By W.A.J. and Robinson, A. (Ed.) Luxemburg

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3. 5. 8 Riemann surface d D a f D c R. M(d/p)id(a) = 1, D. 1. metric d. p d-isometries B w(a) = b. wE z E 52, 52 d p W-invariant {f o w : f E F wE = F. a). 0 = K. D. STROYAN 56 8, conformally invariant. by 0 The following are equivalent for a W-invariant family I;: F is a normal family. M(s/d)f ( z ) is ajinite for every z E *B and every f E *F. M(s/d)f ( z ) < K (a standard constant) for every z E B and f E F. Every f E *F is S-continuous on all of *B and hence uniformly Scontinuous in the metric d.

2 f conditions of the theorem are best possible. 0 Ix 5 1, f(x) = (Ix - yI:y E C ) 0 Ix I1 1. f 3"x - [3"x] E C nEN x EC by p,, = n, nEN p,f(3"x) = 0 x EC p, = + 00, p(e) > 0 =0 0 Ix I3 = (1 - 2x)(1 - x ) t Ix I 1 f ( x + = f ( x ) x, f 0 Ix I 1 3. 3 p. 82) 46 W. A. J. LUXEMBURG References Hardy, G. H. and Rogosinski, W. ). Luxemburg, W. A. , 1972, What is Nonstandard Analysis? Amer. Math. Monthly, to appear. , 1966, Non-Standard Analysis (North-Holland, Amsterdam). , 1948, Theory of the Integral (New York, second rev.

14. 15. LEMMA. For each f F, E C(T) and for each n E *Z we have FOURIER ANALYSIS -1 29 +x E,(n) = 21c 1 F,(t)e-i"'dt = (sin (:)/(:))],(n). -I it follows that Hence, the required result follows. 16. f(n/o) + mef(S). 17. ,(n)l I 3 mF,(n/lnl). 17 we obtain the following special case of the Riemann-Lebesgue Lemma for the o-circle group *T(2no). 18. THEOREM (Riemann-Lebesgue Lemma). For each f E C (T ) and for all 0 # In1 I oo,we have fv particular, ],(n) , = Ofor all in$niteIy large In1 Ioo. W. A. J.