�9 88 �9 J6dar, L. etal~ Best Approximation Matrix Problem

ON THE BEST APPROXIMATION MATRIX

PROBLEM AND MATRIX FOURIER SERIES

L. Jbdar, E. Navarro and E. Defez

( Pol ytechnical U~ffversit y o f Valencia, Spain)

Received Apr. 5, 1996 Revised July 10, 1996

Abstract

In this paper the concept o f positive definite bdiwear matrix moniem functional, acting on tile space o f all

the matrix valued continuous functions defined on a bounded interval [a,b], is introduced. The best approM-

mation matrix problem with respect to such a futwtioTw2 is solved in terms o f matrix Fourier series. Basic prop-

o-ties o f matrix Fourier seri~ such as the Rienwaln--Lebesgue matri~r property and the bessel--~rseual omtri.z

inequality are proved. The concept o f total set with respect to a posith,e definite t~mtria" functimml is intro-

duced, and the totalhty o f an orthonormal sequence o f matrix polynomuds with respect to the functional is es-

tablished.

1 Introduction

General orthogonal polynomials play on important role in numerical analysis, approxima-

tion theory and in the analytic solutions of partial differential problems, see for example [S ] ,

[33,[10], [153.

Orthogonal matrix polynomials appear in connection with representation theory, matrix Cz].-..[z~] expansion problems, prediction theory ,ana m the reconstruction of matrix functions !ZT~hose

orthogonal on the unit circle have been studied in [-263, and orthogonal matrix polynomials on

the real line have been treated in [ 2 4 ] , [ 1 9 3 , [ 2 1 ] , [ 2 2 ] , [ 7 3 , [ 8 3 , [ 9 3 . In [17 ] , [18] and

[203, extension to the matrix framework of the classical families of I.aguerre. Gegenbaner and

Hermite polynomials have been introduced.

In a recent paper Cu3 it has been shown that a positive definite matrix moment functional

defined on the set P [ x ] of all matrix polynomials P ( x ) = A , x " + A , _ ~ x " - l + . . . 4-Ao, where A.

is a matrix in C; *" for O ~ i ~ n , and x is a real variable, induces a matrix inner product. The

aim of this paper is to consider matrix functionals acting on the mt of all s *" valued continuous

functions defined on a bounded interval [a ,b] . The best approximation matrix problem with

respect to a positive definite matrix functional is solved, and its connection with matrix Fourier

series is showed. Some interesting analogies of the Riemann-Lebesgue property, the Bessel-

Parseval inequality and the concept of total set, are extended to the matrix framework, r e h

Approx. Thecny & its Apl~l. , 13,4. Dec. 1997 �9 89 �9

tivized to the matrix functional.

The organization of the paper is as follows. In section 2, the concept of positive definite

matrix functional, acting on the set C([a ,b ] .CyX '~ , of all C "~'" valued continuous functions de-

fined on a bounded interval Ea,b'], is introduced and a matrix best approximation problem is

solved. Section 3 deals with the statement of some important facts about matrix Fourier series,

such as the Riemann-Lebesgue matrix property and the Bessel-Parseval matrix inequality.

Finally, in section 4, the concept of total set with respect to a positive definite matrix function-

al is introduced, and it is proved that an orthonormal sequence of matrix polynomials is total

with respect to the functional.

Throughout this paper, a matrix polynomials of degree n in C "• is an expression P ( z ) =

A,x" +,4 ,_ ~x "-t + "" + Ao with coefficients A, E C "x' , 0 ~ j ~ n. The set of all polynomials of

degree n will be denoted by P.[':r-].

