CHAPTER ONE Notations and Mathematical Preliminaries 1.1 NOTATIONS AND ABBREVIATIONS The notations and abbreviations used in the book are summarized here for ease of reference. = f a (t) :=df a (t)/dt a f—complex conjugate of / / : = f^ f{t)e~ ia)t dt, Fourier transform of f(t) f(t) := j^ /^ f{a))e ia}t dco, inverse Fourier transform of f(eo) || /1|—norm of a function / * g—convolution (/, h) := fj(t)h(t) dt, inner product /„ = O(«)-order of n, 3C such that /„ < Cn C—complex N—nonnegative integers R—real number R n —real numbers of size n Z—integers Z + —positive integers L 2 (R)—functional space consisting finite energy functions / \f{t)\ 2 dt < +00 LP (R)—function space that f\f(t)\ p dt< +00 / 2 (Z)—finite energy series 5^1-oo M 2 < +00 Q—set H S (Q) := W A ' 2 (^2)-Sobolev space equipped with inner product of
Transcript
CHAPTER ONE
Notations andMathematical Preliminaries
1.1 NOTATIONS AND ABBREVIATIONS
The notations and abbreviations used in the book are summarized here for ease ofreference.
= fa(t) :=dfa(t)/dta
f—complex conjugate of // : = f^ f{t)e~ia)t dt, Fourier transform of f(t)
f(t) := j ^ / ^ f{a))eia}t dco, inverse Fourier transform of f(eo)|| /1|—norm of a function/ * g—convolution(/, h) := fj(t)h(t) dt, inner product/„ = O(«)-order of n, 3C such that /„ < Cn
C—complexN—nonnegative integersR—real numberRn—real numbers of size n
Z—integersZ+—positive integersL2(R)—functional space consisting finite energy functions / \f{t)\2 dt < +00LP (R)—function space that f\f(t)\pdt< +00
/2(Z)—finite energy series 5^1-oo M2 < +00Q—setHS(Q) := WA'2(^2)-Sobolev space equipped with inner product of
This chapter is arranged here to familiarize the reader with the mathematical nota-tion, definitions and theorems that are used in wavelet literature and in this book.Important mathematical concepts are briefly reviewed. In most cases no proof isgiven. For more detailed discussions or in depth studies, readers are referred to thecorresponding references [1-5].
Readers are suggested to skip this chapter in their first reading. They may thenreturn to the relevant sections of this chapter if unfamiliar mathematical conceptspresent themselves during the course of the book.
1.2.1 Functions and Integration
A function f{t) is called integrable if
f \f(t)\dt<+oo, (1.2.1)J-OO
and we say that / e Ll(R).Two functions f\ (t) and /2O) are equal in Ll (R) if
/•OO
/ 1/1(0-/2(01^ = 0.J—00
MATHEMATICAL PRELIMINARIES 3
This implies that f\ it) and f2it) niay differ only on a set of points of zero measure.The two functions f\ and f2 are almost everywhere (a.e.) equal.
Fatou Lemma. Let {fn}neN be a set of positive functions. If
Ih^ fnit) = fit)
almost everywhere, then
OO pOO
fiOdtSlm^l fnit)dt.
This lemma provides an inequality when taking a limit under the Lebesgue integralfor positive functions.
Lebesgue Dominated Convergence Theorem. Let fkit) e LiE) for k — 1, 2 , . . . ,and
lim fkit) = fit) a.e.k—>oo
If there exists an integrable function F(t) such that
\fkit)\<Fit) a.e., £ = 1 , 2 , . . . ,
then
lim f Mt)dt= [ fit)dt.
This theorem allows us to exchange the limit with integration.
Fubini Theorem. If
f If f(tut2)dti) dt2 <oo,
then
poo poc roo
/ /poo poc roo po
/ / f(h,t2)dtidt2= / dt2J — OQ J — OQ J — OO J — OO
pOG pOO
= dtx\ f(ti,tz)dt2.J— OO J—OO
This theorem provides a sufficient condition for commuting the order of the multipleintegration.
