Orthogonal polynomials whose zeros are dense in intervals

JOURNAL

OF

MATHEMATICAL

Orthogonal

ANALYSIS

Polynomials

AND

Whose T. S.

24, 362-371 (1968)

APPLICATIONS

Zeros

Are Dense in Intervals

CHIHARA

Seattle University, Seattle, Washington Submitted by R. P. Boas

I.

I~vTR~DuCTI~N

Consider a sequenceof manic polynomials {P,(X)} defined by a recurrence of the form 12= 1, 2, 3,...

P,(x) = (x - 4 ~,-l(X) - 4i~n-‘&>9 P-l(X) = 0,

~o(x> = 1,

c, real,

/\n+1> 0

(n > 1). U-1)

It is well known (Favard [6]) that (1.1) is sufficient (aswell asnecessary) for {P,(X)} to be an orthogonal sequencewith respectto a real distribution d+(x) on the real line. That is, there is a bounded, nondecreasingfunction # with an infinite spectrum (= support of d+(x)) such that

P,(X) has rr real, distinct zeros, xnl < xne < a**< x,, , and there is the familiar separationtheorem (Szegij [12]): %2+1,i

<

%i

<

%+1,i+1

i=1,2

,**a,n

(l-3)

so that & = ki x,i

and

qj = lim n~m x n. n--i+1

w

both exist, at leastin the extended real number system. In particular, (& , Q) is the “true interval of orthogonality. ” the smallestinterval containing all of the zeros, Xni . We alsoclearly have & < ti+i < ~+r < vj SO we put o = lim & i-rm

and

7

=fzlj*

If we set

x = (Xni : 1 < i < ?a;tt = 1,2, 3,... }, 362

(1.5)

ORTHOGONAL POLYNOMIALS WHOSE ZEROS ARE DENSE

363

then it is seenthat u and T are respectively the smallestand largest limit points of X’, the derived set of X. In particular, if the Hamburger moment problem associatedwith (1.1) is a determined one (as is the case,for example, if {cm} and {X,} are bounded), then it follows from a theorem of Stone [lo, Theorem 10.421that u and 7 are the smallestand largest limit points of the spectrum of # (which is uniquely determined up to an additive constant at all points of continuity after specifying its moment of order zero). In 1898, 0. Blumenthal [2] proved the following result: THEOREM

(0. Blumenthal). Let lim c, = c, n-m

lim h, = h > 0, n-tm

wherec and A are finite. Then a = c - 2(h)‘/“,

7 = c + 2(X)i’2

and the set X is densein [a, 71. In this paper, we wish to obtain an anologue of Blumenthal’s theorem for an unbounded caseas well as a mild generalization. Since Blumenthal’s theorem is not readily accessibleand deservesto be better known, we will present a sketch of his proof. II.

PROOF OF BLUMENTHAL'S

THEOREM

According to a classicaltheorem of Poincare [9], under the conditions of the theorem, the sequence(Pn+i(z)/E’~(z)> converges for all complex z not in the closed interval, [c - 2(/\)1/2,c + 2(h)1/2]. Consideration of the behavior of the zeros of P,(x) then leadsto the conclusion that u and T have the values assertedin the theorem. Next, from results from the theory of continued fractions, it follows that {P,+i(z)/P,(z)} is uniformly bounded on every bounded set that is a positive distance from the set X. Finally, assumethat [u, 71 contains a subinterval that is free of the zeros, xsi . Then we can choosea bounded region G that contains an open interval (a, b) C (a, T) and such that G is a positive distance from X. It would then follow by the Stieltjes-Vitali theorem that {P,+l(z)/P,(x)} converges (uniformly) on G, hence in particular on (a, b). From the recurrence formula (1. l), we could then conclude that lim -----x=2 ‘,+l(‘)

n+m P,(x)

x - c f [(x - c)” - 4h]“2 2

a
But this implies the limit is nonreal which is clearly impossible.

364

CHIHARA

(1) Blumenthal also asserted that the spectrum of the distribution function # has at most finitely many limit points in the complement of [u, 71 but his proof is invalid. Indeed, it will be shown in VI (example 1) that under the conditions of Blumenthal’s theorem, it is possible that fi < (3 (i = 1,2, 3,...). Since I/ cannot be constant on (Ei , [?+i) (see Szego [12, Theorem 3.41.21) it then follows that its spectrum has denumerably many points smaller than cr. REMARKS.

(2) If X = 0, the conclusions in Blumenthal’s theorem are still valid (one of them vacuously, of course). This is Krein’s generalization [l, p. 2311 of a theorem of Stieltjes.

III.

