) r^{j}$ , as $r \to 0^{+}$ , then for each N, $⟨ f (λ x), ϕ (x) ⟩ = \underset{j = m}{\sum^{N - 1}} \frac{1}{λ^{j + n}} ⟨ μ_{j} (w), a_{j} (w) ⟩ + R_{N} (λ),$ (5.5) where $R_{N} (λ) = O (λ^{- N - n}), as λ \to \infty .$ (5.6) However, $R_{N} (λ)$ is given as $\begin{array}{l} R_{N} (λ) & = ⟨ f (λ x), ϕ (x) - \sum_{j = m}^{N - 1} a_{j} (x / | x |) | x |^{j} ⟩ & (5.7) \\ = ⟨ f (λ x), P_{N} (x) ⟩ = λ^{- n} ⟨ f (x), P_{N} (x / λ) ⟩, \end{array}$ where $P_{N} (x) = ϕ (x) - \sum_{j = m}^{N - 1} a_{j} (\frac{x}{| x |}) | x |^{j},$ (5.8) belongs to $K_{*}^{[N]} (R^{n})$ , and thus (5.6) follows from the Lemma 4.2. □

It is interesting to observe that the standard moment asymptotic expansion in the space $K^{'} (R^{n})$ can be obtained by applying the projection operator (2.3) to the generalized expansion (5.4). Indeed, it was shown in [41] that $Π (μ (w) δ_{*}^{[k]}) = 0$ if $k < 0$ , while if $k ⩾ 0$ then $Π (μ (w) δ_{*}^{[k]}) = \frac{{(- 1)}^{k}}{C_{n - 1}} \sum_{| k | = k} \frac{⟨ μ (w), w^{k} ⟩}{k!} D^{k} δ (x) .$ (5.9) Thus, using (5.3) we obtain the following corollary [14, Theorem 4.3.1]. Corollary 5.3.

If $g (x) \in K^{'} (R^{n})$ , with moments $μ_{k}$ , then $g (λ x) \sim \sum_{k = 0}^{\infty} \sum_{| k | = k} {(- 1)}^{k} \frac{μ_{k} D^{k} δ (x)}{k! λ^{n + k}}, as λ \to \infty .$ (5.10)

5.1. Expansion of radial distributions

The moment asymptotic expansion takes a rather simple form when applied to radial thick distributions, as we now explain. A radial distribution is one that is invariant with respect to rotations; if $f \in D_{*}^{'} (R^{n})$ is radial then $f (x) = f_{1} (| x |)$ , where $f_{1} \in D_{*}^{'} (R)$ has support in $[0, \infty)$ , but $f_{1}$ is not unique [10].

Lemma 5.4.
Let $f \in K_{}^{'} (R^{n})$ be radial,* $f (x) = f_{1} (| x |)$ , where $f_{1} \in K_{}^{'} (R)$ has support in* $[0, \infty)$ . Then the moment functions of f, $μ_{k}^{{n}} \in D^{'} (S)$ are constants, given by $μ_{k}^{{n}} = μ_{k + n - 1}^{{1}},$ (5.11) where $μ_{k}^{{1}} = ⟨ f_{1} (r), | r |^{k} ⟩$ .
Proof.
Indeed, if $φ \in D (S)$ then $⟨ μ_{k}^{{n}} (w), φ (w) ⟩$ is given as $⟨ f (x), | x |^{k} φ (x / | x |) ⟩ = {⟨ f_{1} (r), | r |^{k + n - 1} ⟩}_{r} {⟨ 1, φ (w) ⟩}_{w} = ⟨ μ_{k + n - 1}^{{1}}, φ (w) ⟩,$ as required. □

If we employ the Theorem 5.2 we obtain the ensuing asymptotic expansion of radial thick distributions.
Corollary 5.5.
If $f \in K_{}^{'} (R^{n})$ is radial, then* $f (λ x) ∽ \sum_{k = - \infty}^{\infty} \frac{C_{n - 1} μ_{k}^{{n}} δ_{}^{[k]}}{λ^{k + n}}, as λ \to \infty,$ (5.12) where the constants* $μ_{k}^{{n}}$ are given in the Lemma 5.4 .

