A generalized approximation theory for quadratic forms: Application to randomized spline type Sturm-Liouville problems

Journal of Theoretical Probability, Vol. 8, No. 3, 1995

A Generalized Approximation Theory for Quadratic Forms: Application to Randomized Spline Type Sturm-Liouville Problems

John Gregory 1 and H. R. Hughes I

Received December 6, 1993; revised April 1, 1994

An approximation theory for families of quadratic forms is given. We show that if continuity conditions for a family of quadratic forms hold uniformly on an index set for the family, generalized signature approximation results hold. We then apply these results to randomized spline type Sturm-Liouville problems and obtain continuity of the nth eigenvalue for generalized Sturm-Liouville problems under weak hypotheses.

KEY WORDS: Random quadratic form; Sturm-LiouviUe problem; continuity of eigenvalues; spline approximation.

1. I N T R O D U C T I O N

Our theory of quadratic forms on Hilbert space began with H~stenes ~7) and his students. The key idea was that if J(x) is an elliptic quadratic form on a Hilbert space ~r a variety of important problems could be studied by using two nonnegative indices: the signature, which is the dimension of the negative space, and the nullity, which is the dimension of the space of J-orthogonal complements. These terms will be defined later but the reader may think of them as the number of negative and zero eigenvalues, respectively, of a symmetric matrix.

An approximation theory for elliptic quadratic forms on Hilbert spaces was given in Gregory. ~3) The theory was inclusive enough to yield a generalized Sturm-Liouville eigenvalue theory and a generalized oscillation/conjugate point theory for self-adjoint differential systems.

'Department of Mathematics, Southern Illinois University at Carbondale, Carbondale, Illinois 62901.

703

0894-9840/95/0700-0703507.50/0 �9 1995 Plenum Publishing Corporation

704 Gregory and Hughes

For example, in the Sturm-Liouville theory, it is usually difficult to find the eigenvalues for the eigenvalue problem

[p(t) x'(t)]'+r(t)x(t)+2q(t)x(t)=O, x(a)=x(b)=O (1.1)

where p(t)> 0 on [a, b]. However, this differential equation is the Euler- Lagrange variational equation of the quadratic form

b b

J(x; 2)=;~ [px'a-rx2] d t - 2 ~ qxa dt (1.2)

If s(2) denotes the signature of J(x; 2), then it is known that s(2) counts the number of eigenvalues less than 2.

Furthermore, in Gregory, t4~ it was shown that s(2) may be approximated if the coefficients p, r, and q are discretized in a reasonable way and x is restricted to a spline space. Discretization yields finite dimensional tridiagonal quadratic forms for which the signature can be efficiently computed. A complete treatment of this work on quadratic forms may be found in Gregory. tSI

The purpose of this paper is to extend many of the above results to random quadratic forms. The main idea is that we can weaken the hypotheses used in Gregory ~3-5~ to include a parameter co in g2 and still get the signature inequalities obtained in Gregory. (3-51

In Section 2, we assume that a is an approximation parameter, J(x; a, co) defined on d ( a ) is taken from a family of quadratic forms indexed by co and a, and J(x; ao) is a fixed quadratic form defined on d (a0) . Then, under reasonable continuity conditions as a ~ ao, we obtain fundamental inequalities involving the signatures and nullities of J(x; a, co) and J(x; ao) for a sufficiently close to ao.

In Section 3, we extend these ideas to quadratic forms H(x; a, 2, 09) = J(x; a, co)-2K(x;a, co) where 2 is a real parameter and K(x;a, co) are compact quadratic forms which include the co parameter. These results combine to give an approximate Sturm-Liouville theory, which is a signifi- cant generalization of those in Gregory. ~3~

In Section 4, we consider, as one application of the results of Section 2 and 3, the setting where the approximating spaces are spline-type spaces and the approximating quadratic forms are finite dimensional approximations. Here a is the step size and J(x; a, co) and K(x; a, co) are the discretized quadratic forms where randomness has been allowed to enter into the discretization process. Thus, we show that efficient procedures for approximating the eigenvalues for Sturm-Liouville problems still apply in this setting.

A Generalized Approximation Theory for Quadratic Forms 705

Finally, we note that many of the ideas and results in this paper hold for problems such as higher order splines, 2nth order integral-differential equations, elliptic partial differential equations, quadratic control problems and oscillation or focal point theory.

