The Cesaro Behaviour of Distributions

The Cesaro Behaviour of DistributionsAuthor(s): Ricardo EstradaSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 454, No. 1977 (Sep.8, 1998), pp. 2425-2443Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/53185 .

Accessed: 06/05/2014 08:45

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings:Mathematical, Physical and Engineering Sciences.

http://www.jstor.org

This content downloaded from 130.132.123.28 on Tue, 6 May 2014 08:45:51 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=rsl

http://www.jstor.org/stable/53185?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


JyalSobt9

The Cesaro behaviour of distributions

BY RICARDO ESTRADA

Universidad de Costa Rica, P.O. Box 276, Tres Rios, Costa Rica

Received 26 November 1996; accepted 25 November 1997

A theory of summability for generalized functions of one real variable is presented; this theory generalizes the classical theory of summability of series and integrals. The relationship between the average behaviour of generalized functions according to this summability theory and the distributional behaviour is studied. Several applications are given.

Keywords: generalized functions; summability

1. Introduction

The purpose of this article is to study the behaviour at infinity of generalized functions of one real variable. The main concept explored is the Cesaro or Riesz behaviour of distributions. It is shown that the notion of Cesaro summability of series and integrals considered in classical analysis (Hardy 1949) admits a direct generalization to distributions and that this generalized notion has many interesting and useful properties.

The usual method of studying generalized functions at infinity is by studying the parametric behaviour. This is also known as the distributional behaviour. It is one of the basic ingredients of the distributional theory of asymptotic expansions as considered by several authors (Estrada & Kanwal 1990, 1994; Pilipovic 1990; Vladimirov et al. 1986). One of the principal results of the present article is the complete determination of the relationship between the two concepts, namely the Cesaro and the distributional behaviours. It is shown that, except for what can be called initial conditions, the two give roughly the same information.

The determination of the relationship between the two types of behaviour at infinity promises to be of great importance in other areas. Indeed, the fact that the Cesaro limits, which are an averaged type of limit, and the distributional limits are basically equivalent allows one to use the simpler ideas of parametric analysis to study complicated averaging schemes. The potential applications in the spectral asymptotics of differential operators were the motivation for this study (Estrada & Fulling 1997), but these ideas have also shown to be rather useful in other contexts (Estrada et al. 1997).

Asymptotic expansions of Green functions and spectral densities associated with partial differential operators are widely applied in quantum field theory and else- where. Starting with the work of H6rmander (1969), the Riesz means have played a central role in the study of these spectral asymptotic expansions. The ideas of the present paper can be applied to the spectral analysis of Green functions, which are to be expanded as a series in a parameter, usually the time (Estrada & Fulling 1997). This analysis permits one to determine whether the dependence of the expansion

Proc. R. Soc. Lond. A (1998) 454, 2425-2443 ? 1998 The Royal Society Printed in Great Britain 2425 T1X Paper



R. Estrada

coefficients on the background geometry is local or global. The analysis also shows when the expansion is genuinely asymptotic in the literal pointwise sense or is valid in a distributional Cesaro-averaged sense: this is the difference between the heat kernel and the Schridinger kernel. Calculations in the physics literature that are sometimes dismissed as formal are provided with a firm footing.

Applications to spectral geometry are considered in Estrada et al. (1997). In particular, the character of the universal bosonic functional, introduced in Chamseddine & Connes (1996) to replace the Yang-Mills action functional, can be understood and the distributional Cesaro-averaged sense of the asymptotic expansion of such a bosonic functional is established.

The last part of the present study is concerned with a question of paramount importance in the theory of distributional asymptotic expansions, namely the characterization of the 'distributionally small generalized functions'. The distributionally small functions are those that satisfy the moment asymptotic expansion, which is the basic building block of the distributional expansions (Estrada & Kanwal 1990, 1994). It has been known that not all generalized functions satisfy the moment expansion, and several classes of distributions where the moment expansion holds have been identified. Here it is shown that the space of distributionally small generalized functions coincides with the space C', dual of the space of the so-called GLS symbols (Grossman et al. 1968). These results generalize and extend those of Estrada (1992), where distributionally small sequences were studied and it was shown that many famous number theoretical residues are distributionally small.

The plan of the article is as follows. In ? 2 a brief summary of the spaces of distributions used and of the theory of distributional asymptotic expansions is given. The concept of Cesaro behaviour is introduced in ? 3 and some of its basic properties are studied. The possibility of using summability ideas to assign a value to distributional evaluations is explored in ? 4. The next section gives the relationship between the Cesaro and distributional behaviours. The last two sections apply these ideas to the characterization of certain classes of distributions, particularly the distributionally small generalized functions.

2. Preliminaries

In this section we explain the spaces of distributions needed in this paper. We also recall the moment asymptotic expansion, the basic result in the distributional theory of asymptotic developments.

The spaces of test functions D, ,, S and the corresponding spaces of distributions D', E' and S' are well known (see Estrada & Kanwal 1994; Horvath 1966; Kanwal 1983; Schwartz 1966). We shall also need the spaces OM,OC, K and their duals. A function of a real variable qf belongs to OM if there are constants 7yk such that

+(k)(x) = O(IxIYk) as IxI - oo. If /yk = a for all k, then q E (Oc. When -k- = - k then b E IC. The topologies of these spaces are given by the canonical seminorms.