Let us now apply the projection operator $Π : K_{}^{'} (R^{n}) \to K^{'} (R^{n})$ , and recall [41] that $Π (δ_{}^{[j]}) = 0$ unless $j = 2 k ⩾ 0$ , while $Π (δ_{}^{[2 k]}) = \frac{Γ (k + 1 / 2) Γ (n / 2)}{Γ (k + n / 2) (2 k)! \sqrt{π}} \nabla^{2 k} δ (x),$ (5.13) where $\nabla^{2 k} = {(\nabla^{2})}^{k}$ is the kth power of the Laplacian. We thus obtain the following form of the moment asymptotic expansion. Corollary 5.6.
Let* $g (x) \in K^{'} (R^{n})$ be radial, $g (x) = g_{1} (| x |)$ , $g_{1} \in K^{'} (R)$ with support in $[0, \infty)$ . Let ${μ_{k}^{{1}}}_{k = 0}^{\infty}$ be the moments of $g_{1}$ . Then $g (λ x) ∽ \sum_{k = 0}^{\infty} \frac{ν_{k}}{λ^{2 k + n}} \nabla^{2 k} δ (x), as λ \to \infty,$ (5.14) where $ν_{k} = \frac{2 Γ (k + 1 / 2) π^{(n - 1) / 2}}{Γ (k + n / 2) (2 k)!} μ_{k + n - 1}^{{1}} .$ (5.15)

6. Finite parts and pseudofunctions

In order to give illustrations of our formulas, we need to consider several important properties of Hadamard finite part integrals and of the associated pseudofunctions.

Let us recall the notion of the finite part of a limit [14, Section 2.4]. Let $G (ε)$ be a function defined for $0 < ε < ε_{0}$ that satisfies ${lim}_{ε \to 0^{+}} G (ε) = \infty$ . Suppose $F$ , the basic functions, is a family of strictly positive functions defined in the same interval such that all of them tend to infinity at 0 and such that given two different elements $f_{1}, f_{2} \in F$ then ${lim}_{ε \to 0^{+}} f_{1} (ε) / f_{2} (ε)$ is either 0 or ∞. Then the finite part of the limit of $G (ε)$ as $ε \to 0^{+}$ with respect to $F$ exists and equals A if we can write $G (ε) = G_{1} (ε) + G_{2} (ε)$ , where $G_{1}$ , the infinite part, is a linear combination of the basic functions and where $G_{2}$ , the finite part, has the property that the limit $A = {lim}_{ε \to 0^{+}} G_{2} (ε)$ exists; such a decomposition is unique if it exists since any finite number of elements of $F$ has to be linearly independent. We then employ the notation ${F . p .}_{F} lim_{ε \to 0^{+}} G (ε) = A .$ (6.1) In the standard Hadamard finite part limit one takes $F$ as the family of functions $ε^{- α} | ln α |^{β}$ , where $α > 0$ and $β ⩾ 0$ or where $α = 0$ and $β > 0$ and uses the simpler notation $F . p . {lim}_{ε \to 0^{+}} G (ε)$ .

Consider now a function f defined in $R^{n}$ that is probably not integrable over the whole space but which is integrable in the region $| x | > ε$ for any $ε > 0$ . Then the radial7

⁷

Other, non-radial finite part integrals can be considered, of course, but the results would be different, in general [40].

finite part integral is defined as

F . p . \int_{R^{n}} f (x) d x = F . p . lim_{ε \to 0^{+}} \int_{| x | > ε} f (x) d x,

(6.2) if the finite part limit exists.

Definition 6.1.

If g is a locally integrable function in $R^{n} ∖ {0}$ such that the radial finite part integral of $g ϕ$ exists for each ϕ belonging to a space of thick test functions $A_{*} (R^{n})$ , then we can define a thick8

⁸

The same notation is employed for standard distributions.

distribution

P f {g (z); x} = P f (g) \in A_{*}^{'} (R^{n})

⟨ P f {g (z); x}, ϕ (x) ⟩ = ⟨ P f (g), ϕ ⟩ = F . p . \int_{R^{n}} g (x) ϕ (x) d x .

(6.3)

The distribution $P f (g)$ is sometimes called a pseudofunction. One may usually employ the notation $P f (g (x))$ when the variable of the pseudofunction is important, but when considering the transformation properties of the association $g \mapsto P f (g)$ it is convenient to employ the more detailed notation $P f {g (z); x}$ . Indeed, since g and $P f (g)$ are not exactly the same object, then applying an operation to g does not give the same result as applying the same operation to $P f (g)$ ; in particular, as we shall presently see, if $λ \in R ∖ {0}$ , then $P f {g (λ z); x}$ and $P f {g (z); λ x}$ are related, but do not coincide, in general.