4 NOTATIONS AND MATHEMATICAL PRELIMINARIES
1.2.2 The Fourier Transform
The Fourier transform pair is defined as
f(w)= / f(t)e~ia"dt,J-OQ
fia))eiajt da).
Rigorously speaking, the Fourier transform of fit) exists if the Dirichlet conditionsare satisfied, that is,
(1) / - o .(2) fit) has a finite number of maxima and minima within any finite interval,
and any discontinuities of fit) are finite. There are only a finite number ofsuch discontinuities in any finite interval.
All functions satisfying (1.2.1) form a functional space L1. A weaker condition forthe existence of the Fourier transform of fit), in replace of (1.2.1), is given as
\f(t)\2dt<+oo. (1.2.2)-OO
All functions satisfying (1.2.2) form a functional space L2.When the Dirichlet conditions are satisfied, the inverse Fourier transform con-
verges to fit) if fit) is continuous at r, or to
if /(f)is discontinuous at t. When fit) has infinite energy, its Fourier transformmay be defined by incorporating generalized functions. The resultant is called thegeneralized Fourier transform of the original function.
1.2.3 Regularity
Lipschitz Regularity. If a function fit) has a singularity at / = u, this implies thatfit) is not differentiable at v. Lipschitz exponent at v characterizes the singularitybehavior.
The Taylor expansion relates the differentiability of a function to a local polyno-mial approximation. Suppose that f is m times differentiable in [i> — h, v + h]. Letpv be the Taylor polynomial in the neighborhood of v:
k=0
MATHEMATICAL PRELIMINARIES 5
Then the error
M O I < lt ~V}n sup \f(m)(u)\m- ue[v-h,v+h]
where
t G [v - A, v + hi ev(t) := f(t) - /?i;(O-
The Lipschitz regularity refines the upper bound on the error sv(t) with nonintegerexponents. Lipschitz exponents are also referred to as Holder exponents.
Definition 1 (Lipschitz). A function f{t) is pointwise Lipschitz a > 0 at t = v, ifthere exist M > 0 and a polynomial /?,,(0 of degree m = [a] such that
Vf G /?, | / ( 0 - p«(OI < Af |r - v\a. (1.2.3)
Definition 2. A function / ( 0 is uniformly Lipschitz a over [a,Z?] if it satisfies(1.2.3) for all v G [a, b] with a constant M independent of v.
Definition 3. The Lipschitz regularity of f(t) at v or over [a, &] is the sup of the asuch that f(t) is Lipschitz a.
Theorem 1. A function / ( r ) is bounded and uniform Lipschitz a over R if
|/(o>)|(l + \co\a)dco < +oo. (1.2.4)
If 0 < a < 1, then /?i;(0 = f(v) and the Lipschitz condition reduces to
WteR, \f{t)-f{v)\<M\t-v\a.
Here the function is bounded but discontinuous at i>, and we say that the function isLipschitz 0 at v.
Proof. When 0 < a < 1, it follows m := [_aj = 0, and pv(t) = / (u) .The uniform Lipschitz regularity implies that 3M > 0 such that
We need to have
Since
6 NOTATIONS AND MATHEMATICAL PRELIMINARIES
f(t) = — / f{co)ela)t dco,
1/(0-/0012JT
dco
(1) For|f -v\~l < \co\,
\t~v\a - \ t - v \ a ~ '
(2) For \t - v\~l > \a)\,
co2 o ( t - v)3
— v) (t — v) — i 1} 2 \ K } 3 !
On the right-hand side of the equation above, the imaginary part
[co(t - v)]3 [co(t - v)]5
I =co(t-v) + + • • • < co(t - v),
and the magnitude of the real part
p _ t[a>(t-v)]2 W(t-v)f I [<o(t-v)2! 4! 2!
Thus
and
\(t-v)(o\<\ and [(t - v)co]2 < \(t - v)co\
[co(t - v)]2
\ela)t -ei0JV\ < ico(t - v) + •2!
