Acoustic and electromagnetic scattering analysis using discrete sources i elements of functional analysis

Acoustic and electromagnetic scattering analysis using discrete sources i elements of functional analysis Acoustic and electromagnetic scattering analysis using discrete sources i elements of functional analysis Acoustic and electromagnetic scattering analysis using discrete sources i elements of functional analysis Acoustic and electromagnetic scattering analysis using discrete sources i elements of functional analysis Acoustic and electromagnetic scattering analysis using discrete sources i elements of functional analysis Acoustic and electromagnetic scattering analysis using discrete sources i elements of functional analysis

I

ELEMENTS OF FUNCTIONAL

ANALYSIS

In this chapter we will recall some fundamental results of functional anal-ysis We firstly present the notion of a Hilbert space and discuss some basic properties of the orthogonal projection operator We then introduce the concepts of closeness and completeness of a system of elements which belong to a Hilbert space The completeness of the system of elementary sources is a necessary condition for the solution of scattering problems in the framework of the discrete sources method After this discussion, we will briefly present the notions of Schauder and Riesz bases We will use these concepts when we will analyze the convergence of the null-field method

We then consider projection methods for the operator equation

Au = / ,

where A is a linear bounded and bounded invertible operator from a Hilbert

space H onto itself We will consider the equivalent variational problem

B{u, x) = J^*{x) for all x e H,

where B is a bounded and strictly coercive sesquilinear form and J" is a linear and continuous functional Convergent projection schemes will be

constructed by appealing on the fundamental theorem of discrete

approx-imation Later, we will particularize these results for the space of square

integrable tangential vector functions We conclude this chapter by

analyz-ing projection methods for a linear operator A actanalyz-ing from a Hilbert space

H onto a Hilbert space G, and for the operator equation

where JB is a compact operator

1 HILBERT SPACES ORTHOGONAL PROJECTION OPERATOR

Let if be a complex vector space (linear space) The function.(, )j^ :

H X H -^ C is called a Hermitian form if

(a) {au 4- /3v, w)j^ = a {u,w) fj -\- f3 {v, w)jj , (linearity)

(b) (w, v)fj = {v, u)]^ , (symmetry)

for all u,v,w e H and all a,/3 E C Here, a* denotes the complex

conju-gate of a A Hermitian form with the properties

(a) ( w , i x ) ^ > 0 , (positivity)

(b) (u, u)ff=Oii and only ii u = 9HI (definiteness)

where ^H stands for the zero element of H, is called a scalar product or an

inner product The vector space with a scalar product specified is called a

inner product space or a pre-Hilbert space

In terms of the scalar product in /if, a norm

can be introduced, after which H becomes a normed space The

follow-ing important inequality is the basis for the statement that inner product

spaces contain all the elements of Euclidean geometry, while normed spaces

have length, but nothing corresponding to angle It is the Cauchy-Schwarz

inequality and is given by

|(u,i;)^| < \\u\\tj \\V\\H for all u.veH (L2)

The Cauchy-Schwarz inequality and the definition of scalar product imply

that the norm properties:

1 HILBERT SPACES ORTHOGONAL PROJECTION OPERATOR 3

(a) \\u\\fj > 0, (positivity)

(t>) ll^ll// = 0 if and only \i u = 0//, (definiteness)

(c) ||aix||^ = \a\ \\u\\jj , (homogeneity)

(d) ||u + v\\f^ < \\u\\ff -f ||i;||^ , (triangle inequality)

for all u,v e H and all a € C are satisfied Therefore any scalar

prod-uct induces a norm, but in general, a norm || ||^ is generated by a scalar

product if and only if the parallelogram identity

ll« + v\\l + \\u - v\f„ = 2 {\\u\\l + Ml) (1.3)

