Showing papers in "American Journal of Mathematics in 1969"

The resolvent of an elliptic boundary problem.

Abstract: This paper and its successor, [10], extend to boundary value problems the analysis of the powers of an elliptic operator given in [9]. (Similar methods are applied in [3], [5], and [6] for the case without boundary; [5a] announces boundary results.) Although [9] treated general elliptic pseudo-differential operators on compact manifolds, we restrict ourselves here to differential operators, except in the final section, where we correct some (fortunately inconsequential) errors in the proof given in [9]. We consider a q X q system A = > aaDa of Cdifferential operators

Analytic extension of the trace associated with elliptic boundary problems.

Abstract: This paper extends to boundary problems the analysis of the powers of an elliptic system of operators given in [6]. The results depend on the analysis of the resolvent given in [7], and this paper is a continuation of that one. We will think of that earlier paper as part I of this one, and not repeat definitions, theorems, notations, and so on here, but simply refer to the earlier paper. For instance, formula 1(29) refers to formula 29 of [7],

A convolution structure for jacobi series.

Abstract: Gangolli [6] discovered this convolution structure for special values of ac and J3 namely /3= 1/2, a = (n -1)/2; 3=0, ac n; and /3=1, a-2n + 1; 1 G= 3, c = 7. n here is a non-negative integer. Let P (a,0) (x) be the Jacobi polynomial of degree n, order (2,/) defined by P(?()are orthogonal onl (-1, 1) writh resplect tO ( 1 0x)at(1 ? x7)~ a ndl

A Characterization of Group Algebras as a Converse of Tannaka-Stinespring-Tatsuuma Duality Theorem

Abstract: Let G be a locally compact group with a left invariant Haar mneasure p. Then one can construct two involutive Banach algebras L -(G, /A) anld Ll(G. a) associated with G, and the former is the conjugate space of the latter as Banach space. By the terminology of group algebra, one has been expressing the algebra L1(G p), and then one's main attention has been traditionally concentrated only on L'(G, u). It should be, however, pointed out that there is a remarkable duality between them not only as Banach spaces but also as Banach algebras. For example, taking an arbitrary pair f, g in the IHilbert space L2(G,u), which may be regarded as a neutral space in the duality between L1 (G, K) and LX (G, p), the product fg belonigs to L1 (G, ,) and the convolution f $ y belongs to LX (G, 4), where v and g' are given by y(s) g(s) and g' (s) = g (s-1). Moreover, in the study of Tatsuuma duality theorem in [29], the duality property between Ll (G,,u) and L(G, u) is implicitly used and plays important r3le. To emphasize the importanec of the duality system {L1 (G, ), LX (G, u) }, let us assunme G abelian for a little while. Then the spectrum of Ll (G, u) forms a locallv compact abelian group G with the normalized dual Haar measure ji, which is called the dual group of G. The Fourier transform 5 carries the algebra Ll (G, u) with the convolution into the algebra L(G, A) with the multiplication. Conversely, the Fourier inverse transfornm 5 carries

The Dual Space of Semi-Simple Lie Groups

Abstract: Introduction. This paper is inspired by Kazhdan's work [8]. In [8], he has studied the structure of lattices, i.e., discrete subgroups with finite invariant measure on the factor space, of a Lie group by investigating a particular topological property of the dual space of a Lie group. Let G be a separable locally compact group and G its dual space. G is said to have property (T) if the class of the trivial representation2 is an isolated point in G. In [8], it is proved that if G has property (T), then any lattice P of G also has property (T) ; in particular, by [8, Theorem 2], P is finitely generated and r/[Jr, r] is finite. Kazhdan has proved that connected simple Lie groups with finite center and R-rank > 2 have property (T) based on his study of SL(3,R). Here by the same approach, we show that SO(2,3) has property (T). Thus we are able to conclude the following theorem:

On an explicit construction of a certain class of automorphic forms.

Abstract: Introduction. In modern terms the problem of construction of autoniorphic functions can be expressed most elegantly in terms of representations of adele groups as the problem of obtaining the explicit spectrum of L2(Gk\GA) where G is an algebraic group defined over a number field k and where GA and G7. are respectively its adele and k-rational points. This space is divided into the space of cusp forms L20(Gk\GA) (with a discrete spectrum) aiid its orthogonal complemeilt. Whereas much is known about the spectrum of the orthogonal complement especially through the work of A. Selberg andc R. P. Langlands on Eisenstein Series, nothing in general is known about the explicit spectrum of the space of cusp forms.

PRINCIPAL SOLUTIONS OF DISCONJUGATE n-TH ORDER LINEAR DIFFERENTIAL EQUATIONS.

Abstract: where P, (t, A) is a polynomial in A of degree n with coefficients which are real-valued, continuous functions of t on some interval I and with positive leading coefficient. Part I deals with disconjugacy criteria for (1); Part II with the definition, existence, and properties of "principal" solutions of disconjugate equations (1); and Part III with the existence of solutions with two-sided restrictions on their logarithmic derivatives. One result in Part I states that if P.n(t, A) has only real zeros, Al(t) ? * * n(t), and these are separated by constants cl1, * n-ly

The Theory of Hyperfunctions on Totally Real Subsets of a Complex Manifold with Applications to Extension Problems

Abstract: Introduction. Suppose K is a compact subset of a Stein manifold. The set K is said to be holomorphically convex if K is homeomorphic to the spectrum of the topological algebra of analytic functions on K (Definition 2. 1). Because of the results proved in iHarvey-Wells [7], the proof of Sato's duality theorem [15] given in Martineau [10] applies to compact holomorphically convex sets (see Theorem 2. 2).