The transformation formulas for dilatations of $P f (| x |^{α})$ were given in [41], namely, $P f {| λ z |^{α}; x} = λ^{a} P f {| z |^{α}; x}$ for any $α \in C$ , but $P f (| λ x |^{α}) = P f {| z |^{α}; λ x} = λ^{a} P f {| z |^{α}; x}$ for $α \notin Z$ only, while if $α = q \in Z$ then $P f (| λ x |^{q}) = P f {| z |^{k}; λ x} = λ^{q} P f (| x |^{q}) + C λ^{q} log λ δ_{*}^{[- q - n]} (x),$ (6.4) where C is given in (2.4). If we take $q = 0$ and evaluate this formula at $ϕ \in D_{*} (R^{n})$ we obtain the ensuing result.9

⁹

Clearly, formula (6.5) will hold under much weaker conditions on ϕ.

Proposition 6.2.

If $ϕ \in D_{*} (R^{n})$ , with $ϕ (x) \sim \sum_{j = m}^{\infty} a_{j} (w) r^{j}$ , as $x \to 0$ , and $υ > 0$ then $F . p . \int_{R^{n}} ϕ (υ x) d x = υ^{- n} F . p . \int_{R^{n}} ϕ (x) d x - υ^{- n} log υ \int_{S} a_{- n} (w) d σ (w) .$ (6.5)

We can also obtain a kind of “parts” formula.

Proposition 6.3.

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 $\begin{array}{rclr} F . p . \int_{R^{n}} ϕ (υ x) ψ (x) d x & = & υ^{- n} F . p . \int_{R^{n}} ϕ (x) ψ (x / υ) d x \\ - \sum_{l = p}^{- n - m} υ^{- n - l} log υ \int_{S} a_{- l - n} (w) b_{l} (w) d σ (w) . & (6.6) \end{array}$

Proof.

Apply (6.5) with $ϕ (x)$ replaced by $ϕ (x) ψ (x / υ)$ . □

We can consider (6.6) as a transformation formula for the pseudofunction $P f (ψ (x))$ . Proposition 6.4.

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 $\begin{array}{rclr} P f (ψ (λ x)) & = & P f {ψ (z); λ x} \\ = & P f {ψ (λ z); x} + C \sum_{j = - \infty}^{- n - p} \frac{log λ}{λ^{n + j}} b_{- n - j} (w) δ_{*}^{[j]} (x) . & (6.7) \end{array}$

Proof.

Indeed, $⟨ P f {ψ (z); λ x}, ϕ (x) ⟩ = λ^{- n} F . p . \int_{R^{n}} ϕ (λ^{- 1} x) ψ (x) d x,$ (6.8) while $⟨ P f {ψ (λ z); x}, ϕ (x) ⟩ = F . p . \int_{R^{n}} ϕ (x) ψ (λ x) d x,$ (6.9) and thus (6.7) follows from (6.6) by taking $υ = 1 / λ$ . □

6.1. Computation of finite part integrals

We shall now compute some finite part integrals whose results are available in the literature [14,30], but for which the computation of the results does not seem generally known.

Suppose, to fix the ideas, that one needs to compute $F_{f . p .} (λ) = F . p . \int_{0}^{\infty} x^{λ} ϕ (x) d x,$ the Hadamard finite part of the (probably) divergent integral, where $ϕ \in S (R)$ and where $λ \in C$ . The definition [14, Section 2.4] requires that we decompose $G (ε) = \int_{ε}^{\infty} x^{λ} ϕ (x) d x$ , $ε > 0$ , as a sum of its infinite and finite parts, and while doing this is many times possible, it is sometimes easier to proceed in a different way, as we now explain.

Indeed, let $F_{0} (λ) = \int_{0}^{\infty} x^{λ} ϕ (x) d x, ℜ e λ > - 1 .$ (6.10) The function $F_{0}$ is analytic in its semiplane of definition and it is well known that it admits an analytic continuation to the region $C ∖ {- 1, - 2, - 3, \dots}$ ; this continuation is unique and we denote it as F. The singularities at $λ = - k$ , $k = 1, 2, 3, \dots$ are simple poles, with residues $R_{- k} = \underset{λ = - k}{Res} F (λ) = \frac{ϕ^{(k - 1)} (0)}{(k - 1)!} .$ (6.11) Observe now that any analytic function $W (λ)$ defined in a punctured disc $0 < | λ - ω | < r$ , with an isolated singularity at $λ = ω$ has a well defined regular part, $W_{reg} (λ)$ – the sum of the terms with positive powers in its Laurent series – which has a removable singularity at ω; the value $W_{reg} (ω)$ is the regularized value of W at $λ = ω$ . Actually $W_{reg} (ω)$ is the finite part of the limit of $W (λ)$ as $λ \to ω$ if the singularity is a pole (but not if it is an essential singularity). For our function F this gives values $F_{reg} (- k) = F . p . {lim}_{λ \to - k} F (λ)$ for $k = 1, 2, 3, \dots$ . We then obtain, $F_{f . p .} (λ) = \{\begin{matrix} F (λ), & λ \neq - 1, - 2, - 3, \dots, \\ F_{reg} (- k), & k = 1, 2, 3, \dots . \end{matrix}$ (6.12)

