Some orthogonal matrices and related orthogonal functions systems

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A R K I V F O R M A T E M A T I K B a n d 6 n r 16 1.65 06 1 Communicated 14 April 1965 by OWTO F~OSW~A~

Some orthogonal matrices and related orthogonal functions systems

By

EDGAR ASPLUND and HAROLD S. SHAPIRO

I n an earlier paper [3], one of the present authors showed that given a n y n mutually orthogonal unit vectors in 12, there exists a uniquely determined in- finite orthogonal matrix (i.e. unitary transformation of 12 onto itself)Haijll having the given vectors as its first n rows, and satisfying a~j= 0 for ] > i > n. This matrix was used to show that every subspace of Ls(F) (where F is the circle group) having finite deficiency has a basis consisting of trigonometric polynomials. I n [3] also several questions were raised concerning the behavior of the Fourier expansion of a smooth function, when the expansion is with respect to a complete orthonormal system of smooth functions and it was pointed out that certain other types of orthogonal matrices, if they could be constructed, might be relevant to these questions. I n the present note these questions are answered, the essence of the results being that no amount of smoothness (not even the requirement that all functions involved be uniformly bounded trigonometric polynomials) can guarantee either smallness of the :Fourier coefficients beyond what is implied b y Bessel's inequality, nor convergence of the series at every point. We also show t h a t the functions of a uniformly bounded complete orthonormal system of smooth functions m a y have a common zero; in such a system the Fourier series of " m o s t " functions converges to the wrong value at a point, and a /ortiori cannot converge uniformly. Questions of almost everywhere convergence in the context of the present investigation we have not, however, been able to settle.

Our main tool is the construction of an orthogonal matrix with n prescribed (orthonormal) columns, and with ai~ = 0 for j > i + n. Actually we only need these matrices for n = 1, but as the matrix theorem has perhaps independent interest, we prove it for arbitrary n, and complex entries.

1. A class of unitary matrices

Theorem 1. Suppose lla~H ( 1 < i < o ~ , l • j < n ) is a complex matrix whose columns are mutually orthogonal unit vectors. Then there exists a unique unitary matrix A = lla~jll ( 1 ~ i < c~, l ~ j < o~) having the given matrix as its /i~wt n col- umns, and satis/ying the additional conditions

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E . ASPLUND, H. S. SHAPIRO, Orthogonal matrices and orthogonal functions systems

(i)

a i j = O / o r i > i ( i ) = i + r a n k J ~ , J k =

**~ a*a~.**

i = k + l

(ii) I / r a n k Ji 1 = r a n k Ji, then a~.j(~ is (strictly) negative.

Here a i denotes the n-dimensional i-th row a~ = (aft . . . a~n ).

Proo/. The heuristic b a c k g r o u n d for the following construction of the m a t r i x A is found in [2]. L e t 1"> n a n d denote b y i(]) the smallest solution i of the equa- tion ~ = i + r a n k J i . The n u m b e r i(]) t h e n satisfies

r a n k Jio) 1 = r a n k J~(j).

P u t i ( ] ) = k. The a b o v e condition t h e n means t h a t if d is an n-dimensional column vector,

J~ d = 0 implies (Jk + a~ak) d = 0. (1)

Condition (1), however, implies b y the n o n - n e g a t i v i t y of t h e matrices involved, t h a t a k d = O if J k d = O .

B u t this is just the condition for solvability of the systems of equations J k d k = a ~ .

Fix a set of solutions dk of these systems a n d p u t Jtkdk = bk, i.e., Jk bk = ~ka~, where the positive n u m b e r s +~k are determined b y

~ = ( l + a ~ d k ) - l = ( l § g~dk) -1, k=i(~), i > n . E v i d e n t l y , 0 < 2k < 1 for all k = i(]).

W e n o w claim t h a t the following m a t r i x (which obviously satisfies conditions (i) a n d (ii) of T h e o r e m 1 is u n i t a r y .

Iil

^a2 ⁰

o

A =

0 r arbi(n+l) ... arb~(i 1) --~r 0 ...

, r = i ( i ) .

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L e t us first verify t h a t t h e j t h column is o r t h o g o n a l to the first n columns;

we h a v e

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A R K I V F O R M A T E M A T I K . B d 6 n r 1 6

--~ra*r§ ~ a * a ~ b r = - ~ r a * r §

i = r + l

For the orthogonality of the columns ?" and h, with h > ] > n, r = i(]), s = i(h), we have

- 2s(asbr)* + ~ (a~+~br)* (as+~bs)

i = l

= - 2tsb~ as + ~ b~ as+~as+~b s

i = l

= - b r ( , ~ s a * - J s b s ) = O .

