Showing papers in "American Journal of Mathematics in 1958"

An Interpolation Problem for Bounded Analytic Functions

Abstract: be possible for a given sequence of points {zv}, I z, 1, and an analytic function f(z) in I z i < 1, I f(z) I _ 1. The result is, however, very implicit ancd gives in a concrete situation very little help in deciding if the interpolatioln is possible or not. The object of the present paper is to give a simple and explicit condition on {z,} under which the interpolation (1. 1) is possible with a bounded function f(z). If we allow {w,} to be an arbitrary bounded sequence, the condition is also a necessary one. It should be observed that even the existence of any infinite such sequence {zv} is non-trivial; this problem was suggested by R. C. Buck and constructions of such examples have also recentlybeen given by G-leason and Newman (unpublished). The proof of the main theorem depends on a reformulation of problem (1.1) which is presented in section 2. It is essentially included in a result by Garabedian [2]; since the discussion there is quite general and the proof complicated, we have included a complete and simple proof here. Section 3 contains an inequality of the Schwarz type, which is the crucial step in our proof. This is then completed in section 4. The last section is devoted to an application to the ideal structure in the algebra of bounded analytic functions.

The Determination of Paracompactness by Uniformities

Abstract: A possible solution to this problem was suggested by Kelley in [3, p. 208], where he conjectured that if X is topologicallv complete, plus another condition, then X is paracompact. This conjecture was shown by Isaac Namioka to be false. However, in the theorem proved below it will be seen that a property very similar to topological completeness is successful in this role. In fact X is paracompact if and only if X admits a uniformity under which every filter satisfying a Cauchy-like condition has a cluster point.

The Suspension of a Loop Space

Abstract: Let i: SX-* 93X be the identification map, where 3X is the reduced suspension. G. WV. Whitehead [17] studied the homotopy suspension E: rn (X) +, (SX) by using the map +(i): X -I?eaX. We consider a dual situation: abbreviate 0 (X, x0) by 2, and let j: Q? -> f2 be the identity. Then the map +-'(j): SQ -X induces homomorphisms of the homology groups which are closely related to the homology suspension r: -in(Q) H,+ (X). It is convenient to convert +-l (j) into an equivalent fibre map. The fibre is of the homotopy type of the join Q2 * Q, and the Serre homology sequence of the fibering is essentially the same as G. W. Whitehead's sequence [18] involving a, but contains an extra term. This gives an alternative proof of Whitehead's main result, and also allows us to extend several of his corollaries by one dimension: e. g. a cohomology operation of type (n, q;7r, G), q < 3n, is additive if and only if it is a suspension. As a further application, in Part II we apply the above fibering to the problem of calculating the Postnikov invariants of the suspension of an Eilenberg-MacLane space K (7r, n).

On the structure of orthogonal groups.

Abstract: This theorem was proved by D. E. Dickson [3] in the case of finite ground field, then by J. Dieudonne [4] in the general case. In this paper we will give another proof of this theorem based on a principle given by K. Iwasawa in his paper [6]. The author wishes to express his hearty thanks to Professor Iwasawa who read the original manuscript and gave him several important suggestions. For our convenience, we will use the same terminology as in C. Chevalley; The algebraic theory of spinors, Chapter I. We will define some notations we will use in this paper. The conjugate of a subspace U will be denoted by U'. If U is nonisotropic, the restriction Qu of Q to U is a quadratic form whose associated bilinear form is nondegenerate. We will denote the index of Qu (sometimes to be referred to as the index of U) by v(U), the orthogonal group of Qu by O(U) and the commutator group of O(U) by Q (U). Since every pu E 0 (U) is extended uniquely to pE 0(V) which induces the identity transformation on U', we can consider 0(U) a subgroup of 0(V). The subspace spanned by vectors xl, ,X will be denoted by . If u is a nonsingular vector, we denote the symmetry with respect to the hyperplane ' by o . In the case of characteristic 2 the term " symmetry " means " transvection orthogonale " defined by J. Dieudonne [4], p. 41.

On Some Invariants of Cyclotomic Fields

Abstract: where A, u and v are integers independent of n, The numbers A and a seem to have deep significance for the arithmetic of the fields Kn. In general, if the invariant 1 ,u(Kf/F) of a so-called r-extension K over a finite algebraic number field F is 0, then the Galois group of the maximal unramified abelian p-extension over K is, up to a finite subgroup, isomorphic with the direct sum of A copies of the additive group of p-adic integers, where A-= X(K/F) denotes another invariant of K/F.1 So, if / 0, we have an analogue, for number fields, of a similar result for algebraic function fields of one variable over algebraically closed fields of constants. For this and other reasons, it seems interesting to know whether a> 0 or , = 0 for a given I-extension K/F, and we shall find ill the present paper necessary aind sufficient conditions for y > 0 when the I-extension is obtained from the cyclotomic fields Kn defined above, namely, when u is given as the second coefficient in the above formula (1).

Conformal Invariants and Prime Ends

Abstract: Introduction. From the point of view of conformal mapping, it is unsatisfactory to consider the individual points of the boundary of a simply connected region as the primitive constituents of this boundarv. When such a region is mapped conformally onto the unit disk, in accordance with the Riemann mapping theorem, the points of the unit circumference correspond, indeed, to the prime ends of the region. These prime ends are here considered as equivalence classes of sequences of points of the region under a relation P*. The principal tool we employ to introduce this relation is extremal length, a conformally invariant quantity associated with a family of curves. Its definition was first given by Ahlfors and Beurling ([2], [3]). The term " prinle end " originated with Caratheodory [4], who initiated the systematic study of the structure of the boundary of a simply connected region. His approach was topological, and it dealt with conceptssubregions, crosscuts, etc.-which are defined with reference to the given region. It is easily seen that with Caratheodory's definition the prime ends of the unit disk correspond to points on the circumference in the sense that an equivalence class is associated with each boundary point and that two sequences belong to the same class if and only if they converge to the same point of the circumference. The problem that arises under his approach, however, and this is solved by one of Caratheodory's fundamental theorems, is that one must show that prime ends are preserved under conformal mapping. Lindelif [6] circumvented this difficulty by defining prime ends by reference to the conformal map of the disk onto the region; namely in terms of the set of indetermination or cluster set. However, his method does not obviate an explicit analysis of the topological situation in the region itself. Lindelof also obtained the following result, which is related to the classification of prime ends. Caratheodory had distinguished two classes of points, principal and subsidiary (see Section 2. 7, below), on the point set of a prime end, i. e. on the