PAPER
Legitimation of the use of fancy tools
I
G.HlRANO1)I
K. TAKAHASH 12)I
T. KA I DA3)I
S. KANEM I TSU4)I
T. MATSUZAK 15)概 要:本 論 文 は 、 筆 者 らが 月1回 開 催 して い る学 際 的 な セ ミナ ー に お い て 工 学 分 野 に お け るデ ィ ラ ッ ク の デ ル タ 関数 の使 用 の 正 当性 に っ い て論 じた も の で あ る。 本 セ ミナ ー は これ ま で に12回 実 施 して お り、 工 学 領 域 にお け る数 学 的 基 盤 の 見 直 しな ど を試 み る こ とに よ り、 大 学 教 育 にお け る新 た な教 科 書 作 成 を 目標 と して い る。 これ ま で議 論 して き た 、 自動 制 御 シ ス テ ム 、 マ ン ・マ シ ン シ ス テ ム 、 離 散 フ ー リエ 変 換 、 符 号 理 論 、 コー ド化 とシ フ トレジ ス タ合 成 の線 形 再 帰 系 列 に つ い て は別 稿 で 報 告 して い る。 本 稿 は第12回 の 内容 に つ い て ま と め た もの で あ る。
キ ー ワ ー ド:デ ィ ラ ッ ク の デ ル タ 関 数 、 測 度 論 、 ボ レ ル 測 度 、 超 関 数 論
Keywords: Dirac delta-function, measure theory, Borel measure, theory of distributions
1. Introduction
We mainly mean Dirac delta-function 8 by a "fancy tool". There are many books treating the same topic from an elementary point of view, in all of which the treatment of the delta-function is not rigorous and sometimes rather fallacious, leading to the misunderstanding.
E.g. the 8-function is introduced, as the "limit" as the width w —> 0 + with the height h satisfying the relation hw = 1 , thus making h = I —> co which appears as (1.4) below. Given a pulse function f(t) which is defined for 0 < t < w(w > 0) with the value h > 0. Its Laplace transform is (for a > 0)
(1.1) L[f] (s) =ie_sthdt
wh=—(1— e-ws).
o By the Maclaurin expansion h1
L[f](s) =expansion(——w2 s' + •••)
(1.2) 1 s2
= 1 — —
2 ws + ••• = 1 + o(1), w —> O.
But this does not lead to the result L[8](s) = 1 nor more in general
(1.3) L[8](s — kT) = e-kTs This leads to the "definition":
o=f (1.4) = 0, w lo0o t which is invalid and is to be properly interpreted. This contradicts the usage of the term "unit" impulse function because of its "value" at t = 0.
As has been expounded briefly in [THKKM1] cf.
also [L], the relationship
(1.5) Y = GU
between the Laplace transforms U = U(s) resp.
Y = Y(s) of the input function u = u(t), resp. of the output function y = y(t) is fundamental, where G = G(s), the transfer function, is the ratio of Y and U when the initial values are assumed to be 0.
Example 1. The first-order element is described by the DE
産 業 理 工 学部 電 気 通信 工学 科 講 師]
産 業 理 工 学部 情 報 学科 講 師ktakaha 産 業 理 工 学部 情 報 学科 准 教 授kaida 産 業 理 工 学部 情 報 学科 教 授kanem士 産 業 理 工 学部 電 気 通信 工学 科 講 師1
(1.6) T y + y = Ku,
where K, T are constants, called the proportion constant and time constant, resp. The Laplace transform of (1.6) reads
(Ts + 1)Y(s) = KU (s), whence the transfer function is G(s) = —.
s+1
We find the output signal when we exert the unit impulse input u = S. Since L[8](s) = 1 by (1.3), the output function and the transfer function are the same and we have Y (s) =K 1whose Laplace inversion
T s+T-1 gives
K t y(t) =
In general, we may say that "the transfer function of a system is the Laplace transform of the unit impulse response (output) of the input function."
Example 2. Let H = H(t) = 1 denote the Heaviside function (or the unit step function) to be introduced in Example 9 below. The limiting case of (1.1) as w 00 with h = 1 leads to the unit step input H = H(t) = 1 for t > 0 whose Laplace transform is L[H](s) = 1.
Hence if this input is exerted in the first-order element in Example 1, then the Laplace transform reads
K
(Ts + 1)Y(s) = — s,
K 1
or Y = T s(s+T-1)'The partial fraction expansion of the
-
right-hand
side is K C1 _i).
s+7-Hence we obtain
y = K (1 —
which is the unit step response or the indicial response.