I n the same way, we see that the ]th column is a unit vector:

~2r§ ~ (ar+ibr)* (ar+~b~)= ~r2 § ar/~rbr :/tr~ (l § ].

Finally, we have to show that the system of column vectors in the matrix A is complete. Assume, then, that

e ~ ( e l , . . . , e . . . }

is orthogonal to all columns in A , and that er is actually the first non-vanishing entry. Then

1 ~ ,

a*

- ~:_ er+~ar+~

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er 1=1

b y the orthogonality with the first n columns in A. If d is an n-dimensional vector in the null space of Jr, we have b y the positivity of the matrices involved

a s d = O for s > r .

But, b y (3), this shows t h a t a r d = O so that, in fact, J ~ d = O implies J r _ l d = O which in turn means that

rank Jr_l = rank J r or r = i ( ] ) for some ] > n .

Now we use the orthogonality of the vector e with the ]th column in A:

0 = --/~re~§ ~ b*a*+ie~. ~

i = l

= - er(2r § b*a*~) = - er~r (1 § d * J k d k ) = - er,~r 1,

This contradiction proves that A is a unitary matrix. To see that A is unique, suppose that we have constructed another matrix A ' which satisfies the conditions of Theorem 1. Assume that this new matrix coincides with A (as given b y

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E. ASPLUND, H. S. SHAPIRO,

Orthogonal matrices and orthogonal functions systems

e q u a t i o n (2)) in all c o l u m n s of i n d e x less t h a n j for some ] > n. T h e n t h e ele- m e n t s in t h e j t h c o l u m n of A ' w i t h r o w i n d e x less t h a n r = i ( j ) m u s t v a n i s h , since t h e s q u a r e s of t h e a b s o l u t e v a l u e s of t h e e l e m e n t s in e a c h of t h e s e rows s u m to one a l r e a d y o v e r t h e first j - 1 columns. T h e e l e m e n t in p o s i t i o n (r, j) of A ' m u s t b e - ~ r since t h e s u b s e q u e n t e l e m e n t s in t h e r t h r o w v a n i s h b y h y p o t h e s i s . F i n a l l y , t h e r e m a i n i n g e l e m e n t s in t h e j t h c o l u m n of A ' a r e d e t e r - m i n e d b y t h e o r t h o g o n a l i t y of t h e c o r r e s p o n d i n g rows w i t h t h e r t h row, a n d hence coincide w i t h t h e i r c o u n t e r p a r t s in A . T h u s

A ' = A ,

as a s s e r t e d .

Remark.

I n t h e f i n i t e - d i m e n s i o n a l case t h i s p r o v e s t h e e x i s t e n c e a n d u n i q u e n e s s (up to p o s t m u l t i p l i c a t i o n b y a d i a g o n a l u n i t a r y m a t r i x ) of a u n i t a r y c o m p l e t i o n of a g i v e n o r t h o n o r m a l set of v e c t o r s w i t h (in v i e w of t h e r e s u l t s of [2]) t h e l a r g e s t possible d o m a i n of zeros in t h e u p p e r r i g h t corner, if one m e a s u r e s t h e d o m a i n b y t h e n u m b e r of t h o s e i n d e x p a i r s (i, j) such t h a t a r s - 0 for r ~ i, s ~ j.

2. Some orthogonal function systems

I n t h e p r e s e n t section we shall e x a m i n e t h e case n = l in m o r e detail; suppose n o w t h a t ai a r e r e a l n u m b e r s w i t h a i ~ 0, at > 0 for i n f i n i t e l y m a n y i a n d

~ c ~ a 2

,=1 ~ = l - W e h a v e in t h i s case

~ _ c~ k>~l, (4)

C k - 1 2

where we h a v e i n t r o d u c e d t h e n o t a t i o n

k> 0 (5)

~ = k + l

W e define ck to be t h e p o s i t i v e n u m b e r d e f i n e d b y (5). N o t e t h a t {ck} is non- i n c r e a s i n g a n d t e n d s to zero. W e h a v e f r o m (2),

bk=ak/(Ck~lCk )

SO t h a t o u r o r t h o g o n a l m a t r i x t a k e s t h e f o r m

A =

c 1

a 1 0 , . .

c o

a l a2 c2

a 2 . . .

c 0 c 1 c 1

a l a n a 2 a n a n l a n Cn

0 , . ,

a n . . .