For a matrix C in C '~ ' , we denote by [I C I[ : its 2 - n o r m defined by

H C.r I1, Ii C II, = s u p ~ ,

" . * 0 [ I z l l z

where for a vector y in C' , ii y II 2 denotes the usual euclidean norm. The set of all the eigen-

values of C is denoted by a(C) . If C is an hermitian matrix, then a(C) is a subset of the real

line and we denote by , lu (C) and , ~ ( C ) respectively, the minimum and the maximum of a

(C). We say that C is positive semi-definite if C is hermitian and ( C x , x ) ~ 0 for any vector

x in C' , where ( , ) denotes the euclidean inner product in C" • If C is hermitian and (Cx,

x ) > 0 for any nonzero vector x in C' . then C is said to be positive definite. If B and C are her-

mitian matrices in C/• we say that B ~ C when B - C is positive semi-def ini te , and B > C

when B - - C is positive definite. If C = (co)x,g..j~,, by E14,p. 57-] one gets

max I%I <~ ! l C l l z ~ < r max [c,il. (1)

If we take into account the scalar factorial function denoted by (z). and defined by ( z ) . = z ( z

+ 1 ) . " ( z + n - 1 ) , n ~ l , and (z0) = 1, then by application of the matrix functional calculus to

this function, for any matrix C in C ~" one gets

( C ) . = C ( C + 1 ) . . . ( C + (n -- 1) I ) , (C)0 = I , n >/1 .

If x and y are complex numbers with positive real part, R e x > 0 , R e y > 0 , then we recall that

the Euler Beta function B ( z , y ) is defined by

f'0u~-'(1 B ( x , y ) = - u ) ' - ~ d u ,

see [28,p. 416-]. To conclude this introduction we recall that by [13, p. 801 - 802], if C is an

hermitian positive definite matrix, then C admits an unique positive definite square root denot- 1

ed by C'/.

�9 90 �9 Jbdar, L. etal, Best Approximation Matrix Problem

2 Matrix moment functionals and the best approximation matrix problem

Throughout this paper we denote by E the Banach space of all the continuous functions f :

[a ,bl--*C "• endowed with the norm

II f 1[ .0 = max{ II f(x) II ,~ a ~< z ~< b}, (Z)

where I'a, b']is a bounded interval of the real line.

Def'mition 2. 1 An hermitian bilinear matrix functional -~' on E is a function -r X E-~

C "• such that

(i) ~ ' ( A f , g ) = A ~ ' ( f , g ) ,

(ii) .~'(f ,Ag) =~'( f ,g) A u ,

(iii) ~ ' ( f + g , h ) = ~ ( f ,h)+~' (g ,h) ,

(iv) .~g(f ,gq-h)=.~. ( f , g ) + . ~ ( f ,h),

(v) .fg, ( f ,g ) =.~ (g , f )u , for A E C "• and f , g , h in E.

The functional ~ ' is said to be positive definite if ~'( f , f )>tO, for every f E E , and .Qa

( f , f ) = O only when f = 0 .

The functional ~ is bounded on E if there exists a positive constant K such that

II -f~'(f,g) II, <~ K II f II - I1 g II * * , f , g E E. (3)

In the following a conjugate bilinear matrix functional will be abbreviated saying only matrix

functional.

Example 2. 1 Let W. [a,b]~C "• such that W(x) is hermitian positive definite for all x

E [a,b],W~x) and W(x)r are integrable in [a,bl, and let ~ be defined by

~':E X E--" C "X',

( f ,g ) -*. ff f (x)W(x)gU(x)dx.

It is clear that ~ is bounded, because

11,>~ ( ~ II w(x) I I zdx)I1 f [I- II g 11 ~.. II ( f ,g)

In order to prove that -~' is positive definite, note that, for any f E E ,

.~ ' ( f , f ) ~ f ( x ) W ( x ) f H ( x ) d x = * ~ ~ ,, = I f ( x )W(x ) W(x) f (x)dx

= ~[f(x)W(x)~z][f(x)W(x)�89

(4)

(5)

Since V(x)~[f(x)W(x)~*][f(x)W(x)~]U>~O, from (5) , for any vector y in C , it follows

that

Yu'~e(f , f ) y = yU( flV (x)dx} y = ; J ' V (x)ydx >~ O, (6)

Approx. Theory & its Appl. . 13, 4, Dec. 1997 �9 91 �9

because yUV (x )y~O.