holds

Given a sequence {un) of elements of a normed space X, we say that

Un converges to an element u of H if \\un — u\\^ -+ 0 as n —• oo A

se-quence (un) of elements in a normed space X is called a Cauchy sese-quence

if \\un — UmWx —* 0 as n , m ^ oc

A subset M of a normed space X is called complete if every Cauchy

sequence of elements in M converges to an element in M A normed space

is called a Banach space if it is complete An inner product space is called

a Hilbert space if it is complete

A sequence (un) in a Hilbert space H converges weakly to u £ H if for

any v E H, {un,v)fj —> (u,i;)^ as n —>> oo Ordinary (norm) convergence is

often called strong convergence, to distinguish it from weak convergence

The terms 'strong' and 'weak' convergence are justified by the fact that

strong convergence implies weak convergence, and, in general, the converse

implication does not hold If a sequence is contained in a compact set,

then weak convergence implies strong convergence Note that every weakly

convergent sequence in a Hilbert space is bounded and every bounded

sequence in a Hilbert space has a weakly convergent subsequence

Two elements u and v of an inner product space H are called orthogonal

if {u,v)fj = 0; we then write u±v If an element u is orthogonal to each

element of a set M, we call it orthogonal to the set M and write u±M

Similarly, if each element of a set M is orthogonal to each element of the set ,

K, we call these sets orthogonal, and write M±K The Pytagora theorem

states that

\\u±v\\l^\\u\\l + \\v\\l (1.4) for any orthogonal elements u and v

A set in a Hilbert space is called orthogonal if any two elements of the

set are orthogonal If, moreover, the norm of any element is one, the set is

called orthonormal

A subset M of a normed space is said to be closed if it contains all its limit points For any set M in a normed space, the closure of M is the union of M with the set of all limit points of M, The closure of M is written M Obviously, M is contained in M, and M = M if M is closed

Note the following properties of the closure:

(a) For any set M, M is closed

(b) If_A/ C K, then M C F

(c) M is the smallest closed set containing A/; that is, '\{ M <Z K and

K is closed, then JI C K

Complete sets are closed and each closed subset of a complete set is complete

Next, we define the orthogonal projection operator Let J/ be a Hilbert

space and M a subspace of H (i.e a complete vector subspace of H) Let u E H Since for any v € M we have ||^/ — ^||// > 0, we see that the set {\\u — vW^^ / t ' G A/} posses an infimum Let d = mi^^^M ||^ ~" ^IIH and let {vn) be a minimizing sequence, i.e (f^,) C M and \\u - VUWH ~^

dasn —V oo Since M is a vector subspace, ^{vn + v^n) G Af, whence

11 !L _!Ii|| > d^ Using this and the parallelogram identity

H

,,||5, = 2(||7i-i;,||^, + | | u - i ; , n | | ^ ) - 4 Vn 4-1;,,

H

(1-5) gives

\\V„ - Vr,,\\l < 2 (||U - VnWl + II" " ^mll«) ' ^d^\ (1-6) whence, by letting n, m —• oo, ||i;„ — I'mll// —* 0 follows Thus, {vn) is

a Cauchy sequence and since M is complete, there exists w £ M such that \\vn — 'w\\f^ —> 0 as n —> oo; moreover \\u — i^n||// —^ ||^ - ?^||// = rfas

n —• 00 Suppose now that there exists another element w' for which the function \\u — u||^ attains its minimum; then d = \\u - if ||^ = ||w — vj'^n • Clearly, \(w •\- w') E M and we have

d — inf \U - l^llrr < w + w

H

(1.7)

Thus, u - w -\- w' — d, and by the parallelogram identity

H

\\W - w'W^j = 2 (||n - xot„ + \\u - w't^^) - 4 I L w ^-w' = 0, (1.8)

1 HILBERT SPACES ORTHOGONAL PROJECTION OPERATOR 5

we find w = w'

The vector w gives the best approximation of u among all the vectors

of M Note that d is called the distance from u to M and is also noted by

p(u, M ) The operator P : H —^ M mapping u onto its best approximation,

i.e

P u = : u s (1.9)

where ||t/ — t/;||j^ = d — miy^M 11^ "" ^11// ' ^^ ^ bounded Hnear operator

with the properties: P^ = P and {Pu.v)jj — {u,Pv)ff for any u,v € H

It is called the orthogonal projection operator from H onto M, and w is

called the projection of u onto M

The following statements characterizing the projection are equivalent:

(a) \\U-W\\H < \\u-v\\ff,

(b) Re{u- w,v - w)fj < 0,

(c) Re{u-v,w- v)fj > 0,

toT ue H, w ^ Pu e M and any v € M,

Let M be a subset of a Hilbert space // The set of all elements

or-thogonal to M is called the oror-thogonal complement of M,

M^ = {ueH/u±M}

Clearly, M-^ is a subspace of H, To show this we firstly observe that A/-^

is a vector subspace, since for any scalars a and /? and any u.v £ A/-^,

{au-\- (3v,(fi)^ = 0 for all (^ G A/; whence au -{- f3v e A/-^ follows To

prove that Af-*- is complete, let us choose a Cauchy sequence {n„) C A/-^;