The moment asymptotic expansion is given by

f(~x)- (-1~'IO )kPk6(k) (x) f(Ax) - k!Ak+ as A- oo, (2.1)

k=O

where

k f(x),xk), k C N (2.2)

Proc. R. Soc. Lond. A (1998)

2426




are the moments of the distribution f. The moment asymptotic expansion holds for distributions of rapid decay at infinity: it holds in the spaces ?', O', ?O and K', but not in D' nor S' (Estrada & Kanwal 1990, 1994). In these last two spaces the moments do not generally exist for all k.

The interpretation of (2.1) is in the distributional sense, that is, in the weak dual topology of the spaces '?, 0(, O or C', namely

(f(Ax), (x)) = k + (0N+2 as A - oc (2.3) k-=0

for each q in the corresponding space of test funtions.

3. The behaviour of distributions in the (C) sense

Let f be a distribution in one variable, f c TD'(R). Our first aim is to study the Riesz or Cesaro behaviour of f(x) as x - oo.

Definition 3.1. Let / E R \ {-1, -2, -3,... }. We say that

f(x) O(x3) (C), as x -oo (3.1) if there exists N c N, a primitive of order N of f, F and a polynomial p of degree N- 1 such that F is locally integrable for x large (x > xo) and the ordinary relation

F(x) = p(x) + O(N+), as x - oo, (3.2) holds.

The notation (C, N) could be used if we want to be more specific. If an order relation holds (C, N) for some N it also holds (C, M) for M > N.

The relation f(x) = o(x) is defined similarly by replacing the O by o in (3.2). The notation f(x) = o(x-~) (C) as x -+ oo means that f(x) = o(x,) (C) for every /3.

Observe that we require 3 E R \ {-1, -2, -3,... . When / = -1, -2, -3,..., a different definition, involving the primitives of x-, x2, x-3,... should be used. Since we shall not need to consider this case to obtain our results, we shall always suppose that /3 = -1, -2, -3,....

Notice that we ask (3.2) to hold for a primitive of order N. However, it is clear that the equivalent of (3.2) holds for all primitives of order N for suitable polynomials of degree N - 1.

Suppose that p(x) = aN_lXN-1 + aN_2N-2 + -- alx + ao and that -(k +1) < /3 -k for some k E Z. Then only the coefficients aj, j N- k are of importance, since the coefficients for j < N - k are arbitrary. In particular, if 3 > -1 then the polynomial p is arbitrary and thus irrelevant.

Observe that the (C) behaviour of a distribution at infinity depends only on the behaviour of the distribution for x large. Indeed, if f(x) = g(x) on (a, oo) for some a then f(x) = O(x3) if and only if g(x) = O(x/) and similarly with the o symbol. Thus, by multiplying by a suitable cut-off function we may suppose when needed that our distribution has support bounded on the left.

We shall consider the case when x -+ oo only, but it should be clear that the case when x - -oo is completely analogous.

Let us start with the basic properties of this notion.


2427



Theorem 3.2. Let f CE D' such that

f(x) = O(x) (C, N), as x -+ oo. (3.3)

Then for k = 1, 2, 3,...

f( k)(x) -

O(x3-k) (C, N + k), as x -+ o. (3.4)

Proof. Indeed, if (3.3) holds there is a primitive of order N of f, F, which is locally integrable for x large and a polynomial p of degree N - 1 such that

F(x) = p(x) + O(X3+N), as x -+ oo. (3.5)

But F is a primitive of order N + k of f(k) and p is also of degree N + k - 1. Hence

(3.5) also yields (3.4). U

The behaviour of primitives is also simple.

Theorem 3.3. Let f C D' such that

f(x) = O(x), (C, N), as x - oo. (3.6)

Let F be a primitive of order n of f. Then there are constants ao,..., an such that

F(x) = anxn + .* * + alx + ao + O(x3+n) (C, M) (3.7)

as x -+ oo, where M = max{N - n, 0}.

In order to give our next result we need some notation. Recall that there is no standard way to single out a particular primitive of a distribution, in general. How-

ever, if the distribution has support bounded on the left, there is only one primitive with support bounded on the left. Thus, we shall use the notation I(f) to denote the primitive of f with support bounded on the left when there is one, and more

generally, In(f) the primitive of order n with support bounded on the left. When f is also locally integrable, then

rx

I(f)= f(t)dt, (3.8) J -00

In() c a (o (n

-

1)! ff(d (3.9)

We can now prove the following.

Theorem 3.4. Let f E D' such that

f(x) = O(x3) (C), as x -+oo. (3.10)

Let a c R. Then if a + /3 -1, -2, -3,....

xaf(x) = O(x"+ ) (C), as x - oo. (3.11)

Proof. Our proof will follow from the following formula, valid if f has support bounded on the left:

In (xf(x)) = (-l)j )C(Ca,j)Ij(x-jfn(x)), (3.12) j=0

where fn = In(f), and where C(a,j) = a(a - 1)... (a - j + 1).