Alternatively [14, Section 2.4], we can say that $x_{+}^{λ}$ , the regular distribution of λ for $ℜ e λ > - 1$ , defined by the integral $⟨ x_{+}^{λ}, ϕ (x) ⟩ = F_{0} (λ)$ , admits an analytic continuation to $C ∖ {- 1, - 2, - 3, \dots}$ , also denoted as $x_{+}^{λ}$ , and given as $⟨ x_{+}^{λ}, ϕ (x) ⟩ = F (λ)$ . The parametrically analytic distribution $x_{+}^{λ}$ has simple poles at the negative integers, with residues ${Res}_{λ = - k} x_{+}^{λ} = {(- 1)}^{k - 1} δ^{(k - 1)} (x) / (k - 1)!$ . The regularized values10

¹⁰

These regularizations are called Riesz regularizations in [30].

{(x_{+}^{- k})}_{reg} = F . p . lim_{λ \to - k} x_{+}^{λ} = lim_{λ \to - k} (x_{+}^{λ} - \frac{{(- 1)}^{k - 1} δ^{(k - 1)} (x)}{(λ + k) (k - 1)!}),

(6.13) are well defined distributions of

S^{'}

. The distributions

P f (H (x) x^{λ})

defined for any

λ \in C

⟨ P f (H (x) x^{λ}), ϕ (x) ⟩ = F . p . \int_{0}^{\infty} x^{λ} ϕ (x) d x,

(6.14) that is, as

F_{f . p .} (λ)

, can also be computed as [12]

P f (H (x) x^{λ}) = \{\begin{matrix} x_{+}^{λ}, & λ \neq - 1, - 2, - 3, \dots, \\ {(x_{+}^{- k})}_{reg}, & k = 1, 2, 3, \dots . \end{matrix}

(6.15)

Interestingly, this approach can also be employed for radial finite part integrals in several variables [12], however, it does not work at the singularities for other shapes [40], not even for principal value integrals, a result that has important implications in the numerical solutions of integral equations [15], and in other calculations in Mathematical Physics [19].

We are particularly interested in the finite part integrals $μ_{λ} = F . p . \int_{0}^{\infty} x^{λ} e^{- x} d x$ (6.16) for any $λ \in C$ . In this case, with the above notation, we have that $F (λ) = Γ (λ + 1)$ , where Γ is the well known gamma function [25, Chapter 1]. We immediately obtain from (6.12) that $μ_{λ} = Γ (λ + 1)$ whenever $λ \neq - 1, - 2, - 3, \dots$ .

Let us now compute the values $Γ_{reg} (- k)$ for $k = 0, 1, 2, \dots$ . Observe first that (6.11) yields that the residue of the gamma function at $z = - k$ equals ${(- 1)}^{k} / k!$ . Thus, the functional equation $z Γ (z) = Γ (z + 1)$ gives $Γ_{reg} (0) = lim_{z \to 0} (Γ (z) - \frac{1}{z}) = lim_{z \to 0} \frac{Γ (z + 1) - 1}{z} = Γ^{'} (1) = - γ,$ (6.17) Euler’s constant, or in other words, $Γ (z) = \frac{1}{z} - γ + O (z), as z \to 0 .$ (6.18)