Hence
[co(t - v)]2 •
< \2co(t-v)\.
k"°'-Vn _ 2\a>\\t-v\
o4(t - v)4
\t~v\a - \t-v\a
MATHEMATICAL PRELIMINARIES 7
Combining (1) and (2) yields
\t — v\a ~~ lit y_oo
It can be verified that if
/-oo
then fit) is p times continuously differentiable. Therefore, if
J—oo
then f^m\t) is uniformly Lipschitz a — m, and hence f(t) is uniformly Lipschitz a, wherem = [a\. •
1.2.4 Linear Spaces
Linear Space. A linear space / / is a nonempty set. Let C be complex, /f is calleda complex linear space if
(1) x + y = y + x.
(3) There exists a unique element 0 e H such that for Vx £ //, x + 0 = 0 + *.(4) For VJC e H, there exists a unique — x such that x + (—JC) = 0.
In addition we define scalar multiplication V(a, x) e C x H such that
(1) a(^jc) = (a^)jc, Va, p eC,Vx e H.(2) 1* = *.
(3) (a + £)JC = ajc + £JC, Va, p eC,Vx e H.a(x + y) = ajc + a j , Va e C, VJC, y e H.
Norm of a Vector
Definition. Mapping of || JC ||: Rn -> R is called the norm of JC on Rn iff
(1) || JC || > O , V J C e Rn.
(2) || ajc | |= |a | || JC | | ,Va e R,x e Rn.
(4) || JC | |= 0 <=> x = 0.
Let JC = O i , A'2, . . •, Jc,2)r € Rn. The following are commonly used norms:
8 NOTATIONS AND MATHEMATICAL PRELIMINARIES
|| x || oo =max|jt/|, £°°norm,
£l norm,
I2 norm,
lp norm.
1.2.5 Functional Spaces
Metric, Banach, Hilbert, and Sobolev spaces are functional spaces. A functionalspace is a collection of functions that possess a certain mathematical structure pat-tern.
Metric Space. A metric space H is a nonempty set that defines the distance of areal-valued function p(x, y) that satisfies:
(1) p(x,y) >0 and p(x,y) = 0 iff JC = y.
(2) p(x,y) = p(y,x).
(3) p ( x , y) < p ( * , Z) + p ( z , y ) , V x , y , z e H .
Banach Space. Banach space is a vector space H that admits a norm, || • ||, thatsatisfies:
(1) V/ G //, || / ||> Oand || / ||= Oiff / = 0.
(2) Wa eC, \\ af ||= |CY| || / ||.
(3) Wf + g\\<\\f\\ + \\g\lVf,geH.
These properties of norms are similar to those of distance, except the homogeneityof (2) is not required in defining a distance. The convergence of {fn}neN to / e Himplies that lim^-^oo || fn — f ||= 0 and is denoted as lim^-xx) fn = / .
To guarantee that we remain in H when taking the limits, we define the Cauchysequences. A sequence [fn}neN is a Cauchy sequence if for Vs > 0, there exist nandm large enough such that || fm — fn \\< e. The space H is said to be complete ifevery Cauchy sequence in H converges to an element of H. A complete linear spaceequipped with norm is called the Banach space.
Example / Let S be a collection of sequences x = (x\, X2, . -., xn, . . . ) . We defineaddition and multiplication naturally as
MATHEMATICAL PRELIMINARIES 9
+ y = (xi + yi,x2 + y2,.'.,xn + yn,
ax = (axi,otX2, . . . ,otxn,...),
and define distance as
It can be verified that such a space S is not a Banach space, because p(x, y) does notsatisfy the homogeneous condition of the norm.
Example 2 For any integer p we define over discrete sequence /„ the norm
\\f\\p =
The space £P = [f : \\f\\p < 00} is a Banach space with norm
Example 3 The space LP(R) is composed of measurable functions f on R that
-If[J-
<oo.
The space LP(R) = {/ :|| / \\p< 00} is a Banach space.