2428 R. Estrada




To show that (3.12) holds, all we should show is that the nth derivative of the right-hand side equals x0f(x), since this expression has support bounded on the left. But,

dx (-l1)j C(a,j)Ij(x-fn(x)) j=0

= E (-I) f) c(vi() dn-j ((x))

E(-1)j (j) ca E j) ) C - - q n x) j= 0 q=O

q=O m=q

= x&f(x),

since n

(mi n^-^^(_l _ 1I dq

2m= q qm) () 1) m q= dxq (1 + x)nlx=-_ = q,n (3.13)

m=q

Suppose now that f(x) = O(x/) (C) as x -+ oo. As we said, we may assume that f has support bounded on the left. Then there exists n such that the ordinary relation In(f) = O(x +~) holds. Thus, Ij(x-jfn) = O(xc+fP+n), and using (3.12), In ,(xf(x)) = O(xo"+/+n), and the desired relation x"f(x) = O(xc+,) (C) follows.

The notion of limit in the (C) sense can be defined in terms of the order relations. We say that limxoo = L (C) if f(x) = L + o(1) (C), as x -4 oo.

Example. We have

sinx = O(x-??) (C), as x - oo, (3.14)

and

cosx = O(x-?) (C), as x - oo. (3.15)

More generally, if f is periodic and if ao is the constant in the Fourier series of f then ao is the mean value of f in the sense that

f(x) = ao +O(-??) (C), as x - oo, (3.16)

a fact that yields

lim f(x) = ao (C). (3.17) x-+oo

4. The Cesaro summability of evaluations

Let f E VD and let 0 E E. Then, in general, the evaluation (f(x), q(x)) does not make sense. In particular, the moments (f(x),xn) are not always well defined. Of


2429



course, if we can find a space X C ? such that f c X' and X c X then there is no problem. Here, however, we consider another approach, based on the summability ideas, which is not so restrictive.

To orientate the discussion, suppose f(x) = n'1 an,(x - n). Then (f(x), ((x)), when defined, should be equal to the sum of the series 'n??= an((n). If the series is convergent, this gives a value (f(x), b(x)), but, more generally, we can use the summability theory of divergent series (Hardy 1949).

Similarly, if f is locally integrable and supported in (a, oo) then the evaluation should be equal to a f f(x)q(x) dx. The problem of assigning a value to the integral can be handled by using the theory of summability of integrals (Hardy 1949).

Our definition applies to a general distribution and reduces to the standard theory of summability of series and integrals in these particular cases.

Definition 4.1. Let f E D' be a distribution with support bounded on the left and let Ec ?. We say that the evaluation (f(x), q(x)) has a value L in the Cesaro sense, and write

(f(x), x(z)) =L (C) (4.1)

if the first-order primitive G = I(g) of the distribution g(x) = f(x)>b(x) satisfies

G(x) = L + o(1) (C), as x- oo. (4.2)

It is not hard to show that

an6(X - n), q(x))=L (C), (4.3) n=l

if and only if 00

E anoq(n) = L (C). (4.4) n=l

Similarly, if f is locally integrable and supported in (a, oo) then

(f(x), (x))= L (C), (4.5)

if and only if

/ f(x)(x) dx = L (C). (4.6) Ja

A similar definition applies if f has support bounded on the right. Suppose now that f is any distribution and X G E. Let f = fl + f2 be a

decomposition of f where fl(x) and f2(-x) have support bounded on the left. Then we define (f(x),)(x)) = L (C) if both (fi(x),0(x)) = Li (C) exist and L = L1 + L2. Observe that this definition is independent of the decomposition: if also f = fi + f2, then h = fi - f1 = f2 - f2 has compact support and thus

(fi(x), )(x)) (C) exists if and only if (fi(x), +$(x)) (C) exists, since / E E, and in that case the sum (fi(x), )(x)) + (f2(x), )(x)) (C) is also equal to L.

Example. Let f = fo be a periodic distribution, of period p, whose mean, the constant in the Fourier-series expansion, vanishes. Let fn be the periodic primitive of order n with zero mean. Then

xnfl(x) - nx- lf2(x) + n(n- 1)xn-2f3(x) - + (-l)-ln!f+l(x)


R. Estrada 2430




is a first-order primitive of xnf(x), and since fj(x) = o(x-??) (C), as x -+ oo, it follows that

(f(x),x) = 0 (C), (4.7)

for n E N.

Example. As we shall see in ? 7, if f E /C' and q E C, then the evaluation (f(x), q$(x)) is always (C) summable. In particular, since E-nL1 6 (x - n) - H(x - 1) belongs to K', where H(x) is the Heaviside function, it follows that the evaluation

/ 00 00 00

8(x - n) - H(x- 1), c(x) = , 0(n) - q0(x) dx (C) (4.8) n=1 n=l 1

is Cesaro summable whenever 0 E /C. If we now take a function qo0 E KC such that

0o(x) = 1 for x > 1, 0o(x) = 0 for x < 2 and apply (4.8) to q((x) = qo(x)xa, which belongs to IC, we deduce that the evaluation

6(x - n) - H(x - 1), x = Z(a) (C), (4.9) n=l

is (C) summable for any a E CC. Actually it is easy to see that since 5, is an entire function of ca then Z(ca) is also an entire function.

We can find a formula for Z(o) if we observe that if Re a < -1 then the evaluation (4.8) is given by the difference of a convergent series and a convergent integral, so that

oo oo 1 Z(a) = Zna - / x = (-a) + 1+ Re < -1, (4.10)

n=l c+ 1'

where C(s) is the Riemann zeta function. If we now use that C(s) is analytic in C \ {1} it follows that (4.10) holds for ca : -1. Alternatively, the fact that C(-a) = Z(a) - (l/(a + 1)) shows that ((s) is analytic in C \ {1} and that the residue at s = 1 is equal to 1.

