On the Behavior of the nth Iterate of a Fredholm Kernel AS n Becomes Infinite

VOL. 18, 1932

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671

straight lines and on one side of a line perpendicular to these. It is natural to classify the half-strips referred to in Theorem 1 as primary, secondary, etc., according to the natures of the critical rays to which they correspond. THEOREM 2. The number of zeros off(z) in a primary half-strip and uithin = r is equal, for r large, to the circle irN

Iz

-

[I + O(l/r)],

where I is the length of the side of P which corresponds to the half-strip in question. THEOREM 3. The number of zeros of ftz) within the circle z = r and in a non-primary half-strip corresponding to a critical ray of f(z) that is a critical ray of a function fa(z), (a = 0 ... M'), is asymptotically equal, for r large, to the number of zeros of fa(z) in the same region. I

Wilder, C. E., Trans. Am. Math. Soc., 18, 415-442 (1917).

2

Tamarkin, J. D., Math. Zeitsch., 27, 1-54 (1927).

P61ya, G., Math. Zeitsch., 2, 352-383 (1918); Sitzungsber, d.Bay. Akad., M1unchen, 285 (1920). 4'chwengler, E., Diss. Zurich (1925). 6 Langer, R. E., Trans. Am. Math. Soc., 31, 837- 844 (1929). 6 Ritt, J. F., Trans. Am. Math. Soc., 31, 680-686 (1929). a

ON THE BEHAVIOR OF THE nth ITERATE OF A FREDHOLM KERNEL AS n BECOMES INFINITE By MAURICE FRiCHET INSTITUIT HENRI PoINcAid, PARIS Communicated October 3, 1932

Recent researches concerning geometrical probabilities which we will call "linked" (French: en chaine) have demonstrated the importance of the study of the asymptotic behavior of the nth iterated kernels which present themselves in this problem. We shall define iteration by the formula Kn + J) (M,P) =fv KE(") (M,Q)K(P) (Q,P)dQ; the kernels to be studied in this problem are those for which K(M,P) ) 0 (P) K(M,P)dP = 1. (T) A summary of the results obtained in this case has been published recently in the C. R. Acad. Sci., Paris.

fv

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PROC. N. A. S.

But the same problem can be extended to the most general kernel, whether or not the conditions (P) and (T) are satisfied. To solve it, it is good to consider first the simpler problem of studying the asymptotic behavior of the solutions of a system of q linear difference equations of the first order with constant coefficients in* q unknowns. When this is done, the only tools needed for the solution of the general problem are the classical Fredholm theory and the notion and theory of the principal kernel due to Goursat, Heywood and Lalesco. For simplicity, we shall consider the case where the iterated kernels are continuous, after a certain rank m, on the domain of integration V, which is assumed bounded. Calling L4 the bound of K()(M,P) -which bound is known to be rshall then be able to classify the several possifinite when n mwe bilities in the following manner. 1. L4 is not bounded when n increases indefinitely (a) if at least one of the fundamental constants Xj of K(M,P) is < 1, in modulus; (b) or if all the X, being, in modulus, ) 1, at least one of X, is = 1 in modulus and is a multiple pole of the resolvent of K(M,P). 2. Ln has a bound independent of n (for n ) m) in all other cases, that is: (c) if all the fundamental constants, X. of K(M,P) be > 1 in modulus; (d) or if theX being ) 1, in modulus, all those whose moduli is 1, be simplepoles of the resolvent. Case (c) may be characterized, against the three other ones, as the case where K() (M,P) converges uniformly to zero. Case (d) is the most interesting, inasmuch as the application to linked probabilities is a particular instance of case (d). It gives rise to several possibilities which we shall now discuss, without limiting ourselves to the application to probabilities. Special Study of the Case Where the Iterated Kernels Have a Common Bound.-The sequence of the K() do not generally converge in the usual sense. But in the present case, it always converges uniformly in the Cesaro way. That is to say, if we put

I

sw,(M,P,n)

=

n

[K( + 1) (M,P)

+ ....

+ K(P + n) (M,P)]

where v is any fixed rank ) r-mso that 9,(M,P,n) be certainly finite and even continuous-then, this arithmetic mean converges uniformly in the usual sense to a continuous limit H(M,P), this limit being independent of v. The difference [9, (M,P,n) - H(M,P) ] is infinitely small with it is of an order which is at least equal to the order of n-

MA THEMA TICS: M. FRkCHET

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673

The generalized limit Il(M,P) is a function of "finite rank" r j=r

H(M,P) =

E Di (M) *j (P)

i-1

where by, *j are any biorthogonal and normal system of fundamental solutions relative to the fundamental constant X1 = 1 of the associated kernels, K(M,P) and K(P,M). (When 1 is not a fundamental constant of K(M,P), ll(M,P) is = 0.) Some particular cases (of special interest in the application to probabilities) shall now be examined. (A). Il(M,P) INDEPENDENT OF M. In order that the generalized limit Hl(M,P), be independent of M, it is necessary and sufficient that the two following conditions be together fulfilled I (P) fW K(M,P)dP = 1 II there be a unique common solution X(P) of both the equations

X(P)

=

fv X(Q)K(Q,P)dQ;

fv X(P)dP

=

1.

And then II(M,P) reduces to this unique common solution. In our general hypotheses, the Fredholm determinant D(X) may not have a definite significance, since K(M,P) may be unbounded or even infinite for some pair (M,P). But in the more particular case where K(M,P) is continuous, D(X) recovers a definite value for each X and the set of conditions I and II may be replaced by the set of conditions I and II'. The unity is a simple root of D(X). (B). ASYMPOTIC PERIoDIcITY IN n OF K( (M,P). If the fundamental constants of K(M,P) whose moduli = 1 are roots of the binomial equation XN - 1 = 0, then K(`)(M,P) is the sum of a periodic function of n, A (M,P,n) with the period N and of a function B(M,P,n), which converges uniformly to zero when n tends to infinity. The converse property is also true. (It may be proved that the kernels verifying the conditions (P)(T) are special instances where the iterated kernels K(") are asymptotically periodic functions of n.) (C). USUAL CONVERGENCE OF K(). More particularly, the necessary and sufficient condition that K* (M,P) converges uniformly, not only in the Cesaro sense, but in the usual way, is that K(M,P) have no fundamental constant of modulus 1 except 1 itself. The limit in the usual sense is still, of course, II(M,P) and may be determined directly in the same way as above. In the present case [K() (M,P) - f(M,P) I is infinitely small with - and its order is greater than that of , where 1 < po < p, p being the smallest modulus of the Xj * 1. The above results shall be proved and completed in a separate memoir.