It is often stated that by the Laplace transform, ODE with constant coefficients may be treated algebraically, while in the sampled-data system, the result reads
00
(1.7) X*(s) = x(2nT)e-"T ,
n=0
where X* = L[x*] and x* is a sampler. Cf. the
1) 2) 3) 4) 5)
passage after Example 12.
Putting esT = z in X* (s) , then (1.7) may be expressed as a "power series" in z-1 not as a polynomial as stated there:
00
(1.8) Z[x](z) = X* (z) = x(2nT)z-n
n=o
called the z-transform of the sample signal x = x(t).
2. Some facts from measure theory Definition 1. Let X be a locally compact Hausdorff space. Let B be the smallest family of subsets of X satisfying
(i) B D A(X), the family of all closed subsets of X (A is for "Ab -geschlossene"-closed),
(ii) B is closed under formation of countable union, (iii) B is closed under complementation Ac.
Each element of B is called a Borel set of X.
By (iii), we may take the family 0(X) of all open sets of X for A(X) in Definition 1.
A non-negative measure it is a set function B -, U f 0) (non-negative reals) which is completely additive, i.e.
li(Uk=1 Ek) = ti(Ek)•
k=1
A measure p on X is called a Borel measure if the Borel sets in X are measurable, which is equivalent to the assertion that all open sets are measurable.
f:X->C is called a Bored function if ri(U) is a Borel set for all U E 0(X).
Let p be a measure on X and let fx f di/ denote a
bounded linear functional defined for bounded continuous functions f with compact support (for these notions, cf. §3). It is called the integral of f over X with respect to p.For 0 < p < 00, let
1
IlfIlp = • IflPdit) •
This II f lip satisfies the conditions for a norm and is called the p -norm. LP (µ) = < 00) forms a Banach space and especially, L2 (p) forms a Hilbert space with respect to the inner product
(f, g) = f f #dit.
L°° (µ) is the space of all bounded Borel functions normed by
IlfIL = ess supxExif (x) I, where
ess sup = min p(x I I f (41 > = O.
3. Basics of the theory of distributions The results in this section are partly modifications of results from Vista II [Vista II]. Good references include
[Don], [Mi, Chapter 2], [Pa], [Yo, Chapter 12].
Papoulis must be popular among the engineers in the US as it is cited in [Dos] together with J. Michener's
"Centennial".
The objective of introducing distributions is to generalize the notion of functions through "integration by parts." It should not be confused with distribution function in statistics. We begin with
Definition 2. Any measurable function on Ikn which is integrable in the neighborhood of each point (in the sense of Lebesgue) is called locally integrable (and sometimes denoted LL)• According to [Mi, p.26] , each point x is classified into two categories: (1) in a sufficiently small nbd, the function is 0 a.e. or (2) for an arbitrarily small nbd V, fv If (x)I dx > 0. The set of all points of the second category forms a closed set, called a support of the function. For an infinitely differentiable function f on an open set U c f E C°°(U), we define
supp f =tx E If(x)I > 0).
Any function f E C°°(U) with compact support c U, fE Co (U), is called a test function. We denote the totality of test functions on U by D:
D=D(U)= [1. : test-function I supp f which forms a linear space over ll or C.
For any n -dimensional index vector (ml, m2, • • • , ni,n), we define
alai Da= „.„7Y1(1),
1 • • • dxn where I al = m1 + • • • +
We introduce a family of (semi-) norms on D:
(3.1) ikoiiN = DiDaViico, N E N U 01,
lal5N
where lif = suPlf(x)1 for a continuous functi Definition 3. A linear functional (a linear mai which assigns a complex number to each set domain of definition)
T:D (U) -> C,
denoted by T E (U), is called a distribution if exist a compact set K c U and a constant C > 0 N E N such that
(3.2) IT (40i CHAN
for all test-functions cp, with tit c K.
If N can be chosen independently of K, and chosen to be the smallest such, then T is said to order N.
a=
for a continuous on f.
!ctional (a linear mapping number to each set in its
is called a distribution if there : U and a constant C > 0 and
K, and N is is said to be of
Remark 3.1. Instead of (3.2), one often adopts the continuity of T as a defining condition in the sense that lim,p 1~a Ttpl = 0 in the norm defined by (3.1).
Definition 4. Let X be a locally compact space and be a Borel measure on X. Then if p, takes on only finite values on all compact subsets of X, then it is
called a Radon measure.