The evaluation (E? 16(x - n) - H(x), xa) is somewhat more complicated because xa does not belong to KC unless a E N. We may avoid this problem by considering the evaluation in the spaces C{xCn ) considered in Estrada & Kanwal (1994, ? 2.11). Or we may use the following approach. Write

/ 00 \ 1

( 6(x- n) - H(x), x = Z(a) + FP x dx, (4.11) n=l O

where FP stands for the Hadamard finite part of the integral (Estrada & Kanwal 1989; Sellier 1994). But

FP/0 xadx=- ,-- a7-1, Jo a+l'

while

r0 FP / x-1 dx = 0. (o


2431



R. Estrada

Therefore if a f -1, 00 /?00

na - FP j ax dx = (-a) (C), (4.12) n=l

in the sense that

D/ N x lim ( n"- FP ta dt) (-a) (C). (4.13)

This formula reduces to the definition of the Riemann zeta function when Re a < -1, but gives a nice new representation when Re a > -1. For instance, if a = 0 we obtain

H] .x

E n?-FP t? dt= x]-x n=l

2 +o(x- 0) (C),

because x ]- x is periodic of period 1 and mean - . The well-known value ((0) = - is recovered.

When a = -1 we obtain

lx x lim (n~n--FP / t-dt) =- (C), (4.14)

where 7y is Euler's constant. Of course the (C) is unnecessary in this case, since

[ Hx M n-1 FP/ t-ldt = En-l - lnx

n=l n=l

= ln[xr + y + o(1) - In x

= + o(l), as x - oo.

Observe now that according to (Gel'fand & Shilov 1964), f0 x0 dx = 0 for any a E C, in the sense of analytic continuation. Thus (4.12) and (4.14) yield the useful formulae

00

n = (-a), a -1, (4.15) n=l

and 00

En-17 (4.16) n=l

for the generalized moments of ?=1 6(x - n).

5. Parametric behaviour

Our next aim is to show that the Cesaro behaviour at infinity and the parametric behaviour at infinity are roughly equivalent.

In what follows, distributional means with respect to the topology of D'.


2432



The Cesdro behaviour of distributions 2433

Lemma 5.1. Let f(x, A) be a distribution that depends on the parameter A for A > Ao. If

f(x, A) = O(A"), as A - oo, (5.1)

distributionally, then

&f(x,A) f = o(A'), as -oo. (5.2)

Proof. Notice that (5.1) holds distributionally if and only if the relation holds weakly, that is

(f(x, A), 0(x)) = O(AX), as A - oo, for all q E D . But this yields

O (x , )(x)) = -(f(x,A),0'(x)) =O (Aa),

since C' c P. T

Lemma 5.2. Let f E D' have support bounded on the left and satisfy

f(x) = O(x) (C), as x -+ oo, (5.3)

where a > -1. Then

f(Ax) = O(A"), as A - oo, (5.4)

distributionally.

Proof. There exists N such that if F is the primitive of order N of f with support bounded on the left, then F is locally integrable and

F(x) = O(xQ+N), as x - oo. (5.5)

Then it follows by Estrada & Kanwal (1994, lemma 4) that

F(Ax) = O(A"+N), as A -+ oo, (5.6) and differentiating N times with respect to x,

ANf(Ax) = O(A+N),

so that (5.4) follows. )

The case when a < -1 is slightly more complicated. Lemma 5.3. Let f E D' have support bounded on the left and satisfy

f (x) = O(x) (C), as x - oo, (5.7) where -(k + 1) > a > -(k + 2) for some k E N. Then the moments

(f (x), xi) = -j (C), (5.8) exist for 0 < j < k and

f(Ax) - E (-) j(x + O(AX), (5.9) j!AJ+l j=o

as A -X oo distributionally.




Proof. Indeed, if f(x) = O(x) (C) then there exists N ) k + 1 and a polynomial p of degree N - 1 such that if fN is the primitive of order N of f with support bounded on the left, then fN is locally integrable and

fN(x) p(x) + O(xO+N), as x - oo. (5.10)

Writing the polynomial as

p(x)= -N ()qqx-l (5.11) q- 0 q!(N - i - q)! ( )

it follows that

fN(x) -E q!(N 1q + O(x+fN) as x - o. (5.12) q=O q!(N - 1 - q)!

Thus (Estrada & Kanwal 1994, lemma 6), k

(-1) q M AN-1-qxN-1-q fNE(Ax k )-1 + o(A-q+)N), (5.13)

f q!(N - 1 - q)! q=-0

distributionally as A -+ oo. Equation (5.9) is then obtained by differentiating N times.

Our proof will be complete if we show that (5.8) holds. If we differentiate (5.12) N - j times, we obtain that the primitive of order j with support bounded on the left has the development

j-1 1--q

f(x) = E q(j - q)! + O(xa+j) (C), (5.14)

for 0 < j < k + 1. When j = 1, (5.14) gives the result (f(x), 1) = /o (C). In the general case we

observe that

(n-! Gn (x) = E ') xn-if+(x) : (5.15)

j=0

is a first-order primitive of xnf(x), i.e. G'(x) = xnf(x), as it is easy to verify. But using (5.14), we obtain for 1 < n < k

Gn =(X) (n-i)j q !q + o(xn+l+c) (C)

n q n n

q=O q jq

= (-l) - (_ pn! , +o(0n+l+a) (C).