2. FUNDAMENTAL INEQUALITIES

Following Gregory (3'5) we consider the following situation. Let 27 be a metric space with metric p. Let ~r be a Hilbert space. We let ~ and - ~ denote strong and weak convergence in ~1, respectively. For each a ~ 27 let ~r be a closed subspace of ~r such that:

(la) If Xke~C(ak), ak~aO, and Xk--~yo, as k ~ , then Yo e ~'(ao);

(lb) If Xoe~(ao) and e > 0 there exists ~ > 0 such that whenever p(a, ao) < ~, there exists x , e ~r satisfying Ilxo - x , [I < e.

The following is proved in GregoryJ 3~

L e m m a 2.1. Condition (lb) is equivalent to the following: Let ~(ao) be a subspace of d ( a o ) of dimension h and e > 0. There exists ~ > 0 such that whenever p(ao, a) < ~, there exists a subspace ~ (a ) of sO(a) of dimension h with the property that if x o is a unit vector in &(ao) there exists x~e~(a) such that Ilxo-x~ll <e.

A quadratic form J(x) is elliptic on d if properties (2b) and (2c) here hold with J(x) replacing J(x; a, co) and ~r replacing d ( a ) . Let J(x, y) be the associated bilinear form. We define the signature s of J(x) on a subspace cg c ~ to be the dimension of a maximal subspace of c~ contained in {x~Cg: J(x) < 0 i f x ~ 0 } and the nullity n of J(x) on a substJace cg to be the dimension of the subspace { x e ~ : J ( x , y ) = O for all yeCg}. These indices are finite if J is elliptic (see Hestenes~7)). Let J(x; ao) be an elliptic quadratic form defined on d ( a o ) and J(x; a, co) be a family of quadratic forms defined on d ( a ) for each a ~ 27 and indexed by o9 e t2. Let J(x, y; ao) and J(x, y; a, 09) be the respective associated bilinear forms. Let S(ao) and n(ao) be the signature and nullity of J(x; ao) on ~r and let s(a; o9) and n(a; co) be the signature and nullity of J(x; a, co) on d ( a ) .

Let K(x; ao) be a compact quadratic form defmed on d ( a o ) and let S(ao, 2) and n(cro, 2 ) be the signature and nullity of H(x;ao,2)= J(x; ao) -- 2K(x; ao) on d(ao) . Under the above hypotheses, H(x; ao, 2) is elliptic and the indices are finite.

Let K(x; a, co) be a family of quadratic forms defined on ~ ' (a) for each a E 27 and indexed by co e g2. Let s(x; 2, co) and n(x; 2, co) be the signature and nullity of H(x; a, 2, co) = J(x; ~, co) - 2K(x; a, co) on d ( a ) .


We say conditions (2a), (2b), and (2c) hold if the respective statements next hold for any Xk, Yk, ak, k = 0, 1, 2 ..... such that ak e X and Xk, y k e dZ](trk) and such that Xk--~ Xo, Y k * YO and ak ~ a o as k ~ ~ :

(2a) limk~ oo J(Xk, Yk; ak, 09) = J(xo, Yo; ao) uniformly in 09;

(2b) lim infk~ oo J(Xk; ak, 09) >~ J(xo; a0) uniformly in 09;

(2c) For any sequence 09k, limk~ oo J(xk; ak, 09k) =J(xo; ao) implies X k ~ X o as k ~ .

We say condition (3) holds if, for any xk, ak, k = 0 , 1, 2,..., such that ak ~ X and Xk e ~r and such that x k --~ Xo and a k -* ao as k ~ 0%

(3) f i m k ~ K(Xk; ak, CO) = K(xo ; a0) uniformly in co.

K and condition (3) will be used in Section 3.

Theorem 2.1. Assume conditions (la), (2b) and (2c) hold for a0e2: . Then there exists 6 > 0 (independent of 09) such that p(a o, a) < 6 implies

s(tr; co) +n (a ; 09) ~< S(ao) + n(ao) (2.1)

for all a~ e s

Proof Assume the conclusion is false. Then there exist sequences {ak} and {09k} such that ak ~ ao and s(ak; COk)+n(ak; COk)> s(a0) +n(ao). Let m = S(ao) + n(ao) + 1. For k = 1, 2 .... there exists m orthonormal vectors Xlk, X2k ..... X,, k in ~r with J(x; ak)<<.0 on span(xtk,..., x,,k). For each i = 1 ..... m the sequence {X~k} is bounded and hence has a weakly con- vergent subsequence. We may assume that this subsequence is {Xik} and {X;k} --~ X; for all i. By (la), x~ is in d ( a o ) .