In fact, $Γ_{reg} (- k) = {(- 1)}^{k} γ (k + 1) / k!$ since $Γ (- k + z) = \frac{{(- 1)}^{k}}{k!} {\frac{1}{z} + γ (k + 1)} + O (z), as z \to 0,$ (6.19) where $γ (z) = Γ^{'} (z) / Γ (z)$ is the digamma function, the logarithmic derivative of the gamma function [25], so that [14, Example 60] $γ (k + 1) = \sum_{j = 1}^{k} \frac{1}{j} - γ .$ (6.20) The proof of (6.19) is by induction. Indeed, (6.18) says that it is true for $k = 0$ , while if we assume that it is true for k then we obtain that $Γ (- k - 1 + z)$ equals $\begin{array}{rcl} \frac{Γ (- k + z)}{z - k - 1} & = & \frac{- 1}{k + 1} (1 + \frac{z}{k + 1} + O (z^{2})) \frac{{(- 1)}^{k}}{k!} (\frac{1}{z} + γ (k + 1) + O (z)) \\ = & \frac{{(- 1)}^{k + 1}}{(k + 1)!} (\frac{1}{z} + γ (k + 1) + \frac{1}{k + 1} + O (z)) = \frac{{(- 1)}^{k + 1}}{(k + 1)!} (\frac{1}{z} + γ (k + 2) + O (z)) . \end{array}$ Hence, for $k = 1, 2, 3, \dots$ , $F . p . \int_{0}^{\infty} x^{- k} e^{- x} d x = Γ_{reg} (- k + 1) = \frac{{(- 1)}^{k} γ (k)}{(k - 1)!} = \frac{{(- 1)}^{k - 1}}{(k - 1)!} (\sum_{j = 1}^{k - 1} \frac{1}{j} - γ) .$ (6.21)

7. Illustrations

We shall now give several illustrations of our formulas.

Example 7.1.

Let $f = δ (S) \in E_{*}^{'} (R^{n})$ be the delta function on the unit sphere of $R^{n}$ , that is, $⟨ f, ϕ ⟩ = \int_{S} ϕ (w) d σ (w)$ . Observe that $⟨ μ_{k} (w), a (w) ⟩$ equals $⟨ δ (S), a (w) r^{k} ⟩ = \int_{S} a (w) d σ (w) = {⟨ 1, a (w) ⟩}_{D^{'} (S) \times D (S)},$ so that $μ_{k} (w) = 1 for all k .$ (7.1) Denote by $S_{t} = t S$ the sphere of center $0$ and radius t. Then, $f (λ x) = (1 / λ) δ (S_{1 / λ})$ , so that the generalized moment asymptotic expansion yields $δ (S_{1 / λ}) \sim \overset{\infty}{\sum_{k = - \infty}} \frac{2 π^{n / 2} δ_{*}^{[k]} (x)}{Γ (n / 2) λ^{n + k - 1}}, as λ \to \infty,$ (7.2) in the space $E_{*}^{'} (R^{n})$ .

The projection $Π (f)$ is the distribution $δ (S) \in E^{'} (R^{n})$ , with the obvious abuse of notation, and thus in the space $E^{'} (R^{n})$ , $δ (S_{1 / λ}) \sim \overset{\infty}{\sum_{k = 0}} \frac{c_{k, n} \nabla^{2 k} δ (x)}{(2 k)! λ^{n + 2 k - 1}},$ (7.3) where $c_{k, n} = 2 Γ (k + 1 / 2) π^{(n - 1) / 2} / Γ (k + n / 2)$ , in agreement with [14, Example 94].

Evaluation of (7.3) at a test function $ϕ \in E (R^{n})$ yields the well known Pizzetti’s formula [1,26], $\int_{ε S} ϕ (w) d σ (w) \sim \overset{\infty}{\sum_{k = 0}} \frac{c_{k, n} \nabla^{2 k} ϕ (0)}{(2 k)!} ε^{n + 2 k - 1},$ (7.4) as $ε \to 0^{+}$ . Therefore, (7.3) can be called the distributional form of Pizzetti’s formula, while (7.2) becomes the thick distributional form of Pizzetti’s formula.

Example 7.2.

If $f (r)$ is a radial distribution of $K_{*}^{'} (R^{n})$ , with constant moment functions ${μ_{k}}_{k = - \infty}^{\infty} = {μ_{k + n - 1}^{{1}}}_{k = - \infty}^{\infty}$ , and $g \in D^{'} (S)$ , then $F (x) = f (r) g (w)$ belongs to $K_{*}^{'} (R^{n})$ and the expansion of $F (λ x)$ as $λ \to \infty$ follows from (5.12) as $f (λ x) ∽ \sum_{k = - \infty}^{\infty} \frac{C_{n - 1} μ_{k + n - 1}^{{1}} g (w) δ_{*}^{[k]}}{λ^{k + n}}, as λ \to \infty .$ (7.5) In particular, if $F (x) = g (w) δ (S)$ , then $μ_{k} (w) = g (w)$ , for all k, and so $F (λ x) \sim \underset{k = - \infty}{\sum^{\infty}} \frac{2 π^{n / 2} g (w) δ_{*}^{[k]} (x)}{Γ (n / 2) λ^{n + k}}, as λ \to \infty,$ (7.6) or, equivalently, $g (w) δ (S_{1 / λ}) \sim \overset{\infty}{\sum_{k = - \infty}} \frac{2 π^{n / 2} g (w) δ_{*}^{[k]} (x)}{Γ (n / 2) λ^{n + k - 1}}, as λ \to \infty .$ (7.7)

