Variational properties of the zeros of orthogonal polynomials

Rend. Sem. Mat. Univ. Poi. Torino
Voi. 53,2(1995)
D. Kershaw
VARIATIONAL PROPERTIES OF THE ZEROS
OF ORTHOGONAL POLYNOMIALS
Abstract. The zeros of P n +i, the Legendre polynomial of degree n + 1 can be
characterized as being the extrema of the quotient:
J
xp 2 {x) dx
f_..p 2 (x) dx
subject to certain constraints. This result is generalized to include polynomials
orthogonal with respect to a general weight function, and with other powers of the
variable x. The results are applied to finding ineqùalities for the largest zero of
orthogonal polynomials.
1. Introduction
Recently, the p-version finite element and spectral methods for approximating partial
differential equations have emerged as powerful methods, sometimes yielding exponential
rates of convergence. The analysis of such methods, and in particular the stability, requires
results on polynomials that seem to be non-standard. One particular stability result arises in
the theoretical analysis of a posteriori error estimators for p-version finite element method
and was to fìnd an upper bound for the quotient:
(1 i) J-i x 2 p 2 (x)dx
S-i P 2 ( X )
where p ranges over ali polynomials with real coeffìcients of degree n — 1. We shall see
that the maximum value is x\ +l where x n+ \ is the largest zero of P n+ i, the Legendre
polynomial of degree n + 1, and is attained when
dx
p(x)
rv*J> rp&
The analysis will be used to investigate the more general problem of finding extrema of
the quotient Q m , given by
, v . f x m p 2 (x) w(x)dx
(1-2) Q m (p) = J \ b : , ' , ' • m = 1,2,
J p 2 (x) w(x)dx

90 D. Kershaw
where w is a nontrivial, non-negative integrable function of [a, b] and p ranges over ali real
polynomials of degree n — 1. The case when m is any positive integer is also amenable
to the analysis, but the results become progresively more complicated and less interesting
and so will not be investigated in detail.
Some preliminary definitions will be made.
N.
DEFINITION 1.1. a) VM is the space of real polynomials of degree not exceeding
b) The orthonormal polynomials associated with the weight function w will be
denoted by 7T/v, where 7TJV € VM for N = 0,1,... .
e) The zeros of TTN, ordered in increasing size, are
r(N) IN) (N)
It will always be assumed that the coefficient of x N in 7T/v is positive. If the
weight function is distinguished by a suffix then this suffix will be attached as a superfix
to 7r. For example ir N is the normalized polynomial of degree N which is orthogonal to
1, x,..., x^"" 1 with respect to the weight function w\.
DEFINITION 1.2. The weights of the N-point Gaussian quadrature formula
associated with the weight function w will be denoted by
r(N) (N) JN) , rr(N) ff(N) rr(N)
(The quadrature points are
T(N) r(N) AN),
The following will be needed, the first is a standard result, and the second is easily
proved.
LEMMA 1.3. a) H^N) > 0, i = 1,2,...,A^,
b) If a 4- b = 0 and w(x) = w(—x) then
x i w +xW j+1 =0,'i = l,2,...,JV.
2. Rayleigh's Principle
The solution of the problem is essentially contained in the following theorem which
follows from Rayleigh's principle, (see for example [2]), however some special cases will
be investigated.

Variational Properties of the Zeros of Orthogonal Polynomials 91
THEOREM 2.1. a) The maximum (minimum) value ofthe quotient
(2.i) Qm{P)=£ *y w ;

92 D. Kersliaw
3. Case m = 1
In this section we restrict our attention to the problem when m — I. The following
sections will be devoted to the case m = 2. The results in the general case become
progressively less interesting with increasing ra, and we shall not dwell on them.
THEOREM 3.1. Let Qi(p) be defined as in (2.1). The we have
(3.1) x^1'
= max Qi(p) = min Qi{q), r = 1,2.... ,n,
p
q
where the maximum is to be taken over ali polynomials p e V n -i such that
(3.2.) / p(x)
Ja
X — Xs
X
(n) w(x)dx = 0, s = r -1-1, r + 2,..., n,
and thè minimum is taken over polynomials q e V n -i such that
(3.3) / q(x) ^ ^ w(x)dx = 0, s = 1,2,... ,r - 1.
«/ a X X g
In each case the extremum is attained when
(3.4) p(x) = q(x) = - n(x)
(n)
Proof. We shall give the proof in the case of the maximum, the one for the
minimum is similar.
Let p be such that (3.2) is true. The evaluation of the integrai in (3.2) by the
appropriate n-point Gaussian quadrature formula gives
ffJ»V(*S">) } = o,
(the prime denotes the derivative) and so, since Ha and ir' n {xs ) do not vanish, we have
(3.5) p(a4 n) )=0, s = r + l,r + 2,...,n.-
With the same Gaussian quadrature formula, the quotient Q\(p) now reduces to
which, since H^1
and p 2 {x^1')
are positive, gives
QX(P) < 4 n \
and equality is attained when p satisfies either (3.2) or (3.5).
It is worth emphasising the result for the extreme zeros.