We use the repeated index summation convention. Let b = (b~ ..... b,,,) be arbitrary. Set Yo = b~x~ and Yk = b~X~k. Since Yk --~ Yo, we have by (2b)

J(Yo; ao) ~<lim inf J(yk; trk, COk) <~ 0 (2.2) k ~ o o

Thus x~ ..... x,, is a linear dependent set. Choose b ~ 0 such that Yo = bixg = 0 and let Yk = biX~k. We have Yk - '~ 0 and

0 = 3"(0; ao) ~< lim infJ(yk; ak, COk) <~ lira sup J(Yk; trk, COk) <<- 0 (2.3) k ~ c o k ~ o o

Hence J(Yk; ak, COk) ~ 0 = J(0; ~0) so that Yk ~ 0 by (2c). However, IlYk II 2 = bibi v ~ 0 for all k. This contradiction establishes the

theorem. []


Theorem 2.2. Assume conditions ( lb) and (2a) hold for a o e Z . Then there exists J > 0 (independent of co) such that p(ao, a ) < 8 implies

S(ao) <~ s(a; 09) (2.4)

for all co ~ ~2.

Proof Let .~(ao) be a maximal subspace of M(ao) such that J(x;ao)<O on ~(ao) if x r Let xl ..... xj, be an orthonormal basis for 9~(ao). It follows from condition (lb), Lemma 2.1, and condition (2a) that there exist subspaces N(a) with bases x~ ...... xh~ such that if

Ao(ao) "- J(x . xj; a0),

Ao(a, og) "- J (x i , . xj .; a, og) (2.5)

then, for i , j = l ..... h, limk_ooAo.(ak, og)=Au(tro) uniformly in 09 if O- k ,-.4 0-0.

For a e ~h let x(a, a)--aixi~,. We may choose M < 0 and J > 0 such that

aiajAii(ao) <~ 2Mala i (2.6)

and, for all co ~ t2,

(Au(a , o9) - Au(ao) ) aiaj <~ --Maiai (2.7)

if P(ao, a) < 6. Therefore J(x(a, a); a, 09) < 0 for arbitrary a :~ 0 and the conclusion follows. []

These results are combined in the following. In particular, when n(a0)=0 , we have that the signature is locally constant, independent of co.

Corollary 2.1. Assume conditions (1) and (2) hold for aoEZ. Then there exists J > 0 (independent of co) such that p(a, a0) < J implies

s(ao) <~ s(a; o9) <~ s(a; co) + n(a; 09) <. s(ao) + n(ao) (2.8)

for all co ~ x"2. Furthermore, if n(ao) = 0 then p(a, ao) < J implies that s(a; co) = S(ao)

and n(a; co) = 0 for all o9 ~ O. We note that even stronger results are obtained in the next section

when we add a parameter 2 for which the signature is monotone.


3. THE EIGENVALUE PROBLEM

Let the metric space (X, p), the spaces ~ ( a ) , and the quadratic forms J(x; a, co) and K(x; a, co) be as before. Define a metric/~ on the space Z" x by

/3(,u ~, ,u2)=p(a~, o'2) + 121--221 (3.1)

if/1; = (ai, 2i), i = 1, 2. For each p = (a, 2), let zar = ~r We have the following result for the quadratic forms

H(x; #, co) - J(x; a, CO) - 2K(x; a, co) (3.2)

Theorem 3.1. If, for aoe22, conditions (1) hold for M(a), conditions (2) hold for J(x; a, co), and condition (3) holds for K(x; a, CO), then, for any 2 0 e R, /z o = (a 0, 20), conditions (I) and (2) hold with (Z', p) replaced by (X x ~,/~), a replaced by/~, and J replaced by H.