it converges to some u £ H because H is complete We must show that

u E M^ Since for any v e H, and in particular for any v e A/, we have

{uny v)fj —> (w, v)fj as n —> oc and (un? ^')H = 0, n = 1,2, , it follows that

(w, i^)// = 0 for any v G A/ Hence, tz G A/-'- and so Af-^ is complete

Now, let if be a Hilbert space, M a subspace of if, and P the

or-thogonal projection operator of H onto M Let u e H From the

prop-erties of the projection we see that Re{u - Pu.v — Pu)fj < 0 for any

t; G Af Choose v = Pu ± (p with (p being an arbitrary element of A/ Then

Re {u — Pu, i ^ ) / f < 0, whence Re {u — Pu, ^)fj = 0 Replacing in the last

relation (f by J V ( J ^ = 1) we get

Re(w ~ Pu,j(f)ff ~ R e [ - j ( w - Pu,(f)fj] = Im(w - Pu,ip)ff = 0 (LIO)

Thus, for a given u e H the projection u; = Pu satisfies u — w 1 M

Therefore, any element u £ H can be uniquely decomposed as

u = w-^w^, ( L l l )

where w e M and w^ G M-^ This result is known as the theorem of

orthogonal projection

The operator Q : H -^ M-^ given by

Qu = u-Pu (1.12)

is the orthogonal projection operator from H onto M^

2 CLOSED AND COMPLETE SYSTEMS IN HILBERT SPACES

BASES

Let X be a normed space and M a subset of X M is dense in X if for

any u e X and any £ > 0 there exist u^ E M such that \\u — u^W^ < e

Equivalently, M is.dense in X if and only if for any u e X there exists a

sequence {un) C M such that \\un — w||x —• 0 as n —> oo

Every set is dense in its closure, i.e M is dense in M M is the largest

set in which M is dense; that is, if M is dense in K, then K C M.li M is

dense in a Hilbert space if, then M — H Conversely, if M = i / , then M

is dense in H

Let H he Si Hilbert space If M is dense in H and u is orthogonal to

M, then u = 6H- Indeed, let uJLM and choose an arbitrary v E H Since

M = H there exists a sequence (vn) C M such that \\vn — v||^ —> 0 as

n —> GO Consequently, {u^Vn)^ —• (^?^)// as n —> oo From {u,Vn)ff =

0, n = 1,2, , it follows that (w, v)^ = 0 for any v e H Thus, u ± H The

element w is orthogonal to any element of H and in particular is orthogonal

to itself, i.e {u,u)ff = ||u||^ = 0 Hence, u =

OH-Elements V^i,^2' "">'^N ^f ^ vector space X are called linearly

depen-dent if there exists a linear combination Yli^i ^i'^i = 0 in which the

co-efficients do not vanish, i.e Yli=i l^^l > ^- ^ ^ ^ vectors are called linearly

independent if they are not linearly dependent, or equivalently, if there

exists no non-trivial vanishing linear combination If any finite number of

elements of an infinite set {t/^J^i is linearly independent, the set {V'Ji^i

is called linearly independent

A system of elements {V^jj^i is called closed in H if there are no

elements in H orthogonal to any element of the set except the zero element,

that means

(w,V^,)^ = 0 , z = 1,2, , implies IX =

0H-A system of elements {ipi}^i is called complete in H if the linear span

CX)

1 = 1

of {ipi}^i or the set of all finite linear combinations of {ipi}^

S p { ^ i , ^ 2 •••} = lu = J ^ a ^ ^ Q i G C,7V = 1,2, i

2 CLOSED AND COMPLETE SYSTEMS IN HILBERT SPACES BASES 7

is dense in H, i.e Bp {-^j, ip2^ •••} = H Equivalently, if {t/^J^i is complete

in H then for any u e H and any e > 0 there exist an integer N = N{e) and a set {ttr}^_i such that \\u — X)i=i ^f^^i "^ ^•

Let us observe that the closure of the linear span of any set {T/^^}^! is

a subspace of H It is a vector subspace by its very definition and it is also

complete as a closed subset of a complete set

Obviously, if the system {V^J^i is complete in H, then the only ele-ment orthogonal to {ipi}^i is the zero eleele-ment of H] thus the set {0i}i^i

is closed in H The converse result is also true To show this let { ^ J ^ i be

a closed system in H Let us denote by W the linear span of {ipi}^i - Then any element u£ H can be uniquely represented as w = Pu 4- Qu, where P

is the orthogonal projection operator from H onto W^ and Q is the orthog-onal projection operator from H onto W Since Qu G W and \l)i € W,