9
Theorem 3.1. ([Don, p.25]) Let X be a locally compact space and let Co(X) be the linear space of all continuous functions on X with compact support. If T (f) is a functional on Co(X) having the property that T(f) > 0 whenever f (x) 0, then there exists a Radon measure on X such that
(3.3) T (f) = 1 f (x) dp(x)•
x Remark 3.2. Since the Euclidean space IV is a locally compact Hausdorff complete metric space, all that precedes applies to it and we may think of the Radon measure simply as something like Jordan measure providing the Riemann integral. We may equate two distributions by their values at all test functions, thus, the "test function go works as a variable" in calculus, as in Examples 9, 10. Since test functions are 0 outside of a compact set, we may explain away by saying "by integration by parts and boundary conditions."
Example 3. Let dpi be a Radon measure on U C Nn.
Then the functional T = Ti, defined by
(3.4) T(cp) = T1,(go) f cp (x) dp(x)
is a distribution of order 0, where the integral is performed over the subregion U of illn, which is not
mentioned at each occurrence.
Indeed, let K c U be compact and let C =
fKiciii(x)1 = total mass of clitt on K. Then
IT (V)I < CIIVII co = Cligolio for all cp E D(U)with supp cp c K.
Example 4. (The Dirac distribution). The Dirac S -distribution is a Radon measure consisting of positive unit mass at the origin,
6(p) = T 8((p) = f go (x) d6(x)
(3.5)
= f go (x) 8 (x)dx = go (0) = cp (0 , • • • ,0)
for test functions cp continuous at the origin. The Dirac distribution supported at a, 8a, is such that
(3.6) 8a(cp) = f go (x)S (x — a)dx = go (a)
for test functions go continuous at the point a. We need only assume go is continuous at the respective points in view of Theorem 3.1. Eq. (3.6) elucidates [FE,
(5.1)].
Example 5. If f is locally integrable on U, then f (x)dx is a Radon measure on U , so that the functional T = Tf defined by
(3.7) T f(go) = f go (x) f (x)dx
is a distribution, which is often identified with the function itself and we often speak of distributions
which are a polynomial, a C°° -function or the characteristic function of a set. In this sense we may express (3.7) as
f (v) = f (p(x) f (x)dx .
Example 9.
function wh x > 0. Then
Indeed,
Let U = ich has the
H' = S.
and value
let H 0 for
be the x < 0
Heaviside and 1 for
Proof. By the convergence theorem, (4.5) leads to Ty(v) = lim Tfl,(v)•
Hence, by the theorem on termwise differentiation with
,_,2
4. On the Dirac delta-function
In this section we follow the argument of Papoulis [Pa]
and Yoshida [Yo] to prove some basic results on the Dirac delta-function.
First we interpret the Dirichlet integral ([Vista I, p.
155]) as a distribution.
Theorem 4.1. (Dirichlet integral) The equality for the Dirichlet integral
roo
Proof. Divide the integral into three parts (-00, —r), (—E, E), (E, 00) with E > 0. Then since the function —(P(x) is integral over two infinite intervals, it follows from the Riemann-Lebesgue lemma [Vista I, Proposition 7.1, p. 137] that the corresponding integrals tend to 0 as t —> Co. On the interval (—E, r), cp(x) can be approximated by cp (0) and so
rE
the limit ft(x) =limt,00 ft t e2rEtxudu does
not exist in ordinary sense. However, we may prove the following
Lemma 4.1. For cp E D = D (IR) , prove that Jo
which is (4.1), completing the proof
For a class of smooth functions f , its Fourier
transform f is defined by
Proof. Proof is similar to that of Theorem 4.2.
write, corresponding to (4.25),
First we
the second
IA 11\
term on the right of which becomes, by
and the left-hand side is to mean the mean at discontinuities.
Proof follows from the special case of (4.27)
holds true, distance from
where 114 = mint' — fx}, fx}} is the
x to the nearest integer.
Exercise 4.3. Prove the Poisson summation formula in the following form for f E Cl ([a, b]) :
M = maxl [ f' (x)I.
a,b1
To estimate the error term R, we entrap all integers in [a, b] (which are —[b — a] in number) in the neighborhood of length —
N1, in which I integrandl 1
to obtain the estimate [b — a] • M N. In the remaining
intervals, we use — N 114toto obtain— 1 —1 irt
x,x = logN.
Hence altogether, the error term R = 0 (MlNogN(b a)).