Variational Properties of the Zeros of Orthogonal Polynomìals 93
COROLLARY 3.2. We have
(3.6) max Q 1 (p) = x^\
pev„.-i
and the maximum is attained when
(3.7) p(x) = K n (x)/(x~xM),
(3.8). min Qi(p) = x ( \
P€Vn-l
and the minimum is attained when
(3.9) p(x) = ir n (x)/(x-x

94 D. Kerskaw
If there is sufficient symmetry the bounds can be attained, as will be seen from the
following result.
Only the proof for the maximum will be given.
THEOREM 4.3. Let a — -b, and w(x) = w(-x), then
(4.4) max Q 2 (p) = (x^Y,
Witti equality when
(4.5) p(x)= ' r " +l(l)
Proof. This follows from theorem 4.1 since the largest zero of F2 n +i will be the
largest zero of 7r n+ i. But more directly by lemma 1.3 (b) we have
(xr i) ) 2 =(-a i) ) 2 .
and the polynomial given by (4.5) when inserted into Qi will give the desired result.
The section closes by giving an upper bound for the quotient Q2.
THEOREM 4.4.
Ifa>0then
(4.6) (^ti 1 ') 2 > ggLQM > ( x " n) ) 2 -
Proof. The leading coefficients of 7r n and 7r n+ i are positive, and so
TT n (x) > 0 for x > x^\
VlW > ° for x > x( n+l\
and because of the interlacing of the zeros of the polynomials ir n and 7r n +i we can conclude
that
7r n+1 (s£°) < 0-
The zeros of each polynomial He in (a, 6), consequently n n {x) and 7r„ + i(x) have opposite
signs for x < 0.
The sign of F2 n +i(xn ) is the same as that of 7r n +i (^n) )7r(-»n n) ), which is
positive.
The sign of F 2n+ i(x (^' 1) 1 ) is opposite to that of TT n+ i{-x^^ )^n{x^\ )» and
so is negative.
It follows that F2 n +i has a zero in (xn jX^+i )• There are no more zeros of F2 n +i
above x^^ ' since it is positive there; and so because ali its zeros are positive the largest
zero will provide the upper bound.

Variational Propertìes of the Zeros of Orthogonal Polynomials 95
5. Comparison Theorems
In this section we shall apply some of the above results to fìnding some inequalitìes
satisfied by the largest zero of orthogonal polynomials. There will be corresponding results
for the smallest zero, the statements and proofs for those will not be given since it will be
clear what they should be.
THEO REM 5.1. Let
(5.1) ^(r)
«"(?) =
if
f xp 2 (t) w r (t)dt
$ h a p>(t) w r (t)dt , r = l,2,
(5.2) Q Ì Ì\P)-Q Ì Ì\P) <
for ali polynomials p of degree not exceeding n — 1, then
(5.3)
.(»»)
(»),
x^(w 1 )-x^(w 2 )
< e.
Proof. The upper inequality gives for p G V
QÌ 1) (p)