Proo f Conditions (1) hold since d ( p ) = ~ ' ( a ) . Suppose that, for k = O, 1, 2,..., /.z k = (a k, 2k) ~ 22 X • and x k, Yk e d(I-Zk). Also suppose that x k - - ~ Xo, Yk--"Yo and/~k ~ # 0 as k ~ oo. To prove (2a), note that

H(Xk, Yk; ilk, CO) -- H(Xo, YO;/-/o, 09)

= J ( x ~ , Yk; ak, CO)- J(xo, Yo; ao, co)]

+ 2o[K(xo, Y0; ao, co) - -K(Xk , Yk; ak, CO)]

+ (20-- 2k) K(xk , Yk; irk, CO) (3.3)

As k ~ 0% the first term converges to 0 uniformly in co since (2a) holds for J and Z, the second term converges to 0 uniformly in co by (3), and the third term does also since K(Xk, Yk, ak, co) is bounded uniformly in 60.

To prove (2b) we have

lim infH(xk;/z k, co) = lim infJ(xk; ak, CO) -- lim 2kK(xk; ak, CO) k ~ o o k ~ o o k ~ o o

~> J(xo; ao)-- , , loK(xo; ao) = H(xo;/Zo) (3.4)

This holds uniformly in co since (2b) holds for J and (3) holds for K. Finally to prove (2c), let cok be a sequence in I2 such that

l imk~oo H(Xk; lZk, COk) =H(x0;/zo). Then by (3), limk-, oo J(xk; ak, Oak) = J(xo; ao) and thus Xk--*Xo as k-~ oo since (2c) holds for J. []


We have the following.

Corollary 3.1. Assume conditions (1), (2) and (3) hold for Sto = (ao, 20) as earlier. Then there exists 6 > 0 (independent of o9) such that, for # = (a, 2), #(sto, St) < 6 implies

S(ao,2o)<.s(a, 2;og)<<.s(a,A;og)+n(a, 2;m)<~S(ao, 2o)+n(ao, 2o) (3.5)

for all 09 ~ s Furthermore, i fn(a o, 2o)= 0 then p(a, ao)< ~ implies that s(a, 2, co)=

S(ao, 20) and n(a, 2; o9) = 0 for all 09 ~ s The following definition gives the relationship between our indices and

the definition of eigenvalues. It is equivalent to the usual definition for linear, compact, self-adjoint operators on a Hilbert space.

Let ao in Z be given. We assume the Hestenes condition that J(x; ao) > 0 whenever K(x; ao) <<. 0 and x # 0 (see Hestenes~7)). A real number 2 is an eigenvalue (characteristic value) of J(x; ao) relative to K(x; ao) on d ( a o ) if n(a o, 4):f-0. The number n(ao, 4) is its multiplicity. An eigenvalue 4 will be counted the number of times equal to its multiplicity. If 4 is an eigenvalue and x :~ 0 in d ( a o ) such that J(x, y; ao )= 4K(x; y; ao) for all y in d (ao) , then x is an eigenvector corresponding to 4. Eigenvalues and eigenvectors of J(x; a, co) relative to K(x; a, co) are defined similarly.

In particular, suppose J(x; ao) has discrete eigenvalues relative to K(x; ao) with finite multiplicities. Let the eigenvalues be ordered 4,(a0)~< 42(ao) ~< ... �9 Then s(ao, 4) is a monotone nondecreasing step function in 4 with jumps n(ao, 4). Since n(ao, 4) r 0 only for isolated values of 4, the existence in a neighborhood of a0 of 4k(a; O9), the kth eigenvalue (counting multiplicities) of J(x;a, og) relative to K(x;a, og), then follows from Corollary 3.1. In addition we have continuity of the k th eigenvalue at ao:

Corollary 3.2. Assume conditions (1), (2) and (3) hold for Sto= (ao, 4o), 4o ~ R. Then, as a --> ao, 4k(a, m) ~ 4k(a0) uniformly in co.

