SIGN CHANGES IN LINEAR COMBINATIONS OF DERIVATIVES AND CONVOLUTIONS OF POLYA FREQUENCY FUNCTIONS

. We obtain upper bounds on the number of sign changes of linear combnatlons of derivatives and convolutions of Plya frequency functions using the variation diminishing properties of totally positive functions. These constitute extensions of earlier results of Karlin and Proschan.

I. INTRODUCTION AND SUMMARY.In Karlin and Proschan [i] and Proschan [2] results are obtained concerning the number of sign changes of linear combinations of convolutions of sign regular functions, while Karlin [3, 4, pp.325-326] has obtained upper bounds on the number of sign changes of linear combinations of first and second derivatives of such functions.(For a function f defined on the real line we denote the number of sign changes of f by S(f) sup S[f(tl), f(t2) f(tm)], where the supremum is extended over all sets t I < t 2 < < t on the real line m is arbitrary but finite, m and S(x I Xm) is the number of sign changes of the indicated sequence, zero terms being discarded.For the definition of sign regularity, see Karlin [4, p. 12.] In the present paper, sharper versions of these results are obtained in addition to con- clusions regarding linear combinations of both derivatives and convolutions of Pdlya frequency functions.
I.I.DEFINITION.A function f defined on the real line is said to be a Pdlya frequency function order n (PF n Note that PF functions possess the sign regularity property.n 2. LINEAR COMBINATIONS OF DERIVATIVES OF P6LYA FREQUENCY FUNCTIONS.The following result is well known (see Karlin, [4], p. 326); but is included since it is the basis for deriving many of our results.A proof is included for completeness and to illustrate our method of approach.
2 1 LEMMA Assume that f is a PF function for fixed n 1 2 Then the n+l th n derivative, f.n (x), changes sign at most n times.When n sign changes do occur, they occur in the order + + +.
where gm ru' m for 0 <_ u <_ 1/m elsewhere Thus for fixed A > 0 and m sufficiently large, the bracketed sum will have sign pattern + + + (the final sign being + or as n is odd or even, respectively).No additional sign changes are introduced as m / and A + O.By the variation diminishing property (VDP) of the PF n+l follows.II function f (see Karlin, [4], Chap. 6),the desired result The following result may be established by essentially the same approach as that above and by using the variation diminishing property of PF functions.
2.2.THEOREM.Assume that f is PFn+1 for fixed n I, 2 Then n g(x) [. b.f (j)(x) possesses at most n sign changes.j=0 3 It is interesting to note that Theorem 2.2 can also be obtained by applying known results.From Theorem 2.1 on p. 50 of Karlin [4], we find that {f, f(_l)., f(_2,..., f[_n} comprises a Weak Tchebychev (.WT) system.That the generalized polynomial, possesses no more thann changes of sign follows from Theorem 4 ,i on p, 22 of Karlin and Studden [9].However, the essential method of proof for Lemma 2,1 may he used to obtain addi S. NAIIMIAS & F. PROSCHAN tional results which are not a consequence of the theory of wr Systems.In particular we have the following.
2.5.THEOREM.Suppose that P0' Pl' Pn are nonnegative integers, a 0, a 1, a are non-zero real numbers, and tô < tl < if Pj is odd and S(aj, aj 1 0 or U(aj, aj+l, Pj) if Pj is even and S(aj, aj+l 1 otherwise. Then g(x) n[a f(PJ) j=0 (x-tj) possesses at most w sign changes.
PROOF.As in the proof of Lemma 2.1, we may write, after interchanging the integral and the summation sign, P. n P. j .P. P. -k gCx) Ak01im lim f_ J=[0(aj/A J) . )(-I) t f(x u)du gm(kA + u where gm (-) has the same definition as in the proof of Lemma 2.1.
As u approaches t.
kA, the bracketed term will have sign pattern + + +/-, the final sign being + if P. is even and if P. is odd.It follows that for u near J t.
kA the term aj -[ will have sign pattern + + + if a. > 0 and P. is even J J + + if a. > 0 and P. is odd + if a. < 0 and P. is even J + + if a. < 0 and P. is odd.J Each term a.-[ will contribute up to P. sign changes.In addition, there will be one more sign change introduced between the final sign of the string of Pj plusses and minuses at tj+1 for 0 _< _< n+l if and only if a.3 and a.3+ 1 have opposite signs for P. even and like signs for P. odd.
The number of sign changes is not increased as A + 0 and m / (R).The result follows by the VDP property of the PP function f ]I w+l An interesting point to note here is that in general it is not possible to deter- mine w from just the knowledge of S(a 0 an).However if all P. 0 < j < n, are n 3' even numbers (zero included), then it is easy to see that w .0Pj+ S(a 0 an), n j= while if the P'3 are odd numbers, then w j=07" Pj + n S(ao, a n).
3. LINEAR COMBINATIONS OF CONVOLUTIONS OF POLYA FREQUENCY FUNCTIONS.
Denote by fn*(x) the n=fold convolution of a PF function f.We have the following result" 3.1.THEOREM.. Let f be a PF k density with f{x) 7. air 1 {x), where n 1 < n 2 < < n k, and the a. are real non-zero constants.Then S(g) <_ S(a_).Moreover, if S(g) S(a_), then the sign changes of (a 1 a k) and of g occur in the same order.
PROOF.The proof is by induction on k.Clearly the result holds for k i.
Suppose it holds for i, is defined in the proog of Lemma 2.1.
By the inductive hypothesis S ak).Since the term algm(U) will introduce no additional sign changes in the bracketed term when S(a 1, a2) 0 and m is sufficiently large, and introduces one additional sign change in the bracketed ter if S(a 1, a2) 1 for m sufficiently large, the result now follows from the variation diminishing property possessed by f.
The proof that if g does possess the full n sign changes then they occur in the same order as in (a I an) is simple and so is omitted.
The following result may now be deduced from Theorem 3.1 and the variation diminishing property of TP k functions.
is TP k in n and x and the variation diminishing property of TP functions.We have shown that the same results may be obtained far more simply by proving Theorem 3.1 directly and using the fact that the variation diminishing property characterizes TP k functions.
When f(x) does not vanish on the negative half line, a proof similar to that of Theorem 3.1 shows that g(x) possesses at most 2 S(a) changes of sign.This is a sharper version of Theorem 8 on p. 730 of Karlin and Proschan [I] or Theorem 6.1 of Proschan [2]. 4. DERIVATIVES AND CONVOLUTIONS.In this section, we extend the results of the previous sections to treat linear combinations of both derivatives and convolutions of Pdlya frequency functions.
Although {f, f(1), f(2), f(n)} constitutes a WT system, one can show that {f, f(1) f(n) f2* fm* will not constitute a WT system when n > 1 and m >_2.Hence the following is not a consequence of the theory of WT systems. 4.1.THEOREM.Let 1 <_ P0 < P1 < < Pk and 1 < n. be sequences of non-zero real numbers.Suppose that f is PFw+1 and f(x) 0 for x < 0, where w P + S(a) + n "'k P_.) U(b k, a I, Pk and U is defined in Theorem 2.3.Then g(x) aif i*(x)+ b.f (x) i=l j=0 J changes sign at most w times. 7. w A+O t -I=0 here w() (bj/Cx J) j=a(g) and a() For every fixed value of A > 0, the sequence Wo(A), WPk(A has at most Pk changes of sign (excluding zeros) starting with sgn(bk).This sign pattern will occur arbitrarily close to u 0 by choosing A sufficiently small m f(ni-l)* and will always dominate the sign of .a (u) at u 0 by choosing t to i=l i be sufficiently large.