Example 7.3.

The distribution $P f (| x |^{α})$ does not belong to $K_{*}^{'} (R^{n})$ for any $α \in C$ , and thus $P f (| λ x |^{α})$ does not satisfy the moment asymptotic expansion, as (6.4) shows. However, if $f_{α} (x) = P f (| x |^{α} H (1 - | x |))$ , where H is the Heaviside function, then $f_{α} \in E_{*}^{'} (R^{n}) \subset K_{*}^{'} (R^{n})$ . Using the Lemma 5.4, it follows that the moment functions are constant functions, given as $μ_{k} = 1 / (k + n + α)$ , if $α \notin Z$ , whereas if $α = q \in Z$ then $μ_{k} = 1 / (k + n + q)$ , if $k \neq - n - q$ , while $μ_{- n - q} = 0$ . Thus, as $λ \to \infty$ , $\begin{array}{rclr} f_{α} (λ x) \sim \sum_{k = - \infty}^{\infty} \frac{C δ_{*}^{[k]} (x)}{(k + n + α) λ^{k + n}}, α \notin Z, & (7.8) \\ f_{q} (λ x) \sim \sum_{k = - \infty, k \neq - n - q}^{\infty} \frac{C δ_{*}^{[k]} (x)}{(k + n + q) λ^{k + n}}, α = q \in Z . & (7.9) \end{array}$ It follows that if $ϕ \in E_{*}^{[m]} (R^{n})$ , $ϕ (r w) \sim \sum_{k = m}^{\infty} a_{k} (w) r^{k}$ , then as $ε \to 0^{+}$ , $F . p . \int_{| x | ⩽ ε} ϕ (x) | x |^{α} d x \sim \sum_{k = m}^{\infty} (\int_{S} a_{k} (w) d σ (w)) \frac{ε^{k + n + α}}{k + n + α},$ (7.10) if $α \notin Z$ . If $α = q \in Z$ , then (6.6) and (6.7) yield $\begin{array}{rclr} F . p . \int_{| x | ⩽ ε} ϕ (x) | x |^{q} d x & \sim & \sum_{k = m, k \neq - n - q}^{\infty} (\int_{S} a_{k} (w) d σ (w)) \frac{ε^{k + n + q}}{k + n + q} \\ + (\int_{S} a_{- n - q} (w) d σ (w)) ln ε . & (7.11) \end{array}$

Example 7.4.

Let us now consider the asymptotic expansion of the integral $I (λ) = F . p . \int_{R^{n}} e^{- λ | x |} ϕ (x) d x,$ (7.12) where $ϕ \in K_{*}^{[m]} (R^{n})$ , $ϕ (r w) \sim \sum_{k = m}^{\infty} a_{k} (w) r^{k}$ . The result is a generalized form of Watson’s lemma, used in the approximation of Laplace integrals [14, Example 58].

Notice that $I (λ) = ⟨ P f {e^{- λ | z |}; x}, ϕ (x) ⟩$ , and while this is not a moment expansion, we can use the Proposition 6.4 with $ψ (x) = e^{- | x |}$ to write $P f {e^{- λ | z |}; x} = P f {e^{- | z |}; λ x} - \sum_{k = - \infty}^{- n} \frac{{(- 1)}^{- n - k} ln λ}{(- n - k)! λ^{n + k}} C δ_{*}^{[k]} (x),$ (7.13) and observe that $P f {e^{- | z |}; λ x}$ can be computed from the moment asymptotic expansion, $P f {e^{- | z |}; λ x} \sim \sum_{k = - \infty}^{- n} \frac{{(- 1)}^{n + k} γ (1 - n - k)}{(- n - k)! λ^{n + k}} C δ_{*}^{[k]} (x) + \sum_{k = - n + 1}^{\infty} \frac{(k + n - 1)!}{λ^{n + k}} C δ_{*}^{[k]} (x),$ (7.14) where we employ (5.12) and (6.21). Thus, $\begin{array}{rclr} I (λ) & \sim & \sum_{k = m}^{- n} \frac{{(- 1)}^{n + k} {γ (1 - n - k) - ln λ} \int_{S} a_{k} (w) d σ (w)}{(- n - k)! λ^{n + k}} \\ + \sum_{k = - n + 1}^{\infty} \frac{(k + n - 1)! \int_{S} a_{k} (w) d σ (w)}{λ^{n + k}} . & (7.15) \end{array}$