SOME

PRELIMINARIES

Poincare’s theorem requires that {cn} and (A,} have finite limits and hence is unavailable for our extensions of Blumenthal’s theorem. We therefore first establish some results which will replace Poincare’s theorem and also provide certain modifications in the previous proof. Our main results will be based on Wall’s concept of a “chain sequence” [13] and its application to the study of (1.1) [3]. In particular, we write

and recall [3, Lemma 51 1. A necessary and su&knt condition for x < [I (x 3 vI) is that x < c, (x > c,) (n > 1) and {cz,Jx)} is a chain sequence. LEMMA

Next we note the identity that follows from (1.1):

44 = LEMMA

2. If x < &

minimal parameter

OY

[l -

(x -$.fjx&x)

I *

(3.2)

x 3 rll , then (~(x)};=~ is a chain sequence whose

sequence, {m,,(x)},“=,, is given by

m,(x) = I PROOF.

p&4

(x - cn) P&x)

P75+&> (x - 4+1) pnw ’

n = 0, 1) 2,... .

(3.2) can be written

S(X) = [I - m,-,(x)1m&4,

n = 1, 2, 3,. .. .

ORTHOGONAL

POLYNOMIALS

WHOSE

ZEROS

ARE DENSE

365

If x $ (4, , or), then by Lemma 1, (a,(x)> is a chain sequence,hence in particular, 0 < a,(x) < 1. Since m,,(x) = 0, it follows by induction that 0 < m,(x) < 1 for n 2 1. Thus {m,(x)} is the minimal parameter sequence. THEOREM 1. (i) If there exists an integer N such that (~:,,,(x)}~=, chain sequence, then x < CTor x > T.

is a

(ii) If x < IT or x > T, then there exists an integer N = N(x) such that {aN+n(~)}~~l is a chain sequence whose (nonminimal) parameter sequence is e%+nwx=o

.

If x = u (x = T), the sameconclusionholds provided at most finitely many fi are smaller than c (finitely many qi are larger than T). (i) Suppose (Q&X)) is a chain sequence.Then ~~+.~(x)> 0 so c, - x does not change sign for n > N. Suppose for definitenessthat PROOF.

c, - x < 0 (n 3 N).

Then for n > N, since 0 < m,(x) < 1, it follows that P,+r(x)/P,(x) > 0. Thus, PN+n(x) has the samesign asP,,,(x). It then follows that there are at most finitely many Q larger than x, hence than x, hence that x 3 7. A similar argument shows that if c, - x > 0 (n > N), then x < u. (ii) Suppose that x < u and let &, = - co. Then if x < u, there is a nonnegative integer p such that 5, -=c$2< E,+1* If x = u, we assume the above holds for somep > 0. Recalling (1.4), we conclude that sgn[ln,(x)] = (- l)n-* for all sufficiently large n. Then Pn+i(x)/Pn(x) < 0 for all n sufficiently large. But u < lim inf %+, c,, [3, Theorem 6l.l We can thus conclude that there is an integer N = N(x) such that for n >, N, an(x) > 0 and also 1 - m,.+(x) =

pm (x - c,) P&?Y)

> O*

It now follows that 0 < m,(x) < 1 (n 2 N) and hence {o~,+,(x)}~~, is a chain sequencewith parameter sequence{m,+,(x)}~~, . We now write Fn(z,

=

(aph;)fw n+1

= [l - m,(z)]-l z

(3.4)

and prove: 1 The theorem in [3 ] is stated with the implicit assumption that the moment problem is determined. IIowever, the proof given yields the same result with our present interpretation of 0 whether or not the moment problem is determined.

366

CHIHARA

LEMMA 2. Assume fl isfinite and let E be any bounded subset of the complex plane which is a positive distance from the set X. Then for each x,, < 6, , there is a constant K, > 0 such that I c&4 PROOF.

I G Ko I Jxx*)

I

(z E E).

(3.5)

Put 6 = inf / z - xni 1

(X E E, xni E X).

From the well-known partial fraction decomposition [12, Theorem

3.3.51

we obtain

where K=

6 max xo - xnk < 1 + lx,---zl lSx
.

Since x0 < [r < c, and E is bounded, (3.5) follows.

IV.

THE UNBOUNDED CASE

We now prove the following analogueof Blumenthal’s theorem. THEOREM 2. In (1 .I), let

h lim ---.n+1

limc,=co ?IZf in addition,

*-tm

wn,,

_

1 4

0 is Jinite, then X is dense in [u, co).

PROOF. By modifying a proof of Stieltjes [II, p. 5611to the context of the Hamburger moment problem, it follows that if u is finite, sois gr . For any is a chain sequenceby Lemma 1 while by hypothesis, x G 5; 9 M%=l an(x) ---f i (n---f co). It then follows that m,(x) -+ Q [3, Lemma 41, hence using (3.4)

ljr$F,(x)

= 2

(for x Q 5,).