If we now use (3.13), namely

(-1)Jn! (I (n j)!(j - q)! = (-1)nn!q,n'

3jq

and observe that n + 1 + a < 0, the required relation

Gn(x) = pn + o(1) (C), as x - oo

is obtained. X


2434 R. Estrada




The following result works for any value of a.

Lemma 5.4. Let f E D' have support bounded on the left. If

f(Ax) = O(AC), as A -+ oo, (5.16)

distributionally, then

f(x) = O(x) (C), as x - oo. (5.17)

In the case -(k + 1) > a > -(k + 2) for some k E N, the moments uj, O j j < k all vanish.

Proof. If (5.16) holds, there exists N such that the primitives of order N (with respect to x) of f(Ax) are bounded by MA' for Ixl < 1 and A > Ao. But if F(x) is the primitive of order N of f(x), then A-NF(Ax) is the primitive of order N of

f(Ax). Thus

IF(Ax)l < MAt+N l , x< A , Ao. (5.18)

Taking x = 1 and replacing A by x we obtain

IF(x)| <Mx"+N, x Ao,

so that

F(x) = O(x++N), as x - oo,

and thus

f(x) = O(x?) (C), as x -oo.

That Lj = 0 for 0 < j < k when -(k + 1) > a > -(k + 2) follows by applying lemma 5.4 and comparing (5.16) and (5.9). U

We may summarize our results as follows.

Lemma 5.5. Let f c )' with support bounded on the left. If a > -1 then

f(x) = O(x0) (C), as x - oo0, (5.19)

if and only if

f(Ax) = O(A"), as A - oo (5.20)

distributionally. When -(k + 1) > a > -(k + 2) for some k c N then (5.19) holds if and only if

there are constants o, . . ., Uk such that

/f(Ax) - 1 + O(A1), (5.21)

= j!Aj+i j=0

distributionally as A -+ oo.

The corresponding result also holds if f has support bounded on the right. The general case follows by using a decomposition f = fi + f2, where fi has support bounded on the left and f2 has support bounded on the right. Then, as it is easy to see, f(Xx) = O(A'), as A -+ oo distributionally if and only if fi(Ax) = O(A') and f2(Ax) = O(A'). Thus, we obtain the following.


2435



Theorem 5.6. Let f .E D. If a > -1 then

f(x) = O(x) (C), as x -+ oo, (5.22 a)

and

f(x) = 0(I xI) (C), as x - -oo, (5.22 b) if and only if

f(Ax) - O(Ac), as A - oo, (5.23)

distributionally. When -(k + 1) > a > -(k + 2) for some k E N then (5.22) hold if and only if

there are constants Mo,..., I,tk such that

f (X) (-

)jA (X) + o(."), (5.24)

distributionally as A -X oo.

Example. Consider the distribution of the eigenvalues {A,},=l of the negative Laplacian -A on a d-dimensional torus. If the {/n}=1I are the corresponding eigenfunctions, then )n can be considered a non-zero smooth function in Rd that satisfies

A(n + An-n 0, (5.25) and the periodic condition

qn(xl + al,..., Xd + ad) = qn(l, . . .,xd), (5.26)

where al,..., ad are the dimensions of the torus. Clearly, the eigenvalues are given by

4,rr2 47r2 Akl,...,kd = +..+ 2 d (kl,..., kd)E ZZ (5.27)

a^ ad

with corresponding eigenfunctions

/kl... kd(X1,.. . ,Xd) = e2 ikl/a . .e27rikdd/ad. (5.28)

Thus the AX are those numbers that admit the representation (5.27), arranged in non-decreasing order. Observe that when al = .* = ad = 27r then the An are the

non-negative integers q that can be written as a sum of d squares and each such value q has the multiplicity rd(q), where rd(q) is the number of integral solutions of the equation q = k2 +... k .

Let us analyse the behaviour of the counting function

N(x)= = 1. (5.29) An (X

The leading behaviour

al... a x/2 as x -+00, (5.30) N(x) (5.30) 2drd/2r(d/2)

is the well-known Weyl formula (Clark 1967; Weyl 1912), which in the case a1 = ... ad = 27r goes back to Gauss. The fact that the error term in (5.30) is rather

complicated (in the ordinary sense) is also well known.


2436 R. Estrada




Our aim is to obtain the expansion of N(x) in the Cesaro sense. Using theorem 5.6, we may consider the distributional behaviour of N(x). Let us start with the derivative N'(x); we have

00

N'(x) = (x - An) = E 28(x -k2 -b2 2), (5.31) n=1 (kl,...,kd)EZd

where bj = 27r/aj. Let 0 E D(IR) and A be a large real parameter and set e = 1/A, so that e - 0+.

Then using Estrada & Kanwal (1994, theorem 53),

(N'(Ax), ?(x)) = E(N'(x), q(Ex)) 00 00

=? E * E '(?(bk +. + b2k2))

kl=-oo kd=-oo r00 rOO

= e? ... ( / (blXl + * + bdx)) dxl ... dXd + o(?0) -00 J -00 -oo -oo

al ... ado1-d/2 2-f0 xI d/2_lq(x) dx ? 0(600). 2dnd/2F(d/2)

x () dx + o(J?).