Naturally the expansion of $F . p . \int_{R^{n}} e^{- λ | x |} ϕ (x) | x |^{α} d x$ also follows from the analysis of the previous example.

Footnotes

Acknowledgements

The authors gratefully acknowledge support from NSF, through grant number 0968448.

References

[1]

M.A.

Aguirre and

C.K.

Li, The distributional products of particular distributions, Appl. Math. Comput. 187 (2007), 120–126.

[2]

Blanchet and

Faye, Hadamard regularization, J. Math. Phys. 41 (2000), 7675–7714.

[3]

Blanchet,

Faye and

Nissanke, Structure of the post-Newtonian expansion in general relativity, Phys. Rev. D. 72 (2005), 044024.

[4]

S.M.

Blinder, Delta functions in spherical coordinates and how to avoid losing them: Fields of point charges and dipoles, Amer. J. Phys. 71 (2003), 816–818.

[5]

J.D.

Bondurant and

S.A.

Fulling, The Dirichlet-to-Robin transform, J. Phys. A 8 (2005), 1505–1532.

[6]

J.M.

Bowen, Delta function terms arising from classical point-source fields, Amer. J. Phys. 62 (1994), 511–515.

[7]

Duoandikoetxea and

Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris 315 (1992), 693–698.

[8]

Estrada, Series of Dirac delta functions, in: Partial Differential Equations,

Wiener and

J.K.

Hale, eds, Pitman, London, 1992, pp. 44–49.

[9]

Estrada, The Cesàro behaviour of distributions, Proc. Roy. Soc. London Ser. A 454 (1998), 2425–2443.

10.

[10]

Estrada, On radial functions and distributions and their Fourier transforms, J. Fourier Anal. Appl. 20 (2013), 301–320.

11.

[11]

Estrada and

S.A.

Fulling, Spaces of test functions and distributions in spaces with thick points, Int. J. Appl. Math. Stat. 10 (2007), 25–37.

12.

[12]

Estrada and

R.P.

Kanwal, Regularization, pseudofunction, and Hadamard finite part, J. Math. Anal. Appl. 141 (1989), 195–207.

13.

[13]

Estrada and

R.P.

Kanwal, A distributional approach to the boundary layer theory, J. Math. Anal. Appl. 192 (1995), 796–812.

14.

[14]

Estrada and

R.P.

Kanwal, A Distributional Approach to Asymptotics. Theory and Applications, 2nd edn, Birkhäuser, Boston, 2002.

15.

[15]

Farassat, Introduction to generalized functions with applications in aerodynamics and aeroacoustics, NASA Langley Research Center, Hampton, VA, 1996, NASA Technical Paper 3248, available at: http://ntrs.nasa.gov.

16.

[16]

C.P.

Frahm, Some novel delta-function identities, Am. J. Phys. 51 (1983), 826–829.

17.

[17]

Franklin, Comment on ‘Some novel delta-function identities’ by Charles P. Frahm (Am. J. Phys. 51 (1983), 826–829), Am. J. Phys. 78 (2010), 1225–1226.

18.

[18]

L.G.

Hernández and

Estrada, Solutions of ordinary differential equations by series of delta functions, J. Math. Anal. Appl. 191 (1995), 40–55.

19.

[19]

Hnizdo, Generalized second-order partial derivatives of

1 / r

, Eur. J. Phys. 32 (2011), 287–297.

20.

[20]

Horváth, Topological Vector Spaces and Distributions, Vol. I, Addison-Wesley, Reading, MA, 1966.

21.

[21]

R.P.

Kanwal, Generalized Functions: Theory and Technique, 3rd edn, Birkhäuser, Boston, 2004.

22.

[22]

A.M.

Krall, Orthogonal polynomials through moment generating functions, SIAM J. Math. Anal. 9 (1978), 600–603.

23.