ORTHOGONAL POLYNOMIALS WHOSE ZEROS ARE DENSE

367

Now suppose[u, co) contains a subinterval that is free of zeros, xni . Then we can choose a bounded region G which (i) contains an interval (a, b) (b < tr), (ii) contains an interval (ar,B) C (0, co) and (iii) is a positive distance from X. Since {F,(z)} converges on (a, b) and by Lemma 2 is uniformly bounded on G, it follows by the Stieltjes-Vitali theorem that {F,(x)} converges uniformly on G to the limit 2. But this implies in particular that for some N = N(x), 1 0 < m,(x) = 1 - F,(x)

< 1

is a chain sequence for OL< x < fl. Since Then h+&)>L o < 01< x < ,kI< 7 = co, this contradicts Theorem 1 (ii) and completes the proof. REMARK. The hypothesis that u is finite is essential.For under the conditions of Theorem 2 on {C,} and {A,}, it possiblethat a = 03 [5, Theorem 3.11 or that u = - CO(seeVI, Example 2).

V.

A GENERALIZATION

Theorem 1 is of someimportance beyond its present use in the proof of Theorem 2. For by virtue of Theorem 1, many resultsconcerningthe minimal and maximal limit points of the spectrum of the solution of a determined moment problem remain valid as results about u and 7 as defined by (1.5). For example, Theorems 6,7, and 8 of [3] and Theorem 2.1 (hence Theorem 3.1) of [5] are valid with the present interpretation of u and 7 independently of the status of the Hamburger moment problem. To illustrate, we reprove one which will be needed in the next theorem. 3 (Theorem 2.1 of [Sj). For ewetypositive chain sequence {/3,},

THEOREM

limsp I c, + cn+l+ [(cn - cn+1)2+ 4 +q”“l PROOF.

>, 27.

If 2.x < c, + c,,, -

[

(c, - cn+1)2+ 4 $q: ?I

(5.1)

368

CHIHARA

then routine calculation shows that (c, - x) (cn+r - x) > 0 and that

Thus if (5.1) holds for n 3 N, then {oI~+~(x)} is a chain sequence by Wall’s “comparison test” for chain sequences [12, p. 861 and it follows from Theorem 1 that x < u. This establishes the first inequality and the second is proved similarly. We will use Theorem 3 to prove a mild generalization of Blumenthal’s theorem. THEOREM 4. Let hi

lime 2n --k n-xo

c2,p1 = k, ,

where k, , k, and h are finite.

2)

lim A, = A, n-K0

Then

u = $ {k, + k, - [(kl - k2)2 + 16X]1’2} i- = 4 {k, + k, + [(k, - k2)2 + 16h]1’2}

(5.2)

and X is dense in [o, u*] u [T*, T], where

u* = min(k, , k,)

and

T* = max(k,

, k,).

PROOF. According to Theorem 3, we have upon choosingpn = $ , A = 8 {k, + k, -

[(kl - k,)z + 16h]1’2} < u

B = 4 {k, + k, + [(k, - k2)2 + 16)1]1’2} 3 T.

Now limn+mc1,(x) = h/[(k, - x) (k, - x)] = a(x) and 0 < a(x) < 2 if and only if x < A or x > B. Since the limit, L, of a convergent chain sequence must satisfy 0 < L < 4 [3, Lemma 41, it follows that {oI~+~(x)}~“,~cannot be a chain sequencefor A < x < B. It then follows from Theorem 1 that cr < A and B > 7, which establishes(5.2). As before, {olJx)> is a chain sequencefor x < 4, and {m,(x)) is its minimal parameter sequence.Since {an(x)} converges, so does{m,(x)} [3, Lemma 41, hence {F,(x)) converges for x < fr . As in the previous proofs, if [a, u*] contained a subinterval free of zeros, we could invoke the Stieltjes-Vitali theorem to conclude that {F,(z)} converges to a realF(x) for somex E (a, u*). But this implies l which is impossible.

1 ___ F(x)

1_ (x -

k,;(x

- k,) > + ’

ORTHOGONAL POLYNOMIALS

WHOSE ZEROS ARE DENSE

369

REMARKS. (1) If u* < x < 7*, U(X) < 0 sono contradiction is arrived at if {F,(x)} convergesfor o* < x < T*. (2) If X = 0, th en u = u* and 7 = T*. In this case,a recent theorem of D. Maki [7, Theorem 5.41saysthat u and G-are the only limit points of x’.

(3) In the special case,c, = (- 1)” cO, c,, f 0, a theorem of the author [4, Theorem 21 combined with Blumenthal’s theorem saysthat X is nowhere dense in (u*, T*) = (- ( c0( , j c,, I). (4) The preceding remarks suggestthe conjecture that, in general, under the conditions of Theorem 4, (X’)’ = [u, u*] u [T*, T].

VI.