Consequently,

Furthermore, if f E E satisfies s ( f , f ) = 0 , from (6) one gets

Sly, I 0 = ' [ f ( x ) W ( x ) T ] [ f ( x ) W ( x ) ~ ] " y d x

b 1 I

= f[W(.r)rf(.r)Uy']H[WCr)~f(x)Uy]dx,

0 = II W ( x ) � 8 9 II :dx, y E C'. a

(7)

t

W ( x ) T f ( x ) " y ~- O, Y 6 C', a ~ x ~ b. 1 1

Hence W ( z ) - f f ( z ) n= 0 for a ~ z ~ b , and by the invertibility of W (x)i" one concludes that f

= 0. Thus M is positive definite.

Definition 2.2. Let {P.(x) 1o0 be a sequence of matrix polynomials in P [ x ] such that

P . ( x ) has degree n. We say that {P,(x)}~0 is an orthonormal sequcnce of matrix polynomials

(abbreviated OMSMP) with respect to the functional &a defined by (4) , if P, (x ) is a matrix

polynomial of degree n with invertible leading matrix coefficient, and .fW(P, , P , ) = ~ . . , I , for

all n,m~O. If f E E , the k-- th matrix Fourier coefficient of f with respect to (P,(z)}.;,0. is

denoted by

C , ( f ) = . ~ ( f ,P,). (8)

The Fourier series of f E E , with respect to {P.(x)}.;~0, is defined by

S ( f ) = E C b ( f ) P ~ . (9) , ~ 0

For the sake of clarity in the presentation we recall the Courant-Fisher theorem, whose proof

may be found in [1 ,p. 100]

Theorem Z. 1. [~J Let C and D be hermitlan matrices, and let ,~ ( C ) , ~.( C + D ) denote the

i--th eigenvalue, where we order ttue eigomalues in an increasing order. Then

2,(C) + 2~.(D) ~ ~(C + D) ~ a,(C) + ~,x(D). (10)

Lemma 2.1. Let A , B be hermitian positive semi--definite matrices in C'• that B>.~

A~O. Then II B II ,>~ !l AIi z.

Proof. Note that B - A ~ O , and ~,(B--A)~O. Taking C . . . . A and D = B , by theo-

rem 2.1, it follows that

~(B -- A) ~ ~ ( - A) + 2~. (B) , (11)

where ~ denotes the i - t h eigenvalue ordered in an increasing order. By (11 ) it follows that

0 ~ ).x(B - A) ~ a , ( - A) + ~ ( B ) . (12)

Since B is hermitian, by r25,p. 23-], one gets

~ . ( B ) = [I B [[,, (13)

and as A is hermitian, by (12), (13), it follows that

II B l!z = ~t~.(B) ~ - - ,1.~(-- A) = - min(a:a 6 a(-- A)} = 2,.~(A) = II A II ~.

Hence the result is established.

�9 92 �9 J6dar, L. etal: Best Approximation Matrix Probiem

Now let us address ourselves to the following best approximation matrix problem. Given

f E E and .~' be positive definite, we seek the matrix polynomial Q(x) of degree n, so that it

minimizes the set

{ [I . ~ ( f - Q , . f - Q) II ,~Q E P.[x]l. (14) Let us suppose that {P.(x) }.;~0 is an OMSMP for ~r and let Q(x) be a matrix polynomial of

degree n. By theorem 2. 4 of [22], there exist matrices A~,At... ,/1. in U ~', uniquely deter-

mined, such that

Q(x) = ~-]A,P,(x). J--D

From the properties of M', we can write

Q.: - : - _

(15)

(16) i - -0 i --0

i --0 i ra0

If we denote by C~ the i - t h Fourier coefficient of f with respect {P.(x)}o0,

Ci --~ . f~( f ,Pi) --~ . ~ ( P , , f ) " , i ~ O, (17)

then by ( 1 6 ) - (17), one gets

o < ~(:- Q,: - Q) = ~(:,:) + Z.~:.. - _~c::' - ~,/~c,", i--O i~O i~O

�9 ~'(./ - a,: - Q) = .~(f,f) - ~c,c:' + ~(c, - :.,)~c, - ~)',>~ o. (18) J--O ~--0