m (ni-l)*
The term aif (u) possesses at most S(a_) sign changes as u traverses i=l + R. commencing with sgn(al).The final sign of the first group of terms will differ from the first sign of the second group if Pk is even and S(a I, b k) 1 or if Pk is odd and S(a I, b k) O.
When f does not vanish on the negative half line, we have the following" 4.2.THEOREM.Assume f is PFw/ 1 and f(x) # 0 for some x < 0. Suppose that g(x) is as given in Theorem 4.1 and w Pk + 2 S(a_) + c k, where c k 1 if Pk is odd and 2 if Pk is even.PROOF.As in the proof of Theorem 4.1, the derivative terms in the integrand change sign at most Pk times in a neighborhood around 0. However, in this case the convolution terms may change sign 2 S(a_) times.If Pk is odd then one additional m (ni-l)* is introduced independent of the sign pattern of .air (u).sign change i=l However if Pk is even, two additional sign changes may be introduced if the of the convolution terms at zero differs from sgn(bk).II sign Using essentially the same technique we could determine the maximum number of sign changes of an expression of the form n m g(x) [ a f(PJ (x tj) + .bi f i (x yi ), j=O i=l where t o < t I < < tn and Yl < Y2 < < Ym" The exact upper bound will depend upon the relative magnitudes of the tj's and Yi'S and the sign patterns of (a 0 an) and (b 1 bin).where U(aj, aj+I, ai)kis as defined in Theorem 2.3.Assume that f(x) 0 for x < 0. Then g(x) [.lajh(x, nj, pj) possesses at most w sign changes commencing j= with sgn(a 1) PROOF.The proof is by induction on k.For k 1 the result follows from Lemma 2.1 and the fact that the variation diminishing property is possessed by the convolution of PF functions Suppose now the result holds for [lajh(x, nj, P)j changes sign at most w times.Assume it holds for argument 1, 2 k 1.Then g(x) may be expressed in the same manner as in the proof of Theorem 4.3 except that h(u, nj n 1, Pj) does not vanish for u < 0. When Pl is odd, the first set of terms has sign pattern + for a 1 > 0, and + + for a 1 < 0 which occurs arbitrarily close to u 0. In either case the second group of terms introduce exactly one additional k k-1 sign change.Since by induction the second term has J=[2PJ + J-2Cj and Pl + 1 additional changes are introduced, the result follows.When Pl is even the first group of terms has sign pattern + + or + In either case the second group of terms may introduce two additional sign changes if their sign at zero is or + respectively.II Note that when f(x) is a probability density that vanishes on the negative half line, f(J)(x) will possess the full changes of sign.(See Karlin, [4], p. 326.) i=l i d pDefine h(x, n, p) (x).Then we have the following result which is dx p similar to Theorem 2.3.4.3.THEOREM.Suppose that Pl Pk are non-negative integers,