EXAMPLES

We present three examples to illustrate various aspectsof the preceding results. We will need the following simple result which provides an upper bound to the amount by which the terms of a chain sequencecan exceed a. THEOREM

5.

If{an}

is a chain sequence such that 01, > $ , then

(where m, is the nth minimal parameter). PROOF.

In general, for any chain sequence(ar3, [Q/2

= [(I - maMl)mn]li2 < “En*,

hence

il ([41’2 - t) < i Cm,- m,) = 3m, . Now if [Q2 - Q > 0, then since [a*] l/2 + 4 < Qthe theorem follows. (Note alsothat m, -+ 4.) COROLLARY.

If a, = t + E, >, a

(n 3 I>,

where C Ed= 00, then {u~+,}~=~ is not a chain sequence for any A? 40912412-9

370

CHIHARA

EXAMPLES.

(1)

Let

c?z+ c,

&2+1= X(1 -I-$,

According to Blumenthal’s

(A > 0, n 2 1).

theorem, CI = c - 2hi/a and

so by the above Corollary, (cuN+n(u)> is not a chain sequence for any N. By Theorem 1, .$i < a for i = 1, 2, 3 ,... . (2)

Let c, = 2n,

An+1 = n2 + ny

(1 < y < 2).

Then for every real X, an(x) = $ + A,(x)

7F2,

where limn+mA,(x) = 1. Thus {arN+,(x)} is not a chain sequencefor any N and by Theorem l(ii), u < x. Thus a = - co. (3) Let c, = an + 6

(a > 0),

h, =dn2

+fn

+g

>O.

If g = 0, the resulting polynomials coincide with the classof orthogonal polynomials characterized by J. Meixner [8]. Since A lim n+l = $ , n’m GAL,, we have: (i)

If 4d < u2, u = cc [3, Theorem 81;

(ii)

If 4d > u2, u = - 03 by Theorem 1;

(iii)

If 4d = u2, write a,(x) in the form 1 44 n t B(x) an(x) = 4 - 4dn2+ C(x) n + D(x) ’

where A(x) = (b - d1j2 - fd-l/” - X) d112, and B(x), C(X), and D(X) are independent of n. Then for x < b - d1j2 - fd-1/2, 0 < an(x) < & for n sufficiently large. Hence {Ann} is a chain sequencefor someN and by Theorem 1 x < u. Thus b - dlf2 - fd-‘1” < (I.

ORTHOGONAL

POLYNOMIALS

WHOSE

ZEROS

ARE

DENSE

371

On the other hand, for x > b - al/2 - fd-1/2, G(X) > $ for 71sufficiently large and

lflb.(X, - a, = co, hence by the corollary, {cL~+J x )} is not a chain sequence for any N. Thus we can conclude that

u = b - d1’2- fd-112. According to Theorem 2, the zeros of the corresponding nomials are dense in [a, co).

orthogonal poly-

REFERENCE3

1. N. I. AHIEZER AND M. KREIN. “Some Questions in the Theory of Moments.” Translations of Mathematical Monographs, Vol. 2, American Math. Sot., 1962. 2. 0. BLUMENTHAL. tfber die Entwicklung einer wilkiirlichen Funktion nach den Nennern des Kettenbruches fur sTW [~(O/(Z - t)] d& Inaugural Dissertation, Gottingen, 1898. 3. T. S. CHIHARA. Chain sequences and orthogonal polynomials. T7ans. Amer. Math. Sot. 104 (1962), 1-16. 4. T. S. CHIHARA. On kernel polynomials and related systems. Boll. Un. Mat. Ital. 19 (1964), 451-459. 5. T. S. CHIHARA. On recursively defined orthogonal polynomials. PYOC. Amer. Math. Sot. 16 (1965), 702-710. 6. J. FAVARD. Sur les polynomes de Tchebicheff. C. R. Acud. Sci. Paris 200 (1935), 2052-2053. 7. D. P. MAKI. On constructing distribution functions: a bounded denumerable spectrum with n limit points. Pacific J. Math. 22 (1967), 431-452. 8. J. MEIXNER. Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. 1. London Math. Sot. 9 (1934), 6-13. 9. H. POINCAR~. Sur les equations lineaires aux differentielles ordinaires et aux differences finies. Amer. J. Math. 7 (1885), 213-217; 237-258. 10. M. H. STONE. “Linear Transformations in Hilbert Space.” Amer. Math. Sot. Colloq. Publ., Vol. 15, New York, 1932. 11. T. J. STIELTJFS. “Recherches sur les fractions continues.” pp. 402-566. (Euvres, Tome II, Noordhoff, Groningen, 1918. 12. G. SZECB. “Orthogonal Polynomials.” Amer. Math. Sot. Colloq. Publ., Vol. 23, New York, 1939. 13. H. S. WALL. “Analytic Theory of Continued Fractions.” Van Nostrand, Princeton, New Jersey, 1948.