CoC 1 C l C 2 C n - 2 C n 1 Cn 1

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A c t u a l l y it is m o r e c o n v e n i e n t to e x p r e s s t h e e l e m e n t of t h e m a t r i x in t e r m s of o t h e r p ~ r a m e t e r s d e f i n e d b y

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ARKIV F/:iR MATEMATIK. Bd 6 nr 16

a n

P n - , n~>l; P 0 = l " (7)

Cn - 1 Cn

2 2

Cn 1-- Cn 1 1

N o t e that, since P ~ = ~ ~ - 2 2 ,

Cn 1 Cn Cn O n - 1

we h a v e p ~ = c n 2 - 2 _ - a , ( 8 )

i=O \ i = n + l /

a n d so ~ p ~ = o~. (9)

~=0

A n y sequence of non-negative n u m b e r s p~ satisfying (9) t h u s uniquely determines a n orthogonal m a t r i x of the form (6), where an a n d cn are t h e n determined b y (8).

We shall n o w use the m a t r i x (6) to define a n orthogonal function system, as was done similarly in [3]. L e t

~/~ ~ / 2 (10)

/ o ( X ) = , / n ( x ) = ~ c o s n x , n = 1, 2 , . . . .

This system is a complete o r t h o n o r m a l system (CONS) on [0, z], a n d hence so is the s y s t e m {gn(x)}, n = 1, 2, ... where

n + l

g n ( X ) = ~ a n ] / j _ i ( X ). ( 1 1 )

y = l

Here we use II~JII to d e n o t e t h e m a t r i x A. More concretely, we h a v e

n 1

gn(x)=an ~ p~/~(x)- cn /n(x), n>~l.

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~=0 C n - 1

N o t e t h a t

g,(x)

is a trigonometric (cosine) polynomial of order n. The necessary a n d sufficient condition t h a t the gn be u n i f o r m l y h o u n d e d is clearly the existence of a positive M with

n 1

an ~ p~ ~< i . (13)

i - 0

L e t n o w

/(x) EL2[O, ze]

a n d denote b y

Sn(X), tn(x)

respectively the n t h partial sum of the Fourier d e v e l o p m e n t of / in the /-system a n d in the g-system. Now,

sn

is the orthogonal projection of / on

Sn,

the span of ]0,/1 . . . . /n a n d tn is the o r t h o g o n a l projection of / on Tn, t h e span of gl . . . . gn. Moreover, t h e function

p ~ / ~ ( x )

i = 0

(14)

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E . ASPLUND, H. S, SHAPIRO, Orthogonal matrices and orthogonal functions systems is in S~ a n d o r t h o g o n a l to T=. Since T~ is a subspace of S~ of deficiency one, we conclude

s=(x)=t~(x)+ </, h=>h=(x), (15)

where <[, g> denotes the inner p r o d u c t ~ [g dx.

B y means of (15) it is quite simple to compare the behavior of the t~(x)with t h a t of the o r d i n a r y Fourier partial sums sn(x).

Theorem 2. There exists, /or each o/ the /oUowing properties a), b), c), d), e), a complete orthonormal system {g~}, n = 1, 2 . . . . on [0, ~] such that g~ is a cosine polynomial o/ order n, and moreover with respect to this system:

a) The Fourier coe[/icients of /o(x)==-l~ 1 ~ are a prescribed unit vector in 12.

b) The Fourier series o/ / 0 ( x ) ~ l diverges to - c ~ at x=O.

c) The Fourier series o/ / 0 ( x ) - - I oscillates between /inite limits at x = 0 . d) The g~(x) all vanish at x=O.

e) The Fourier series o/ / 0 ( x ) - - I diverges on a dense set having the cardinality o/ the continuum.

Moreover, in eases b), c), d), e) we may take g~(x) uni]ormly bounded.

Proo/. a) is just the observation t h a t the first c o l u m n in A m a y be a n y u n i t vector. Now, for / o ~ 1 , we have, taking / = 1 in (15), a n d noting t h a t s~(x)--=l,

(

+ P l cos x + ... p~ cos nx)

tn(X) = 1 - -~2 _ (16)

po

If we choose p~=n- 89 we h a v e tn(O)-->- c~, proving b). On t h e other hand, the ratio

P0 ~_ Pl + ... P~

m a y be m a d e to oscillate between finite positive limits a a n d b, b y choosing p~ = a -1 for a long block of n, t h e n p= = b -1 for a suitably long block of n, a n d so on alternately. This proves c). As for d) we h a v e simply to satisfy the equations (see (12)).

a~ + P l + . . . P ~ 1 - - - , n>~l

Cn ]

or, in terms of the Pi (see (7), (8))

Pn(P~2+Pl+ . p n - 1 ) = p ~ + P ~ + . . . . . Pn 1 2

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ARKIV FOR MATEMATIK. B d 6 n r 16 the solution of which is P0 = 1, P l = P 2 = . . . = [/~" F o r this choice we h a v e c ~ - 1 / ( 2 n + 1), a~ = 2 / ( 4 n z - 1), a n d

V 2 ( 4 n ~ 1) 8 9

g~(x)=

)oo nx in x]

- - ~ sin 89 x '

Of course, t h a t these special {g~ (x)} are a CONS could also be verified b y direct c o m p u t a t i o n . Thus d) has been proved. As for e), let nk d e n o t e a n increasing sequence of positive integers, a n d define

p~ = O, n # a n y nk

=1/~' n=n~.