As (C , -A , ) (C, -&)n>~0, by (18), the minimum of M ' ( f - Q , f - Q ) is achieved when C~=

A,, for O~i~.~n. As . q t ~ ( f - Q , f - Q ) is hermitian positive semi-definite, by lemma 2. 1, if

the minimum of . ~ ( f - Q , f - Q ) is gained at Q = S . ( f ) , the minimum of [t . f E ( f - Q , f -

Q) II 2 is also reached at Q = S . ( f ) .

Summarizing the above deduction we have established:

Theorem 2.2. Let {P.(x)}.>0 be an OMSM P with respect to a positive definite func-

tional M" on E. For a fixed f E E , and a positive integer rn~O, the matrix polynomial QE P.

[x ] , which minimizes (14), is given by the m--th partial sum o f the Fourier series o f f with

respect to {P.(x) }oo,

Q = S . ( f ) = ~ C , ( f ) P , . i--O

3 On matrix Fourier series.

The Riemann--Lebesgue matrix property and

the BesseI--Parseval matrix Inequality

In this section we prove some analogy of the Riemann-Lebesgue property and the Besse]

--Parseval inequality. The next result will play a fundamental role in the following.

Approx. Theory ~ its Appl. , 13,4, Dec. 1997 �9 9 3 �9

Theorem 3. 1. Let {S. }.~o be a sequence o f herTnitian positive definite matrices bz C "•

such that O ~ S . ~ S . + l a n d let us suppose that there exists a subsequence {S~ }.;~o o f {S. }.;Jo,

with k,<k,+~ and lim . . . . k .= 4 - ~ , such that

limSj =- S.

Then

liraS. = S. j - , . o o

Proof. Given r By (19) , there exists a positive integer no such that

E

II s , - s II, < ~ , n 1> ,,o,

and

(19)

(20)

(21)

4s II S,. - S , . I[, < y , m >~ n >f ,,o. (22~

Let n, =k.o and m~nx. As k.<k.~a and lim...~k. = + ~ , there exists a positive integer i such

that

k0+. ~ m ~< k,o+,+j. (23)

By the hypothesis O<~S,<~S,+I, and (23), it follows that

S,.,+, ~< S,, ~ S~.o+,, (24)

S , - S,,o+, <~ S~,~ - S,,o+,. (25)

By (25) and lemma 2 .1 , one gets

II s . - S,.o+, Ilz ~< II s , . , . . - s,.o+, 11.., (26)

and by (22) and (26) it follows that

II s - s . l! ~ < li s - S,.o+, I], + II s,.o+, - s . I1~

~< I1 s - S ~ + , I1, + II S,.o., - S._o+, 11=

<-~+ ~=~.

Hence, the result is established.

Corollary 3.1. (Matrix Monotone Convergence Theorem) Let IS.}.a 0 be a sequence o f

hermitian positive semi--definite matrices in C "x', such that O~-~S.~S.+~ a~ul { II S. I! 2i.~o is

bounded. Then (S. I.~o converges to a matrix S in C "~'.

Proof. By (1) . from the hypothesis, each entry component sequence ~S.( i , j ) }.a0 for

l~ i , j<.~r , is a bounded sequence in C. After taking r • subsequences, we can guarantee the

existence of a subsequence of S. . say f,%}.~o, such that each entry sequences is~ ( i . j ) t .~o ,

for l<-~i,]~r, converges to a complex number S ( i , j ) .

Then the matrix S = (S(i,j))~<,.,,~. is the limit o1" {s~.}.~0. Now, by theorem 3, 1, it fol-

lows that S:=lim._=S,, Thus the result is established.

