9 C~.~+~ (-- i)~ + 0 (n"~-q : + 0 (n~-~-q. ~=0 (2k + t)! (sin (zt/(2n))) ='~'+1 From this the statement (14) of the theorem follows. i. LITERATURE CITED V. N. Gabushin, "On the best approximation of the differentiation operator on the half line," Mat. Zametki, 6, No. 5, 573-582 (1969). CONTINUATION OF ANALYTIC SETS ACROSS REAL MANIFOLDS S. I. Pinchuk and A. B. Sukhov Let ~C C n be a region; let M C~ be a real manifold in ~, and let A be a purely m- dimensional set in ~ M. We establish sufficient conditions for analytic continuation of A through M. This equation was previously considered in [1-7]. The results which were ob- tained were formulated in terms of CR-dimension of M and did not take its "curvature" into account. First investigation in this direction was given by one of the authors in [3]. Study of the "curvature" of a manifold M allows one to strengthen some of the above-mentioned re- suits in the class generated by C2-smooth manifolds. In what follows, we assume that M be- longs to this class. Suppose codim RM = k ~ 2. For an arbitrary point ~ M and r 0 > 0 we consider a ball B(r r 0) = {z~: I z - ~I < r0} and local generating functions pj, j = i, 2, .... k such that M QB(;,ro) = {z~B($,ro): PJ(z)=0, 1= t, 2 ..... k} and ~pz/k ....ASpk z 0. i}, we set and where For ~ = (~z ..... ~k) belonging to the sphere S k-z = {~ Rk: l~I = pc~ = ~'1 czypy E~ = {z ~ T~ (M): H (p=, $., z) : 0}, is the Levi form of the function p~ at the point ~. Also, let s~ be the largest dimension of all the complex linear subspaces L~ of set E~ and put s = max ~. The number s = s(~) [~I=, does not depend on the choice of coordinates and defining functions, and is one of the charac- teristics of degeneracy of the Levi form of the manifold M at the point ~. Thus, for instance, if M lies in a real plane, then s assumes its maximal value equal to n - k. THEOREM i. Let for some point ~ A ~ M. of the point ~. m - I > s (g) (1) Then the set A can be analytically continued in a neighborhood 40th Anniversary of October Bashkir State University. Translated from Matematicheskie Zametki, Vol. 41, No. 3, pp. 320-324, March, 1987. Original article submitted January 17, 1986. 182 0001-4346/87/4103-0182512.50 9 Plenum Publishing Corporation