1

Choosing nk r a p i d l y increasing (e.g. nk = 6 ~) it is a simple exercise to show t h a t there is a dense set E h a v i n g the cardinality of the continuum, such t h a t for x in E, cosn~x~>21- for almost all indices k. F o r such x, t ~ ( x ) - + - c ~ f r o m (16) (similarly, we could m a k e t~(x)-->+ ~ on a n o t h e r such set E'). This proves e).

There remains only t h e question of uniform boundedness. B y (13) the uniform

a "<~n- 1

boundedness of the {g~} is implied b y the boundedness of the sequence n/-~=0 P~.

N O W ,

a 2 = Pn 2 ~ P~n

( p ~ + 2 2 2

9 .' P n - 1 ) ( P o - k ' . . p n ) ( p ~ - k . . . p 2 n 1) 2,

hence a sufficient condition for the uniform boundedness of the {g=} is:

Pn(Po § + " " Pn-1) <~ M ( p ~ + . . . P ~ - I )

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a n d the proof is n o w completed b y r e m a r k i n g t h a t (17) holds for the systems we h a v e c o n s t r u c t e d in b), c), d), e).

R e m a r k s 1. N o t e t h a t the choice Pn = 1 / J / ~ 1 (which satisfies (17)) leads to an which are a s y m p t o t i c a l l y n -} log n. Thus we see t h a t even for a u n i f o r m l y bounded, s m o o t h CONS the c o n s t a n t function m a y have Fourier coefficients a n satisfying ~ a ~ l o g 2 n = c~, i.e. violating the hypothesis of the M e n w

m a c h e r t h e o r e m ([1], p. 76). This suggests the possibility t h a t in such a system the Fourier series of 1 m i g h t diverge everywhere, b u t we h a v e n o t been able to construct a n example giving divergence even on a set of positive measure.

2. B y adjoining to t h e {9~} t h e functions V 2 / z ~ s i n n x we can get CONS on t h e circle g r o u p which exhibit the same pathologies.

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E. ASPLUND, H. S. SHAPIRO, Orthogonal matrices and orthogonal functions systems 3. I f we d r o p t h e r e q u i r e m e n t of u n i f o r m b o u n d e d n e s s d) b e c o m e s q u i t e t r i v i a l to p r o v e : s i m p l y t a k e f u n c t i o n s c o m p l e t e in L2[O, 7~], all v a n i s h i n g a t 0, a n d o r t h o n o r m a l i z e t h e m . I n t h i s w a y we can also c o n s t r u c t a C O N S consisting of C ~ functions, all of w h i c h v a n i s h on a p r e s c r i b e d closed set of m e a s u r e zero.

4. T h a t s m o o t h n e s s alone c a n n o t i m p l y r a p i d decrease of t h e F o u r i e r coefficients m a y be seen r e a d i l y b y s i m p l y p e r m u t i n g t h e o r d i n a r y t r i g o n o m e t r i c func- t i o n s to f o r m a n e w CONS. O n t h e o t h e r h a n d , in all of t h e s y s t e m s so o b t a i n e d a t w i c e d i f f e r e n t i a b l e f u n c t i o n has a n a b s o l u t e l y a n d u n i f o r m l y c o n v e r g e n t F o u r i e r series.

R E F E R E N C E S

1. ALEXITS, G., Konvergenzprobleme der Orthogonalreihen, Budapest (1960).

2. ASPLUND, E., Inverses of matrices {at~ ) which satisfy a~j~ 0 for ] > i + p . Math. Scand. 7, 57-60 (1959).

3. SEAPIRO, H. S., Incomplete orthogonal families and a related question on orthogonal matrices. Michigan Math. J. 11, 15-18 (1964).

Added in proo]: Paper [4] contains information on and applications of finite dimensional matrices of the type studied in [3].

4. LANCASTER, H. O., The Helmert matrices. Amer. Math. Monthly. 72, 4-12 (1965).

Tryckt den 16 december 1965

Uppsala 1965. Almqvist & Wiksells Boktryckeri AB