Hilbert Space. A Hilbert space is an inner product space that is complete. Theinner product satisfies:
(1) (af + 0g,h)= a(f, g) + fi{g, h) for a, ^ G C and / , ^, A e H.
(2) (/,g>=¥T7>.(3) {/, /> > 0 and (/, / ) = 0 iff / = 0. One may verify that
is a norm.
(4) The Cauchy-Schwarz inequality states that
where the equality is held iff / and g are linearly dependent.
In a Banach space the norm is defined, which allows us to discuss the convergence.However, the angles and orthogonality are lacking. A Hilbert space is a Banach spaceequipped with an inner product.
10 NOTATIONS AND MATHEMATICAL PRELIMINARIES
1.2.6 Sobolev Spaces
The Sobolev space is a functional space, and it could have been listed in the previ-ous subsection. However, we have placed it in a separate subsection because of itscontents and role in the text.
On many occasions involving differential operators, it is convenient to incorpo-rate the Lp norms of the derivative of a function into a Banach norm. Consider thefunctions in the class C°°(Q). For any number p > 1 and number s > 0, let us takethe closure of C°°(Q) with respect to the norm
\a\Ss
The resulting Banach space is called the Sobolev space WS'P(Q). For p = 2 wedenote Ws (Q) = Ws'2(£2), which is a Hilbert space with respect to the inner product
= J2 fSometimes WS(R) is also denoted as HS(R). Note that the differentiation in (1.2.5)can be of a noninteger.
Recall that the Fourier transform of the derivative f (t) is / (of{a)). The Plancherel-Parseval formula proves that f\t) e L2(R) if
/»OO 1 pOO
/ \f'(t)\2dt = — H2|/(a))|2 <+00.J—oo £K J—oo
This expression can be generalized for any s > 0,
roo\co\2s\f(co)\2dco< +oo
-oo
if / G L2(R) is s times differentiate.Considering the summation nature of (1.2.5), we can write the more precise ex-
pression of Sobolev space in the Fourier domain as
FJ - a
(l+co2Y\f(co)\2da)<+oo.
For s > n + \, f is n times continuously differentiable. The Sobolev space Ha,a e R consists of functions f(t) e S' such that
oof(oi>)(l + a)2)01 dco < oo.
—oo
Fora = 0, the Ha reduces to L2(R). Fora = 1, 2, ..., Ha is composed of ordinaryL2(R) functions that are (a — 1) times differentiable and whose ath derivative are
MATHEMATICAL PRELIMINARIES 11
in L2(R). For a = - 1 , —2, . . . , Ha contains the — ath derivatives of L2(R) and alldistributions with point support of order < a.
It can be seen Ha D J7^ when a > p. The inner product of / , g e Ha is
(/, 8)* = ̂ J f(co)g(a))a + co2)a day
and is complete with respect to this inner product. Therefore it is a Hilbert space.
1.2.7 Bases in Hilbert Space H
Orthonormal Basis. A sequence [fn}neN in a Hilbert space H is orthonormal if
(/»!) Jn/== °m,n-
If for / G H there exist <xn such that
N
lim \\f-Tanfn\\=0,
then {/„ }flG/v is called an orthogonal basis of H.For an orthonormal basis we require ||/n | | = L A Hilbert space that admits an
orthogonal basis is said to be separable. The norm of / e H is
n=0
fl/esz Basis. Let {/„} be linear independent and complete in I?(a, b), meaningthat the closed linear span of [fn] is L2(a, b). The set is called a Riesz basis if thereexist A > 0 and B > 0 such that
for each sequence {Q} of complex numbers. The Riesz representation theorem guar-antees the existence of the dual {fn} in L2(a, fr) such that:
(1) {fn} is the unique biorthogonal sequence to {/n}; namely (/m, /rt) = 5m,rt.(2) If {cn} G £2, then £ w c n / n converges in L2(a, b).
