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By V. I. Smirnov and A. J. Lohwater (Auth.)

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Formula (56) is therefore proved. On applying Hadamard's theorem to the determinant in (56), we get the following inequality: n+i \dn(8,t)\<(n+l) 2 Mn+\b-a)n, and we can prove from this, precisely as for (50), t h a t series (53) gives an entire function of λ and that, for any A, it is absolutely and uniformly convergent with respect to (s, t) in the square Jc0. On taking into account t h a t we have, under condition (40): B(s, t; λ) Ώ(λ) = D(s, t; λ), we can write for these values of λ: JR(s, t; λ) = Ώ ^ λ ) .

Is an entire function of λ, and secondly, t h a t we obtain an entire function of λ on multiplying series (45) by series (50). Let us write an inequality for the coefficient dn. A determinant of order n stands under the integral sign in (51), each element of which K(ti, tk) has a modulus not exceeding the positive number M. On applying Hadamard's theorem [IIIu 16] and the usual upper bound for an iterated integral, we obtain: \dn\

We thus obtain the following result: 1. The right-hand side of (103) transforms continuous functions u(N) into continuous functions v(M). If the functions u(N) are bounded in modulus by the same number Gv the class of equicontinuous functions v(M) obtained consists of functions which are bounded in modulus by the same number. e. we put LEMMA [ K(M; N) with r > γ Ky(M;N)^HMtN) withr

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