4. SPLINE TYPE STURM-LIOUVILLE PROBLEMS

In this section, ~ ' will denote the totality of arcs x in (t, x~,..., x p) space defined by a set of p real-valued functions

x:xJ(t) (a<~t<<.b;j=l ..... p) (4.1)

which are absolutely continuous and have square integrable derivatives ~J(t) on a<~ t<~b. The inner product is given by

b

(x, y) - xJ(a) yJ(a) + f Yd(t) )J(t) dt (4.2) Ja


The norm is thus given by Ilxll 2 ~ (X, X). Weak and strong convergence on the Hilbert space d is characterized by the following lemma given by Gregory. c5)

Lemma 4.1. The relation Xq ~ Xo holds if and only if XJq(a)~ XJo(a) and ~(t)---, ~ ( t ) in the mean of order two. Similarly, Xq--~ xo if and only if XJq(a)---,x~(a) and ~ ( t ) ~ ( t ) weakly in the class of Lebesgue square integrable functions. In either case, XJq(t)~ XJo(t) uniformly on a ~< t ~< b.

In this section, 27 will denote the set of real numbers of the form

b - a tr = (n = 1, 2,...) (4.3)

n

and zero. The metric is the absolute value function, d will be denoted by d(0) .

To construct d ( a ) , let a = ( b - a)/n define the partition

zc(a)- (ao=a<al <a2 < ... < a n = b ) (4.4)

where

b - a a k = k - - + a (k=0,.. . , n) (4.5)

n

The space d(cr) is the set of all piecewise linear polygons with vertices at zr(a). This is a p(n + l)-dimensional space whose j t h component x~(t) has as basis the n + 1 functions

(1 -[ t - -akl /a , yk(t)-- ~!al--t)/tr:,

( t - - a , , _ 1 ) ~ o r ,

(0,

if t e [ a ~ _ l , a ~ + l ] , k v L 0 , a n d k ~ n ;

if te[ao, a~)andk=O; if t ~ (a ,_ , , a , ] a n d k = n ;

otherwise.

(4.6)

We now consider the following quadratic form

J ( x ) - 2q[x(a), x(b)] + [PijxixJ + 2QikXi~J + Ru~i~ j] dt (4.7)

where

2q -- Ao.xi(a) x-i(a) + 2B#xi(a) x-i(b) + Co.xi(b) xJ(b) (4.8)


A #=Aj i , Bo., and Cij=Cjg are constants; Po.(t)=Pjt( t ) , Q,v(t); and Ro.( t ) = Rj~( t ) are piecewise continuous functions on a <<. t <~ b ( i , j = 1,...,p; i , j summed). In addition, we assume

Ro. ( t) (~,~j >1 hq~ickj (4.9)

holds almost everywhere on a<~ t<<.b, for every ~b=(~b~ ..... ~bp)~ R p, and some h > 0. It is well known that Eq. (4.9) implies that J(x) is an elliptic form on ~r We note that J(x) is the quadratic form whose bilinear form is the second variation associated with standard problems in the calculus of variations.

Let a = (b - a)/n and suppose the quadratic form J(x; a) for x in ~ ( o ) is as follows. Let Puo,o(t) be defined such that on each interval [ak_~, ak) it is constant and

Po~o,(t) e [ inf P~j(t), sup Po.(t)] (4.10) t E [ a k - I,ak) t E [a k_ 1, ak)

Suppose also Po.~o,(b)=Pu, o~(a,_ O. Let Qu~o,(t) and Ru~,o(t ) be defined analogously. Finally, let

r h J(x; o, co) - 2q[ x(a), x(b)] + J~ [ Pr162 k + 2Qykr k + Rjk,,~o2J2 k] dt

(4.11)

In this Pooo, is any of a set of choices in the interval, indexed by co in s for example, a value chosen at random from the interval. A similar state- ment holds for Qu~,o and Ruo,o.

It has been shown in Gregory c5) that conditions (1) hold in this setting. We now show that conditions (2) hold for forms J(x; a, co). Thus, Lemma 2.1, Theorems 2.1-2.2, and Corollary 2.1 hold.

Theorem 4.1. If we define J(x; O ) - J ( x ) , then the family J(x; a, co) defined as earlier on ~r satisfies conditions (2).