Hence, distributionally

N'(Ax) = al ... ad Ad/2-1xd/2-1 + 0(A00), as A - 0 (5.32) 2drd/2F(d/2)

+

and upon integration

N(Ax) = al... ad AXd/2/2X + o(A-), as X - o. (5.33) 2d7rd/2F(d/2 + 1)

Observe that the constant of integration p0 = (N'(x), 1) vanishes, as do all the other moments/1, Li, 2, ...

Theorem 5.6 then yields

) 2dal ... ad xd/2 + o(x-?) (C), as x - o (5.34) N(x)

2drd/2I(d/2 + 1) Hence the error term, although definitely not small in the ordinary sense, is of rapid decay in the (C) sense. The fact that higher-order terms in the asymptotic expansion of the distribution function of the eigenvalues of the negative Laplacian (for other boundary conditions) are to be understood in an 'averaged' sense was proved by Brownell (1957) several years ago. The emphasis in recent years has been on Abelian- type expansions, the so-called heat kernel techniques (Gilkey 1975). Notice, however, that the Abelian expansions do not allow one to obtain the higher-order terms of the expansion of N(x), because Tauberian theorems apply only to the leading-order term.

Example. Let us consider the Green function G(t, ,, 71) corresponding to the oper- ator g(-tA) that acts on the cylinder (0, oo) x T, where T is the d-dimensional torus of the previous example. The integrated trace of G has the form

oo

TG(t, ,Od = g(tA), (5.35)

n=l


2437



R. Estrada

or

G(t, , ) d - (N' (A), g(t)). (5.36)

Therefore, if g 1CK then we obtain the development

al ... ad fo xd/2-1g(x) dx d/2 It a ad/ P(d ) t-d/2 + o(t?), as t - 0. (5.37)

For instance, for the heat kernel K, corresponding to e-t, we have g(x) = e-x for x > 0. The function e-x does not belong to KC, but we may modify g for the irrelevant interval x < 0 to have g E C. Hence (5.37) can be applied, to obtain

I (t , ) d = ; 2d.d./2td/2 + o(t'), as t 0+. (5.38)

These are ordinary expansions. However, when g ? KC then (5.37) cannot be applied. For instance, if g = X[0,I],

the characteristic function of the unit interval, then the integrated trace is equal to the distribution function N(x) with x = 1/t. So formally (5.34) would follow from (5.37), but, of course, (5.34) is not an ordinary expansion.

6. Characterization of tempered distributions

Our results allow us to characterize several spaces of distributions by their Cesaro behaviour at infinity. Tempered distributions are easy.

Theorem 6.1. Let f C D'. Then the following statements are equivalent. (a) f is a tempered distribution, i.e. f c S'. (b) There exists a E IR such that

f(x) = O(x) (C), as x -+ oo, (6.1 a)

and

f(x) = O(Ixl) (C), as x -+ -o. (6.1 b)

(c) There exists a CE R such that

f(Ax) = O(A), as A - oo, (6.2)

distributionally.

Proof. It is easy to see that it is enough to consider the case when f has support bounded on the left. To see that (a) -+ (b) we use the well-known result that says that if f C S' then there is a primitive F of some order N which is of slow growth at infinity (Horvath 1966; Kanwal 1983; Schwartz 1966), i.e. there exists a E R such that F(x) = O(x"+N) as x -+ o, and it follows that f(x) = O(xz) (C). Conversely, (b) implies that some primitive of some order is of slow growth and differentiating, that f is a tempered distribution. The equivalence of (b) and (c) follows at once from theorem 5.6. a

It is interesting to observe that, in the Cesaro sense, the tempered distributions behave at infinity like the elements of KC.


2438




Theorem 6.2. Let f E S' with support bounded on the left. Then there exists a E R such that

f(n)(x) = O(x-n) (C), as x -oo. (6.3)

Proof. Indeed, there exists ac such that f(Ax) = O(A'), as A -+ oo distributionally. But according to lemma 5.1, distributional order relations can be differentiated at will: (6.3) is then obtained immediately from theorem 5.6. X

7. The space 1C'

The space ICq is formed by those smooth functions 0 such that q(n)(x) = O(lxlq-n) as xl -+ oo for each n E N. A topology is generated by the seminorms

ql0lln,q = max{ sup {q(n)(x)}, sup {|lx-1-q(n)(x)}}. Ixl<A1 Ixl1

The space /C is the inductive limit of the spaces ICq as q -+ oo. It is known (Estrada & Kanwal 1994) that the elements of IC' satisfy the moment

asymptotic expansion, i.e. they are distributionally small at infinity. In this section we show that the converse also holds: every distributionally small generalized function belongs to C'.

Lemma 7.1. Let f E D' have support bounded on the left and satisfy

f(x) = o(x-?) (C), as x -+ oo. (7.1) Then f E C'.

Proof. Since IC' is the projective limit of the spaces ICK as q -+ oo, we should show that if (7.1) holds then f E IC for each q. But from (7.1) it follows that f(x) =

O(x-q-2) (C) as x -+ oo. Thus, there exists N such that the primitive of order N of f with support bounded on the left, fN, is a locally integrable function that satisfies fN(x) = p(x) + O(x-q-2+N) as x - oo, where p is a polynomial of degree N - 1. Observe that p vanishes if and only if the moments (f(x), xi), 0 < j < N - 1 vanish. Let g be a compactly supported continuous function whose moments of order up to N - 1 coincide with those of f. Then if gN is the primitive of order N of g with support bounded on the left, then fN(x) - gN(X) = 0(x-q-2+N).