[23]

A.M.

Krall,

R.P.

Kanwal and

L.L.

Littlejohn, Distributional solutions of ordinary differential equations, Canad. Math. Soc. Conf. Proc. 8 (1987), 227–246.

24.

[24]

H.L.

Krall and

Frink, A new class of orthogonal polynomials: The Bessel polynomials, Trans. Amer. Math. Soc. 65 (1949), 100–115.

25.

[25]

N.N.

Lebedev, Special Functions and Their Applications, Dover, New York, 1972.

26.

[26]

C.K.

Li and

M.A.

Aguirre, The distributional products on spheres and Pizzetti’s formula, J. Comput. Appl. Math. 235 (2011), 1482–1489.

27.

[27]

L.L.

Littlejohn and

R.P.

Kanwal, Distributional solutions of the hypergeometric differential equation, J. Math. Anal. Appl. 122 (1987), 325–345.

28.

[28]

Łojasiewicz, Sur la valeur et la limite d’une distribution en un point, Studia Math. 16 (1957), 1–36.

29.

[29]

G.F.

Paskusz, Comments on “Transient analysis of energy equation of dynamical systems”, IEEE Trans. Edu. 43 (2000), 242.

30.

[30]

Paycha, Regularised Integrals, Sums and Traces, AMS, Providence, 2012.

31.

[31]

Pilipović,

Stanković and

Takači, Asymptotic Behaviour and Stieltjes Transformation of Distributions, Teubner-Texte zuer Mathematik, Leipzig, 1990.

32.

[32]

Pilipović,

Stanković and

Vindas, Asymptotic Behavior of Generalized Functions, World Scientific, Singapore, 2011.

33.

[33]

Sellier, Asymptotic expansion of a class of multi-dimensional integrals, Canad. J. Math. 48 (1996), 1079–1090.

34.

[34]

Sellier, Hadamard’s finite part concept in dimensions

n ⩾ 2

. Definition and change of variables, associated Fubini’s theorem, derivation, Math. Proc. Cambridge Philos. Soc. 122 (1997), 131–148.

35.

[35]

S.M.

Shah and

Wiener, Distributional and entire solutions of ordinary differential and functional differential equations, Internat. J. Math. Sci. 6 (1983), 243–270.

36.

[36]

Trèves, Topological Vector Spaces, Distributions, and Kernels, Academic Press, New York, 1967.

37.

[37]

Vibet, Transient analysis of energy equation of dynamical systems, IEEE Trans. Edu. 42 (1999), 217–219.

38.

[38]

V.S.

Vladimirov,

Y.N.

Drozhinov and

B.I.

Zavyalov, Tauberian Theorems for Generalized Functions, Kluwer, Dordrecht, 1988.

39.

[39]

Wiener, Generalized function solutions of differential and functional differential equations, J. Math. Anal. Appl. 88 (1982), 170–182.

40.

[40]

Yang and

Estrada, Regularization using different surfaces and the second order derivatives of

1 / r

, Applicable Analysis 92 (2013), 246–258.

41.

[41]

Yang and

Estrada, Distributions in spaces with thick points, J. Math. Anal. Appl. 401 (2013), 821–835.

42.

[42]

Yang and

Estrada, Extension of Frahm formulas for

\partial_{i} \partial_{j} (1 / r)

, Indian J. Math. 55 (2013), 1–9.

43.

[43]

Yang and

Estrada, The dual of the space of regulated functions in several variables, Sarajevo J. Math. 9 (2013), 197–216.

44.

[44]

A.H.

Zemanian, Generalized Integral Transforms, Interscience, New York, 1965.

Asymptotic expansion of thick distributions

Abstract

Keywords

1. Introduction

1 It is not clear to which singular test functions the formula of [17] applies.

4 In the sense of Zemanian [44]; we assume that D ( R n ) ⊂ A ( R n ) ⊂ E ( R n ) densely and continuously and that differentiation is a continuous map of A ( R n ) .

3.2. The spaces K ∗ , a ( R n ) and K ∗ , a ′ ( R n )

Footnotes

Acknowledgements

References

¹
It is not clear to which singular test functions the formula of [17] applies.

⁴
In the sense of Zemanian [44]; we assume that $D (R^{n}) \subset A (R^{n}) \subset E (R^{n})$ densely and continuously and that differentiation is a continuous map of $A (R^{n})$ .

3.2. The spaces $K_{, a} (R^{n})$ and $K_{, a}^{'} (R^{n})$