Proof For (2a) assume x , , y,, in ~ ( a , ) ,

b l a x, w, x0, y,, ~ Yo and a , -" - - -~ 0 (4.12)

n

Now

IJ(x., y . ; a,,, co) - J(xo, Y0)l

~< IJ (x . , y,,; a,,, co) - y (x~ y . ) l + [J(x,,, y,,) - J(Xo, yo)l (4.13)

860[8/3-17


The second difference converges to zero as J is an elliptic form on ~r The first difference is bounded by

If f {[Pu(t)-P~J~"~ ~ j x~y,, i - -

+ [Qo( t ) - , .j ., [ Qo~,,~o(t)][x.y,,+2'.y{] + [ R u ( t ) - R0..o,(t)] x . 3~} dt

~< ~k(a.) Ma Ilx. ll Ily.[I ~<M2[k(~ (4.14)

where

~O(tr.) - sup sup { [P0.(t) - P u ( u ) [ , 2 IQv(t) - Q0.(u)l, t , U E [ a k - - 1 , a k )

[Ro.(t) - Ro.(u)[} (4.15)

and the first supremum is taken for k = 1 ..... n and i , j= 1,...,p. Thus the first difference in Eq. (4.13) tends to zero as a . ~ 0 by the continuity of Pg, Q,7, and R U, and the fact that both weak and strong convergence imply boundedness.

For (2b), assume x . - - ~ Xo, and note that

J(x , , ;a~,co)-J(xo)=J(x . ;a , , ,co)-J(x . )+J(x . ) - -J(xo) (4.16)

and as before, IJ(x.;a.,co)-J(x.)l<~M3~k(a.) can be made arbitrarily small. The result now follows as J is elliptic and hence weakly lower semi- continuous.

For (2c), suppose x,, '% x0 and J(x.; a,,, co,,)--. J(xo) for a sequence (co,,). We note that

I J ( x . ) - J(xo)[ ~< ] J ( x . ) - J ( x . ; a., co.) I + IJ(x.; a,,, co,,)- J(xo)[ (4.17)

and as before, J ( x . ) - J ( x . ; tr., co,,)l ~ 0 so that J(x,,)~J(xo). But since J(x) is elliptic, x.--* x0, []

The final results in this section are to extend these ideas to the eigenvalue problem in this setting. In particular, we obtain using rather weak hypotheses, the continuity of the nth eigenvalue for the Sturrn-Liouville problem.

The most general compact form in our setting is given by

K(x) -Aijxi(a) xJ(a) + 2Buxi(a) xJ(b) + Cijxi(b) xJ(b)

+ f [ [Pu(t) xix j + 2Qu(t ) xi~ j] dt (4.18)


where A0.=.4ji and /~v are constants; Pu(t)=ff j i( t) and ~)v(t) are continuous functions on a <~ t <<. b ( i , j= 1 ..... p). We also assume that K(x) <<. O, x # 0, x ~ M, implies J (x )> 0. Gregory (3) or Hestenes (7) explains in detail the relationship between our problem and the eigenvalue problem for linear, compact, self-adjoint operators.

Let cr = (b - a)/n and

K(X; tT, CO) "-" A#xi (a) xJ(a) + 2/~vxi(a) xJ(b) + Cijxi(b) xJ(b)

+ I f [/5'~'~176176 dt (4.19)

where P~o,o and O_voo~ satisfy conditions analogous to Eq. (4.10). The following holds by the same reasoning as in the proof of (2a) in

Theorem 4.1.

Theorem 4.2. If cr o -" 0, K(x; cro) - K(x) and K(x; a, co) are defined by Eqs. (4.18) and (4.19), respectively, then condition (3) holds in this setting.

Proof. Suppose x n --~ x0 as n -o or. We have

[K(x, ,;cr, , ,co)-K(xo)l<.[K(x, ,; tr , ,ca)--K(x, ,)[+lK(x,)--K(xo)[ (4.20)

The second difference converges to zero since K(x) is compact. The first difference is bounded by M~(an) where

~(e . ) - sup sup {]f f i j ( t ) - f f i j (u)[ ,2[Qij( t ) -Qij(u)]} (4.21) t,u~[ak-l,ak)

and the first supremum is taken for k = 1,..., n and i , j= 1,...,,p. Thus the first difference tends to zero as or,, ~ 0 as before. []

We have shown then that Lemma 2.1, Theorems2.1-3.1, and Corollaries 2.1-3.1 apply to this situation. In particular, let 2k(0) be the kth eigenvalue of J(x; 0) with respect to K(x; 0) and ,tk(~r n, ca) be the k th eigenvalue of J(x; tr,,, 09) with respect to K(x; tr,, ca). We have our major result on the continuity of the kth eigenvalue for Sturm-Liouville problems.