If ( E KIq-N then the integral f_ (fNj(x) - gN(x))q0(x) dx converges and thus fN -9N E Kq_N. Hence f = f(N) g E 'C. :

We immediately obtain the following. Theorem 7.2. Let f E D'. Then the following are equivalent. (a) f E '.

(b) f is distributionally small, i.e. there exist constants u-o, /l,, p 2,... such that

fo5(x) AIJ('(X) -U2611(x) f(Ax)- ++A2... , (7.2) A A2 ? 2!A3 ...,

as A -+ oo distributionally. (c) f satisfies

f(x) = o(x-') (C), as x -X oo, (7.3a)

and

f(x) = o(Ixl-0) (C), asx - -oo. (7.3 b)


2439



Using this characterization of the distributionally small generalized functions, we can obtain several interesting results. Let us see first of all that the evaluations

(f(x), q((x)), where f C IC' and q E IC, are always (C) summable.

Theorem 7.3. If f E CK' and q E IC then the evaluation (f(x), O(x)) is Cesaro summable. In particular, the moments of f are (C) summable.

Proof. It is enough to do it if q = 1, since qf E KC' if f E C' and q E IC. But according to theorem 7.2, if f E C' then f(x) = O(x-"0) (C) as x -X oo. Thus, by using lemma 5.3, (f(x), 1) is (C) summable. I

Example. The Fourier transforms of tempered distributions are defined by dual-

ity. In general, if f E S', then f(u) cannot be computed as the evaluation (eix, f(x)), because this evaluation is not defined. However, if b E IC and u f 0, theorem 7.3 guarantees that the evaluation (eixu, ((x)) (C) is well defined. Thus

(eixu, )(x)) = ( (u) (C), q EC, u 0. (7.4)

Thus all the distributions of C' have Cesaro summable moments. Interestingly, the existence of the moments in the (C) sense implies that a distribution belongs to K'.

Theorem 7.4. Let f E i'. If all the moments

(f(), xn) - t, (C) (7.5)

exist for n E N, then f E 1'.

Proof. We may suppose without loss of generality that f has support bounded on the left. Relation (7.5) means that for each n c N we have

I(xnf(x)) = n + o(1) (C), as x -+ o. (7.6)

Thus, I(xnf(x)) = n, + O(x1/2) (C) and it follows that

xnf(x) -O(x-l/2) (C), as x -- o, (7.7)

so that

f(x) = 0(x-n-l/2) (C), as x -+ oo. (7.8)

Since n is arbitrary, we obtain f(x) = O(x-??) (C), and theorem 7.2 yields that

f EIC'. E

Example. The result of theorem 7.4 does not hold if we consider Abel summabil-

ity. For instance, the function 0oo

f(x) = Z(-l)ke/kS(x- k) (7.9) k=1

has moments of all orders in the Abel sense: 00

(f(x),xn) = (-l1)ke\/kkn (7.10) k=1

is (A) summable in the sense that limr,l Ek= (-l)keV/kknrk exists for each n E N

(Hardy 1949). However, f ? KC'.


2440 R. Estrada




Example. Let q be a smooth periodic function of period 27r. Let 00

ao + E (an cos nO + ,n sin n0) n=l

be its Fourier series. Then the series converges to 0, but more than that, the rate of convergence is rather strong. Indeed,

2aol + (an cosn0 + 3nsin n) = X(0) + o(x-~), as x - oo. (7.11) n(x

Similarly, if F is a periodic distribution of period 27r and 00

ao + Z (an cos nO + bn sin nO) n=l

is its Fourier series, then the series converges to F distributionally, but more than that

la0 + Z(ancosnO + bsinnO) = F(O) +o(-xo), as x - oo, (7.12) nrx

in the topology of D'(IR). The convergence in (7.12) is very strong but it is distributional with respect to 0. Both (7.11) and (7.12) are ordinary asymptotic expansions with respect to x.

Our aim is to study the pointwise behaviour of (7.12) at points 0 where F is smooth, that is points that do not belong to the singular support of F. A simple example, as the series

1 100 00 +- E cos nO= E 6(0-2k7),

27r 7T n=l k=-oo

shows that (7.12) does not hold pointwise: if 0 7 2klr then

1 c (1 + cos 0) cos x) co + sin 0 sin Mx 0 27r 7r z< 7 (1 - cos 0)

is oscillatory as x -+ oo and thus is not of the form o(x-~), actually not even of the form o(l). However, this oscillatory partial sum is o(x-~) in the Cesaro sense. Actually the same is true in general, that is if 0 q singsup F then (7.12) holds pointwise in the (C) sense:

aoH(x) + (an cosn + bn sin nO) = F(0) + o(x-) (C), as x -oo. (7.13) n?x

To prove this consider the derivative

f(x) = d ( aoH(x) + (an cos nO + bn sin nO)) n?x

oo

= 2ao6(x) + Z (an cos n + bn sin n0)(x - n). n=l


2441



When 0 , singsup F, then all the moments exist in the (C) sense (Estrada 1996; Walter 1966; Zygmund 1959):