Let f 6 E and let us suppose that f P . ( x ) }.a0 is an OMSMP with respect to a matrix func-

tional .c/. By the proof of theorem 2. 2, see (18) , if S . ( f ) is the n - t h partial sum of the

�9 9 4 �9 J6dar, L. etal: Best Approximatmn Matrix Problem

Fourier series of f with respect to {P.(x) t .~0, then it follows that

0 ~ M ' ( f -- S . ( f ) , f -- S . ( f ) ) = f z " ( f , f ) -- ~aC, C]', n ~ O, (27)

where C,=.6r Hence

~-~C,C~ ~ M ' ( f , f ) , n ~ 0. (28) I r a 0

Since C,C u is an hermitian positive semi-defini te matrix, if we denote T. = ~,,~.,C,C~, one

gets O ~ T . ~ T . + ~ , and by (28) and lemma 2. 1, it follows that

II T . II, = II ~-]C,C, n 112 ~< I1 f~ ( f , f ) II , , n >t 0. (29) i - - 0

By corollary 3.1 and (29) one gets the convergence of the series ~-],~oC~C, u and

~,,C,CJ, , < ~. ( f , f ) . (30) ,'~o

In particular,

limC, C, ~ = 0. (31) i ~ o o

By['25,p. 23]we have II C,C~ tJ z = ( }1Ci I1 2) 2, and by (31) one gets

limC, = 0. (32) p-,* oo

Summarizing the above deduction we have setablished.

Theorem 3. 2 Let .~ be a positive definite matrix functional defined an E = C ( [ a , b ] ,

C" x') and let I P. ( x ) }.;,0 be an O M S M P with respect to .~. I f f E E and C, = f,C/ ( f , P, ) , then

limC, =- 0 (Matrix Riemann - Lebesgue Property),

and

Z C , C~ ~ . f~ ' ( f , f ) (Bessel -- Parseval Matrix Inequality). ~o

4 Total sets with respect to a matrix functional

We begin this section with the definition of total set with respect to a positive definite ma-

trix functional.

Definition 4.1. Let .~ be a positive definite matrix functional on E . and let v be a sub-

set of E. We say that u is total in E with respect to M/, if the only element fC:: E such that -cE

( f , g ) = 0 for all g E u , i s f = 0 .

The next result proves a matrix analogy of the Parsevai identity, and shows that an

OMSMP with respect to ~ is total in E.

Theorem 4.1. Let .~. be a bounded positive definite matrix functional on E = C ( [ a , b ] ,

C "x') and let {P.(x)}.~o be an O M S M P with respect to ~ . I f f E E and C , = ! E ( f , P , ) , i ~

O, then it follows that

. ~ ' ( f , f ) :-- Z C , C , n , (Bessel - Parseval Identity)

Approx. Theory & its Appl. , 13: 4, Dec. 1997 �9 95 �9

a,ut {P.(x) }.~o is total m E with respect to s

Proof. Let f E E and E=-I- for n>0. By the application of the Weierstrass approxima- n

tion theorem [-11 ,p. 133-135] , to each component of f and by (1) , we can assure the exis-

tence of a matrix polynomial Qj. Cr) of degree k,, such that

[ I f - - Q , [I = s u p { l l f ( x ) - Q , ( x ) l l 2 ; a ~ x ~ b } < 1 ~ (33)

where k > 0 satisfies (3). By theorem 3. 1, if S , . ( f ) denotes the k . - t h partial sum of the

Fourier of f with respect to {P.(x)}o0:

S , . ( f ) = E C , P,, (34) i - - O

one gets

II .o~"( f - s , . ( f ) , f - s ~ ( f ) ) llz ~ It ,.-qf(f - Q , , f -- Q , ) II ;. (35)

By (3) and (33), it follows that

1 11.9~(f -- Q, , f - Q , ) r[: • -t7

and by ( 3 5 ) - ( 3 6 ) ,

n:>O, (36)