2 = 1,2, , we get {Qu, il^^)u = 0, i = 1,2, The closeness of {V^J^i in H implies Qu = 6H-> and therefore for any element u € H we have u = Pu €

W Thus, H C W, and since W C H we get W = H; therefore W is dense

in H We summarize this result in the following theorem

T H E O R E M 2.1: Let H be a Hilbert space A systetn of elements {'4^i}^i

is complete in H if and only if it is closed in H

A set {ipi }?=i is called a finite basis for the vector space X if it is linearly independent and it spans X A vector space is said to be n-dimensional if

it has a finite basis consisting of n elements A vector space with no finite

basis is said to be infinite-dimensional

Let HN be a finite-dimensional vector subspace of a Hilbert space H with orthogonal basis {<f>i}^-i Then the orthogonal projection operator from H onto HN is given by

N

PNU = Y2 (^' ^t)H ^i^ ueH

t = l

For the time being we note a simple but important result characterizing

the convergence of the projections Let {ipi}^i be a complete and linear independent system in a Hilbert space /f, let H^ stand for the linear span

of {ipi}i^i, and let us denote by PN the orthogonal projection operator from H onto HN We have

\\u - P N + I ^ I I H = inf \\u - t;||^

Vfc/lN + l

< inf ||u - v\\fj = ||u - PNU\\H

v€riN

(1.13)

for any u e H; thus the sequence ||n - PN'^WH ^^ convergent Since { ^ J ^ i

is complete in H we find a subsequence (uiv„) C HN^ such that ||u - UNn \\H

—> 0 as n —* oo Then, from 0 < \\u — PN^'^^'WH ^ N — ^iVnll// we get

||u — PNa'^Wn —^ 0 as n —>• oc; thus the convergent sequence \\u — PNU\\H

possesses a subsequence which converge to zero Therefore, for any xi £ H

we have

\\u - PNU\\U -^Oas N -^OO. (1.14)

A map i4 of a vector space X into a vector space Y is called linear if A

transforms linear combinations of elements into the same linear

combina-tions of their images, i.e if ^ ( a i ^ i -]-a2U2 + ) = aiA{ui)-ha2A{u2)-{- "

Linear maps are also called linear operators In the linear algebra one

usu-ally writes arguments without brackets, A{u) = Au Linearity of a map, is

for normed spaces, a very strong condition which is shown by the following

equivalent statements:

(a) A transforms sequences converging to zero into bounded sequences,

(b) A is continuous at one point (for instance at tx = 0),

(c) A satisfies the Lipschitz condition ||i4u||y < c||u||;^ for all u e X

and c independent on tx,

(d) A is continuous at every point

Each number c for which the inequality (c) holds is called a bound for

the operator A

Let C{X, Y) be the linear space of all linear continuous maps of a

normed space X into a normed space Y The norm of an operator

uex,uy^0x \m\x l|u|lx=i

satisfies all the axioms of the norm in a normed space, whence the linear

space £(X, Y) is a normed space Note that the number \\A\\ is the smallest

bound for yl It is not difficult to prove that the space C{X, Y) is complete

if the space Y is such

A map of a vector space into the space C of scalars is called a functional

The above statements are valid for linear functional The space £{X^ C)

is called the conjugate space of X and is denoted by X* It is always a

Banach space

A system {xpj}^^ is called minimal if no elements of this system belongs

to the closure of the linear span of the remaining elements In order that

the system {V^l^i be minimal in a Banach space X, it is necessary and

sufficient that a system of linear and continuous functional defined on X

exist forming with the given system a biorthogonal system; that is, a system

of Hnear and continuous functionals { ^ j j ^ j such that ^j ( ^ J = 6ij^ where

6ij is the Kronecker symbol If the system {V^^ j ^ j is complete and minimal,

then the system of functionals {^j}Jli is defined in a unique manner In

2 CLOSED AND COMPLETE SYSTEMS LN HILBERT SPACES BASES 9

a Hilbert space H, by Riesz theorem (see section 1.3), there exists ipj such

that J'j (u) = (^,^j) for any u € / / ; therefore (tj^^j)^

= 6ij In this case the system ^^^^^ _ is called biorthogonal to the system {t''j}^_|