00

la0o + (a cos n + bn sin nO)nk = Uk (C), (7.14) n=l

where

P2k = (-1)kF(2k)(0), U2k+1 = (-l)kG(2k+1)(0) (7.15)

and G is the conjugate function n=_ l (an sin nO - b cos nO). Therefore, using theorem 7.4 it follows that f c 1C' and

f (x ) (_1)kF(2k)((2)( k) () (-l)k+lG(2k+1) (O)j(2k+1) () (716) f(Ax) E L (2k)!A2k+l (2k + 1)!A2k2

( ) k=O

as A -+ oo. Relation (7.13) follows immediately. We also record the expansions

00

2aoO(0) + J (an cos nO + bn sin nO)O(ne) n=l

[(-l1)kF(2k)(0)0(2k)(0)22k (_i)k G(2k+l) (0)(2k+1)()62k+ 717) k~= ~ (2k)! (2k + 1)!

as E -X 0+, for 0 E S and 0 0 sing sup F. When q(x) = e-x, for x > 0, we obtain the expansion of Abel-type means:

00

a0 + E (an cos nO + bn sin nO)r n=l

ft (-1)kF(2k)(0)(lnr)2k (-1)k+l G(2k+l) (0)(lnr)2k+ 1 as r

=[ (2k)! + (2k + 1)! as

(7.18)

It is interesting to observe that (7.17), and (7.18) in particular, have a non-local character: the coefficients depend not only on the values of F near 0, but also on the values of the conjugate function G, which is a global object.

Most of the ideas in this article were developed while the author was a visiting professor at Texas A&M University. I thank Professor Steven A. Fulling, who introduced me to several problems in the spectral asymptotics of differential operators whose solutions depend on the understanding of the relationship between the Cesaro-Riesz and the distributional behaviours.

References

Brownell, F. H. 1957 Extended asymptotic eigenvalue distributions for bounded domains in

n-space. J. Math. Mech. 6, 119-166.

Chamseddine, A. H. & Connes, A. 1996 Universal formula for noncommutative geometry actions: unifications of gravitation and the standard model. Phys. Rev. Lett. 77, 4868-4871.

Clark, C. 1967 The asymptotic distribution of eigenvalues and eigenfunctions for elliptic boundary value problems. SIAM Rev. 9, 627-646.


R. Estrada 2442




Estrada, R. 1992 The asymptotic expansion of certain series considered by Ramanujan. Appli- cable Analysis 43, 191-228.

Estrada, R. 1996 Characterization of the Fourier series of a distribution having a value at a point. Proc. Am. Math. Soc. 124, 1205-1212.

Estrada, R. & Fulling, S. A. 1997 Distributional asymptotic expansions of spectral functions and of the associated Green kernels. Preprint, College Station, Texas.

Estrada, R. & Kanwal, R. P. 1989 Regularization, pseudofunction, and the Hadamard finite part. J. Math. Analyt. Appls 141, 195-207.

Estrada, R. & Kanwal, R. P. 1990 A distributional theory for asymptotic expansions. Proc. R. Soc. Lond. A 428, 399-430.

Estrada, R. & Kanwal, R. P. 1994 Asymptotic analysis: a distributional approach. Boston: Birkhauser.

Estrada, R., Gracia-Bondia, J. M. & Varilly, J. C. 1997 On summability of distributions and spectral geometry. Commun. Math. Phys. 191, 219-248.

Gel'fand, I. M. & Shilov, G. E. 1964 Generalized functions, vol. 1. New York: Academic.

Gilkey, P. B. 1975 The spectral geometry of a Riemannian manifold. J. Diff. Geo. 10, 601-618.

Grossmann, A., Loupias, G. & Stein, E. M. 1968 An algebra of pseudodifferential operators and quantum mechanics in phase space. Ann. Inst. Fourier 18, 343-368.

Hardy, G. H. 1949 Divergent series. London: Clarendon.

Hormander, L. 1969 On the Riesz means of spectral functions and eigenfunctions for elliptic operators. In Belfer Graduate School of Science A. Science Conf. Proc.: Some recent advances in the basic sciences (ed. A. Gelbart), pp. 155-202. New York: Yeshiva University.

Horvath, J. 1966 Topological vector spaces and distributions, vol. 1. Reading, MA: Addison- Wesley.

Kanwal, R. P. 1983 Generalized functions: theory and technique. New York: Academic.

Pilipovic, S. 1990 Quasiasymptotic expansion and the Laplace transform. Applicable Analysis 35, 247-261.

Schwartz, L. 1966 Theory of distributions. Paris: Hermann.

Sellier, A. 1994 Hadamard's finite part concept in dimension n ) 2, distributional definition, regularization forms and distributional derivatives. Proc. R. Soc. Lond. A445, 69-98.

Vladimirov, V. S., Drozhinov, Y. N. & Zavyalov, B. I. 1986 Multidimensional Tauberian theorems for generalized functions. Moscow: Nauka.

Walter, G. 1966 Pointwise convergence of distributions. Studia Math. 26, 143-154.

Weyl, H. 1912 Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differen- tialgleichungen. Math. Ann. 71, 441-479.

Zygmund, A. 1959 Trigonometric series. Cambridge University Press.


2443



The Cesaro Behaviour of Distributions

Documents

Transcript of The Cesaro Behaviour of Distributions