Theorem 4.3. If 2k(0) exists then 2k(tTn,(-/) ) exists for a . in a neighborhood of 0 and ,~.k(O'n, 60) converges to 2k(0 ) uniformly in co.

5. CONCLUDING REMARKS

We now briefly describe how our results can be extended or efficiently applied,


In the latter case, the indices s(a; co) can be easily found for p-= 1 as indicated in Gregory, ~5) p. 94. Thus, if we define the tridiagonal matrix D = (dk.i) by dk, k-- J(Yk; a, 09) and dk.k_ 1 -- dk-I ,k-- J(Yk, Yk- l ; a, co) where Yk is given in Eq. (4.6) for k = 0, 1 ..... n, and if Pr denote the principal minors of D then the signature s(a; co) is the number of sign changes in the sequence {P~} given by P 0 = l , Pl=d0,0 and P~=d~_l,~_,P~_~-- d~_ ,~_2P~_z (r = 2, 3 ..... n + 1 ). Similar results hold for s(a, 2; 09) with the parameter for Sturm-Liouville problems or with p > 1.

The quadratic forms in Eq. (4.7) in this paper can be extended to quadratic forms whose (real) bilinear form is given by J(x) = J(x, x) where

rb J(x, y) -- q(x, y) + JQ R~p(t) x(~i~(t) y~J)(t) dt

q(x, y) " ~l (k) ~t) kl ~k) = A ~px~ (a) yp (a) + B~p[x~ (a) y~t)(b) + x~k)(b) y~t)(a)]

kl (k) (I) +C~px~ (b)yp (b) (5.1)

k l Al~et~ k l - - Ik k l A~a= C~p-Car , and B~p are constant matrices; R~p(t)= Y Rp,(t) are (for purposes of simplicity) continuous functions on a<~t<~b; and the inequality

holds almost everywhere on a<. t ~ b , for every ~b= (~b 1 ..... ~bp) in •P, and some h > 0. In this ~, f l= 1 ..... p; k, l = 0,..., m - I; i , j = 0,..., m; and repeated indices are summed.

In this case the Euler-Lagrange equations become a 2n th order, linear, self-adjoint system. The spline approximations y(a) in d ( a ) are now spline functions of degree 2 n - 1 (or order 2n) having nodes at the points a k in Eq. (4.5). We note that even these results can often be generalized to include 2nth order, integral-differential systems of Fredholm type (see Gregory, ~5) Ch. 4).

Finally, we note that our requirement in Eq. (4.10) is consistent with choosing Pg,,o(t) (and Q~a,o(t) and Rg,,o(t)) at random from the values of P#(t) (respectively Qg(t) and Rq(t)) in the interval [ak, a~+l), but this requirement can be weakened. It is enough that Pu~o,(t) (and similarly for Q and R) be constant on [ak, ak+~) such that [Pg,~o(t)-Po.(t)[ is uniformly bounded by ~k(a) where ~b(a)~ 0 as a ~ 0.

REFERENCES

I. Boyce, W. E. (1968). Random r problems. In Bharucha-Reid, A. T. (ed.), Prob. Methods in Appl. Math., Academic Press, New York, pp. 1-73.


2. Boyce, W. E. (1980). On a conjecture concerning the means of the eigenvalues of random Sturm-Liouville boundary value problems. Quart. Appl. Math. 38, 241-245.

3. Gregory, J. (1970). An approximation theory for elliptic quadratic forms on Hilbert spaces: application to the eigenvalue problem for compact quadratic forms, Pacific J. Math. 37, 383-395.

4. Gregory, J. (1972). A theory of numerical approximation for elliptic forms associated with second order differential systems: application to eigenvalue problems. J. Math. Anal Appl. 38, 416-426.

5. Gregory, J. (1980). Quadratic Form Theory and Differential Equations, Mathematics in Science and Engineering, Vol. 152, Academic Press, New York.

6. Gregory, J., and Hughes, H. R. (1995). Random quadratic forms. Trans. Amer. Math. Soc. 347, 709-717.

7. Hestenes, M. R. (1951). Applications of the theory of quadratic forms in Hilbert space in the calculus of variations. Pacific J. Math. 1, 525-582.

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A generalized approximation theory for quadratic forms: Application to randomized spline type Sturm-Liouville problems