96 D. Kershaw
It follows that
(5.7)
x n (w 2 ) x n [w 1 )>
The right hand side of (5.7) is positive if
which can be written out as
This is equal to
(xq ì q)2{q ì q)i >
(q ì q)2/(yq ì q)i ì
PO '6 nb no
/ / {x-y)q 2 (x)q 2 (y) w 2 (x)wi(y)dijdx.
Ja Ja
fa
nx
(x - y)q 2 {x)q 2 W2{x) W2(y)
{y) dydx,
wi{x) wi(y)
which is positive because of (5.4). Consequently vve have the required result. •
An inclusion theorem is possible when the range of integration is in [0, oo). In
particular it is applicable to Laguerre polynomials.
THEOREM 5.3. Let. a> 0, and W2(x) = xwi(x), then
(5.8) v&RXwx) > 4 n) (-W2) > a£ n) (fi)-
» , .(2)
Proof. Now Xn (^2) is the largest zero of ir n , which is such that
*6
/ n^(x)x r xwi(x)dx = 0, r = 0,1,... ,71 - 1.
J a
.(2)
Since xir n e V n +i and is orthogonal to ali polynomials of degree n — 1, it must be that
linear combination of ir n and ir n +i which vanishes at x = 0. Hence we can write
As in the proof of theorem (5.2) we see that 7Tn (0) and ^^(0) have opposite signs, and
the proof continues as before. •
THEOREM 5.4. Let a — —b and wi,W2 be weight functions sudi that
wi(-x) =wi(x), w 2 {-x) = w 2 (x).
If
(5.9)
then
(5.10)
w 2 (x)
wi{x)
w 2 {y)
> 0, 0 < y < x < ò,
wi(y)
x(w 2 ) > 4 n) K)-

Variational Propertìes of the Zeros of Orthogonal Polynomials 97
Proof. Just as in the proof of theorem 5.2, in order that
should be true we must have
(5.11) f (\x-y)q 2 {x)q 2 {y)
J-bJ-b
The left hand side of (5.11) can be written
«S"W > sW(toi)
w 2 (x)
wi(x)
w 2 (y)
wi(y)
dydx > 0.
-0 px
x PÒ PU PÒ PX
rb p0 pb px
w 2 {x) w 2 (y)
dydx.
-b /
wi(x) wi(y)
+ + (x-y)q 2 (x)q 2 (y)
The middle term -b of JO (5.12) J-b is always JO Jo positive because of (5.9). The outer terms can be
combined to give the expression
'6 px
0 JO
(x - y) [q 2 {x)q 2 (y) - q 2 {-x)q 2 (-y)}
w 2 (x)
wi(x)
The determinant here is positive, and so it remains to show that
q 2 (x)q 2 (y)-q 2 (-x)q 2 (-y)>0.
Now
w 2 (y)
wi(y)
dydx.
7T (1)
q[x) = r^
(n) / \
X-Xh, J {Wi)
where Xn (wi) is the largest zero of nii .
Suppose first that n is even, then since the zeros are symmetrically placed about the
origin (lemma 1.3 (b)) we can write (superfìxes will now be omitted)
q{x) = (x 2 - x\){x 2 -x 2 2)... {x 2 - X^_!)(.T + x m )
=
Jm\X)\X -\- X m ))
where f m is an even function and x m > 0. Then
q(x)q(y) + q(-x)q(-y) = 2f m (x)f m (y) (xy -f xty ,
and
It follows that
q{x)q{y) - q(-x)q(-y) = 2f m {x)f m {y)(x + y)x rn .
q 2 (x)q 2 {y) - q 2 {-x)q 2 (-y) = if^{x)f^{y){xy + x 2 m)(x + y)x mì
which is non-negative for 0 < x,y < b. Hence for n even the sum of integrals in (5.12) is
positive, and so
x^(w»)
> 4*»(«n).

98 D. Kershaw
The proof when n is odd is similar, the only difference being that we have to use
the representation of q given by
where n ~ 2m + 1.
q(x) = x{x 2 - x\){x 2 -xl)... (x 2 - a4_i)(x + x m ),
•
Acknowledgments. My thanks are due to Dr M. Ainsworth of Leicester University for
drawing my attention to the problem, and also to the referee whose helpful comments have
improved the presentation.
REFERENCES
[1] A. ERDÉLYI (et al.) Higher Transcendental Functions, Volume IL McGraw Hill (1953).
[2] J.N. FRANKLIN, Matrix Theory. Prentice-Hall (1968).
[3] G.SZEGÒ, Orthogonal Polynomials. A.M.S. Colloquium Publications, second edition, volume
XXIII (1959).
Donald KERSHAW
Department of Mathematics and Statistics, Fylde College,
Lancaster University, Lancaster LAI 4YF, United Kingdom.
Lavoro pervenuto in redazione il 20.10.1994 e, informa definitiva, il 10.4.1995.