(3) For each / e L2(a, b), {{/, fn)} e I2.(4) For each/ e L2(a,b),
i=0 i=0
12 NOTATIONS AND MATHEMATICAL PRELIMINARIES
A Riesz basis of a separable Hilbert space H is a basis that is close to being or-thogonal. The right inequality in (1.2.6) is essential. It prevents the expansion fromblowing up. The left inequality in (1.2.6) is important too, since it ensures the exis-tence of the inverse.
1.2.8 Linear Operators
In computational electromagnetics, the method of moments and finite elementmethod are based on linear operations. An operator T from a Hilbert space H\to another Hilbert space H2 is linear if
Vai, a2 e C, V/i, h e HU r(«i/i + <*ih) = <*\T(fi) + a2T(f2).
Sup Norm. The sup operator norm of T is defined as
||r||5= sup M . (1.2.7)1111
If this norm is finite, then T is continuous; namely \\Tf\ — Tf2\\ becomes arbitrarilysmall if ||/1 — /2II is sufficiently small.
Adjoint The adjoint of T is the operator Ta from H2 to H\ such that for any/1 e Hi and f2 e H2
When T is defined from H into itself, it is self-adjoint if T = Ta. A nonzero vector/ € H is a called an eigenvector if there exists an eigenvalue X e C such that
Tf = A/.
In a finite-dimensional Hilbert space, meaning that Euclidean space, a self-adjointoperator is always diagonalized by an orthogonal basis {en}o<n<M of eigenvectors
/ en = A/2 ^ w •
For a self-adjoint operator T, the eigenvalues A.w are real, and for any / e H
N-\ N-\
n=0 n=0
In an infinite-dimensional Hilbert space, the previous result can be generalized interms of the spectrum of the operator, which must be manipulated with caution.
Orthogonal Projector. Let V be a subspace of H. A projector Py on V is a linearoperator that satisfies V/ e H, PyfeV and V/ e V, Pvu = / .
MATHEMATICAL PRELIMINARIES 13
The projector Py is orthogonal if
VfeH,VgeV, (f-Pvf,g)=0.
The following properties are often used in the text:
Property 1. If Py is a projector on V\ then the following statements are equivalent:
If {̂ nlwew is a Riesz basis of V and { ^ J ^ N is the biorthogonal basis, then
+00 +OO
n=0 n=0
Density and Limit A space V is dense in H if for any / e / / there exist {/m }m€yvwith /m G V such that
lim | | / - / O T | | = 0 .m—>+oo
Let {r n} n e^ be a sequence of linear operators from H to // . Such a sequence ccw-verges weakly to a linear operator Too if
V / e t f , ^ U r n ^ l l ^ / - T o o / 1 | = 0 .
To find the limit of operators it is preferable to work in a well chosen subspaceV C H which is dense. The density and limit are justified by the property below.
Property 2 (Density). Let V be a dense subspace of H. Suppose that there exists Csuch that \\Tn\\s < C for all n e N. If
V/ € V, ^irn^ | | r n / - 7oo/|| = 0,
then
VfeH, wJimo o l | rn / -7 'oo/ | | = 0 .
For numerical computations, an operator is often discretized into a matrix. Only thendigital computers can be utilized.
14 NOTATIONS AND MATHEMATICAL PRELIMINARIES
Norm of a Matrix. For a matrix A e Rn xn, the norm of A is defined, similarly to(1.2.7), as
A Y
A \\= max •x\\
In particular, the commonly used norms are as follows:
(1) The column norm (l{ norm)
II A ||i =
(2) The row norm (l°° norm)
|| A ||oo=max{||fll-.||i} =
(3) The spectral norm (I2 norm)
where ^^TA is the maximum eigenvalue of AT A.
(4) The Frobenius norm
V=i /=i
BIBLIOGRAPHY
[1] R. L. Wheedan and A. Zygmund, Measure and Integration, Marcel Dekker, New York,1977.
[2] A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, Dover, New York, 1970.
[3] F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publication, Inc. New York, 1990.
[4] M. S. Berger, Nonlinear and Functional Analysis, Academic Press, New York, 1977.
[5] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, New York, 1998.