Now the first term on the right of (4.33) becomes
—2 Eri=1fcb, f (x)cos27rnxdx, whence (4.33) leads to
fBi(X)1
a(X) dx = f f(x)cos27rnxdx
n=-N a+ R.
Substituting this in (4.32) completes the solution.
The following example is from Papoulis [Pa, pp.
50-52]
Example 12. (Sampling theorem, Shannon 1949) If the Fourier transform f (z) is zero for Izi > T > 0, then f is given by
sin (Tt — 7rn) (4 .34) f (t) =
Tt — 7rn where
7E\
(4.35)= f/T).
In particular, f is uniquely determined by its values (samples) at a sequence of equi-distant points,
distance 71 apart.
Solution Since the Fourier inversion formula [Vista I,
(7.2)] takes the form
of l.i.m.
If e. g. f , f' are piecewise smooth and f E L', then the Fourier inversion formula holds
100
([Vista I, (7.24)]) f(t) =—e-itz f (z)dz,
_oc,
where the left-hand side at jump discontinuities is to mean the mean -1 (f (t +) + f-)). Hence we have
2
1^
the Fourier transform pair f(t)<->—2 7,1(z) and its reflection f (t) H 2rcf (—z).
If r2T signifies the rectangular pulse function in Example 11, then
sinTz
e-itoz,sin(Tt-n
nitit)) r2T(t - to)2 (4.12)Tr(Tt-
r2T(z)e-tqz By (4.1) and (4.11), S(t) = 1 or more generally
(4.13) 6(t - to) =
This suggests the validity of the following lemma which has been extensively used in our investigations on functional equations (cf. [Vista I, p. 75]). Let (T > 0)
00
(4.14) P2T(t) = 6(t - 2nT)
n=-00
be the pulse train consisting of a sequence of equidistant pulses 6(t - 2nT) distance 2T apart.
This is a Radon measure in the sense of Definition 4.
Lemma 4.3. The Fourier transform of the pulse train is again a pulse train
00
(4.15) 13240 = T (t n
T
n=-co
which in the range -T < t < T amounts to the Fourier expansion of the Dirac delta function
00
11
(4.16) —2T>et-Tnt= lirrlDN(t) = 8(0,
2TN->co n=-co
.77
where DN(t) =ieTnt.
Proof. The Dirichlet kernel DN(t) is expressed by [Vista I, (7.13)] as
sin(27—N-Et)
(4.17)1 — DN(t) —12T7t
2T2T 7Tn-• sint — 2T
Now the third factor of the right-hand side of (4.17) is bounded on (-T, T) and the second factor tends to 6(t) by (4.10), we obtain
7t7t
(4.18)1
2T— lim DN(t) =—2T(sin2T) 8(t) = 8(0,
where we used the equality valid for a function (,o continuous at the origin
(4.19) (t)co(t) = (p(0)8 (t).
(4.18) now leads to (4.16), which in turn leads to the Fourier inversion for (4.15) in view of the periodicity of p , completing the proof.
Now we may state a generalization of the above lemma.
Theorem 4.2. The Fourier transform f of a periodic
function f (t) = f (t + 2T)) is given by a sequence of equidistant pulses
00
(4.20) f (z) = it c7,6 (z - T n),
n=-co
distance7r- apart, where cn is the Fourier coefficient given by
T
([Vista I, (7.2)]) cfe
n2Tf(t)-in;t
7,=- and conversely.Definition 5. For f,g E L (IR) , we define their convolution f * g by
co
(4.21) (f * g)(x) = J f (x - t)g (t)dt.
Then f*gEL(N) and f*g=g*f and further Exercise 4.1. Prove that
(4.22) f *g =1":"
Example 11. (i) If H is the Heaviside function (or the unit step function)in Example 9, then
f * H (t) =f f (x - t)dt.
(ii) If rT(t) = H (t + T) - H (t - T) is the rectangular pulse function, then
x+T
(4.23) (f * rT)(x) = f (x)dt x-T
known as the smoothing.
(iii) Let
(4.24) fo(t) =(t), 0 Iti < 2T, , iti > 2T,
then
(4.25) f = 1.0 * P2T • Proof. Equation (4.25) follows from
f (t) = fo(t + 2nT),
n=-00
the definition (4.14) and the evenness of the delta-function.
We are now ready to prove Theorem 4.2.
Proof of Theorem 4.2. By (4.25), (4.22) and (4.15),
co
f (z) = fo(z)1327-(z) = fo(z)67-y,n\
n=-00
which is 7 _ „, fc,(n) - 7,7t n).