1 (37) It ~ ( f - s , ( f ) , f - s , . ( f ) ) , 2 ~< - . - r l

By the proof of theorem 2. 2, see (18), it fol[ows that h R

. ~ ' ( f - Sj ( f ) , f - S~ . ( f ) ) = s ( f , f ) - EC;Cf f , (38)

and by (37) - (38), i m

1 n > 0. (39) II ~ e ( f , f ) - ~ c , c , " 112 < g , i - -O

Hence the subsequence T , = ZI:oC,Cff of the sequence T . = Z/.0C,C, n, converges to ~ . ( f ,

f ) in C'*', and by the Bessel- Parseval matrix inequality, see theorem 3. 2, the sequence

{T,}.;~o is bounded in C'*'. By corollary 3.1 one concludes that

S c ; c " = r (40)

Furthermore, if f E E and C , = . ~ " ( f , P , ) = 0 for every i ~ 0 , then by (40) one gets A P ( f , f )

=0. As ~ is positive definite it follows that f = 0 . Thus the result is established.

Example 4.1. Let D be an hermitian matrix in C'. x" such that

a(D) = max{z;z E a(D)} < - 2, (41)

and let W ( x , D ) be defined by

W ( x , D ) = I (1 - xz)-�89176 Ixl < I, (42)

L I , x ==1= 1.

As D is hermitian, for each x in [ - 1,1 ] , W ( x , D ) is also hermitian and by [25 ,P. 23] it fol

lows that

�9 96 �9 J6dar, L. etal: Bert Approximation Matrix Problem

11W(x,D) I I ,= max{ [zl := E o ( W ( x . D ) ) t .

By the spectral mapping theorem [6,p. 569], 1

o ( W ( x , D ) ) = {(1 -- xZ)-~"+: ' ;z ~ a tD)~ , if I:r] <~ 1.

H e n c e ,

and similarly

Otherwise

[t W ( x , D ) 11, = (1 - - x~)-�89 '~ Ixl < 1,

(43)

( 4 4 )

(45)

II W ( x , D ) � 8 9 II, = (a - x=)-,~'"~'+2'0 Izl < ]. (46)

J" I' II W ( x , D ) II , d x = 2 II W ( x , D ) I1 ,dx - 1 0

I' - :)-�89 = 2 I I ( t 0

~- (1 - t)-�89189 =- B( ~ , a (D) ). (47) 2

In an analogous way, it is easy to show that

II W(x,D)) -~ II ,dx = B( - ~ + ). (48) -1 ' 4 2

1 Thus the matrix functions W ( x , D ) and W ( x , D ) T are integrable in [ - 1,13. Under the h y -

pothesis (41 ), by (44) one gets that the eigenvalues of W ( x , D ) are positive for ] x [ ~ 1. As

W ( x , D ) is hermitian, by [1 ,p. 106] it follows that W ( x , D ) is positive definite for Ixl~<l.

Let E = C ( [ - 1,1"] ,C -y• and let us consider the matrix functional ~ : E X E-~C "• defined by

s 1 6 2 = f ( x ) W ( x , D ) g ( x ) U d x . - 1

By example 2. 1, .Qr is positive definite, and by [18] , the Gegenhauer matrix polynomials

q~ [ ( - D h ] - ' ( z , _ 1).~, . -~ , P. ( x , D ) = ( - - I-- D ). ~ a -~ -(~n - - -~k ~I 2b- ,, ~ 0 ,

are orthogonal with respect to f~'. If we normalize the matrix polynomials P.(x,D) with re-

spect to .fi#. we obtain

Q , ( x , D ) = K = I P , ( x , D ) , n ~ O,

where

K . = . ~ e : ( P . ( x , D ) , P . ( x , D ) )

1 1 1 1 ) ) x~ ( - D - l ) . P ( - - �89 ~ ( I + D) ) ( - -~D + (n - - '

nl , , ~ > 0 ,

Then { Q , ( x , D ) },;~0 is an OMSMP with respect to ~r By theorem 4.1 , the sequence {Q,(x,

D) },;,0 is total in C ( [ ' - 1 , l ' ] ,C ' • with respect to -Q,P.

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