A system { ^ J ^ j is called a Schauder basis of a Banach space X if

any element u e X can be uniquel}^ represented as u = X]^^l ^?^^n where

the convergence of the series is in the norm of X Every basis is a complete

minimal system However, a complete minimal system may not be a basis in

the spacẹ For example, the trigonometric system I/JQ (t) = 1/2, 02n-i(^) =

sin(n;^), ^2M (^) — cos(?if),n = 1,2, ,is a complete minimal system in

the space C([—TT, TT]) but it does not form a basis in it In an arbitrarily

separable Hilbert space if, every complete orthogonal systems of elements

forms a basis Thus, the trigonometric system of functions forms a basis in

L2([-;r,7r])

The system { 0 j ^ j is called an unconditional basis in the Banach

space Á if it remains a basis for an arbitrary rearrangement of its elements

Let T : X -^ X hea bounded linear operator with a bounded inversẹ If the

system {ipi}^i is a basis, then the system { T ^ j j ^ j is a basis If {u%}^^

is an unconditional basis, then {Túi}^i is an unconditional basis In a

Hilbert space, every orthogonal basis is unconditional It can be shown that

an arbitrary unconditional basis in a Hilbert space is representable in the

form { T 0 ^ } ^ j , where {0^}J^i is an orthonormal basis oi H Such bases are

called Riesz bases If { ' 0 j ^ i is a Riesz basis then the biorthogonal system

\pi > is also a Riesz basis A complete system {i^i}^i forms a Riesz

basis of H if the Gramm matrix G = [Gij], Gjj = {^i^'^'j)ff > generates

an isomorphism on /^ The system {t'^jj^i forms a Riesz basis of H if the

inequalities

N

ci^\ôif <

i=l

l]«^i^^

N

< C 2 ^ K | ' (1.15)

hold for any constants QJ and for any iV, where the positive constants cj

and C2 should not depend on  and ậ Equivalently, { ^ J ^ i forms a Riesz

basis of H if there exist the positive constants ci and C2 such that

oo oo

\'<\\u\\l<C2'£\{u,i>,)i/ (1.16)

? = 1 1 = 1

for arbitrary u E H Note that if {t\}^i is a Riesz basis, then sup^ 11^/11// <

C2, infj ll^^llj:^ > ci and similar inequalities hold for the biorthogonal system

{*'}:,•

3 PROJECTION METHODS

One of the most important result of the theory of Hilbert spaces is the Riesz

theorem This theorem states that if J^ is a hnear and continuos functional

on a Hilbert space i/, then there exists an unique element u/ e H such

that

J^{v) = {v,Uf)fj for all v e H, (1.17)

and IIJ^II^

— 11^/11// • Conversely, any element u e H define a linear and

continuous functional on H by the relation

J^u{v) = {v,u)fj for all v e H, (1.18)

and we have ||^W|IH* ~

\W\\H-Let ^ be a linear and continuous functional defined on a Hilbert space

H and {un) a weak convergent sequence with limit u e H Then by the

Riesz theorem, J-^{un) —• ^{u) as n —• oo

Let H he a, Hilbert space The function B : H x H -^ C is called a

sesquilinear form on H if it is linear in the first argument and antilinear in

the second one, i.e

B{ax-\-py,z) = aB(x,^) +/?B(y,^),

(1.19)

B{x,ay-\-f3z) = a'B{x,y)-h0'B(x,z), for all x^y^z G H and all a,/? G C A sesquilinear form B is bounded, if

there exists a constant M > 0 such that

\B{x,y)\ <M\\x\\ff \\y\\f, for all x,y e H, (1.20)

and strictly coercive, if there exists a constant c > 0 such that

ReB{x,x) > c||x||^ for all x e H (1.21) Let H he a Hilbert space and B a bounded sesquilinear form on H With

X being an element of H we define the functional T: i / —• C by J^{y) =

B*{x^ y) Then f E H^ According to Riesz theorem there exists an unique

element x/ G i / , such that J-^{y) = {yi^f)f{ for all y £ H We define the

operator A : H -^ H hy Ax — x/ Then B*{x^y) = {y,Ax)ff , and further

B{x^y) = {Ax^y)ff for all x^y e H Let us now prove that A G C{H,H)

The linearity of -<4 is a consequence of the linearity in the first argument of

B The boundedness of A follows from the boundedness of B, i.e from

\\Ax\\l = {Ax, Ax)f, = B{x, Ax) < M ||a:||^ \\Ax\\^ (1.22)

we obtain ma;||;^ < M||a:||^ for all x e H The operator A is uniquely

determined To show this we suppose that there exist two operators Ai and