Noting that r.
.10(z) = e-izt fo(t)dt = e-izt f ( t)dt,
-co-T we have
1 „fir\
-
Tfo) = cn'
which proves (4.20). The reverse implication being trivial, this completes the proof.
By the above proof, the Fourier series (cf. e. g. [Vista
I, Theorem 7.2, p. 141]) for a periodic function f (t)
may be written as
n=-.
with the rectangular pulse function r2T. Recalling the pair in (4.12), we immediately conclude (4.34).
Now we turn to (1.7) and the z-transform. A causal continuous function (defined for t > 0) works as a sampler x* extracting the sample-data at each 2nT, n E N U f 0) if it is multiplied by the pulse train:
x*(t) = x(t)p,T(t)
ry,
x-e-Thm-x
Substituting this in (4.39) (and writing —n for n) completes the proof.
We state the record of the seminars.
The 5th seminar (May 25): Checking the joint paper
"The Weber-Fechner law", which has appeared in Siaulai Math. Sem. 6 (2011), 85-91
The 6th seminar (June 22): Moja naukovaja zhyz'ni The 7th seminar (July 27): "Hamming distance correlation for q-ary codes with constant weigh" by Prof. T. Kaida
The 8th Seminar (August 8): "Zeta-functions associated with elliptic curves" by Prof. J.J. Urroz
The 9th Seminar (October 26): "Scrutinizing our
coming paper and critical reading II"
The 10th Seminar (November 23): "Around the Catalan constant" by Professor Y. Tanigawa
The 11th Seminar (January 25, 2012): Expounding some results from "Math-Phys-Chem approaches to life"
by H. Kitajima and S. Kanemitsu, to appear in Intern. J.
Math. Math. Sci. 2012,
In our previous paper [THKKM2] published in Kayanomori, there is found a mistake by a kind courtesy of Professor Y. Nakano. (1.5) should read
(s2 + w2,-1 ) = B(s — w)-1 + P(s + wirl
(1.5) = B (w)n + P(—w)n
References
[ATW] L. Auslander, R. Tolimieri, and S. Winograd, Hecke's theorem in quadratic reciprocity, finite nilpotent groups and the Cooley-Tuckey algorithm, Adv. Math. 42 (1982), 123-172.
[Br] E. 0. Brigham, The fast Fourier transform, Prentice-Hall. New Jersey 1974.
[Vista II] K. Chakraborty, S. Kanemitsu and H.
Tsukada, Vistas of special functions II,
World Sci.. New Jersey-London-Singapore etc. 2009.
[CT] J. W. Cooley and J. W. Tukey, An algorithm for machine calculation of complex Fourier series, Math.
Comp. 19 (1965), 297-301.
[Don] W. J. Donoghue Jr., Distributions and Fourier transforms, Academic Press, New York-London, 1969.
[Dos] J. D. Doss, The Shaman Sings, Avon Books, New York 1994, p. 115, 11. 20-21.
[FF] T. Fukami and T. Fujimaki, Control engineering (revised ed.) Part I, Tokyo Denki Univ. Press, Tokyo 2010.
[Vista I] S. Kanemitsu and H. Tsukada, Vistas of special functions, World Sci.. New Jersey-London-Singapore etc. 2007.
[Pa] A.—Papoulis, The Fourier integral and its applications, McGraw-Hill, 1962.
[L] F. -H. Li, Control systems and number theory, Intern. J. Math. Math. Sci. (2012), to appear.
[Mi] S. Mizohata, The theory of partial differential equations, Iwanami-shoten, Tokyo 1965.
[Rad] H.—Rademacher, Topics in Analytic Number Theory, Springer-Verlag, Berlin, 1973.
[THKKM1] K. Takahashi, G. Hirano, T. Kaida, S.
Kanemitsu, H. Tsukada and T. Matsuzaki, Record of the second and the third interdisciplinary seminars, Kayanomori 14 (2011), 64-72.
[THKKM2] K. Takahashi, G. Hirano, T. Kaida, S.
Kanemitsu and T. Matsuzaki, On linear recurrences and their applications, Kayanomori 15 (2011), 13-20.
[Wea] H. J. Weaver, Applications of discrete and continuous Fourier transforms, Wiley, New York etc.
1983.
[Yo] K. Yoshida, Modern analysis, Kyoritsu Shuppan, Tokyo (in Japanese) 1956.
