Showing papers in "Crelle's Journal in 1972"

Weak-star density of polynomials.

Abstract: The purpose of this paper is to investigate the following approximation problem: For which finite positive measures μ in the complex plane are the polynomials weak-star dense in L (μ) ? By a polynomial we mean an analytic polynomial, that is, a function p of the form p (z) = CQ + ολζ + · · · + ν> where c0, . . ., cn are complex constants. For our question to make sense it is necessary that the polynomials belong to L (μ), in other words, that μ be a Borel measure of compact support. All measures considered in this paper will be finite Borel measures. We shall obtain here a necessary and sufficient condition on μ for the required density to hold. Although the condition cannot be called simple, it seems appropriate to the problem, for the following two reasons. (1) Together with its proof, it makes transparent, in case density fails, what it is that causes this failure. (2) The reasoning that leads to the condition yields, in addition, a description for arbitrary μ of the weak-star closure of the polynomials in LTM (μ). The Statement of the condition requires considerable preparation; it is therefore deferred until later. Our treatment of the approximation problem makes extensive use of known results on rational approximation and function algebras (especially Dirichlet algebras). Some of the needed results are set forth in the preliminary §§ 3 and 4. We assume the reader is famili r with the most basic terminology and results in the theory of function algebras; the book of Gamelin [7] is an excellent reference. Although our bibliographical references are mainly to the original sources, most of what we need can be found in Gamelin's book. § 2 is also preliminary; it contains a few simple but useful remarks on balayage. The approximation problem is attacked in §§ 5—7. In § 5 we study a notion of convexity with respect to families of bounded holomorphic functions. The main result here is the recognition of a class of Dirichlet algebras. § 6 deals with a related notion, that of the h ll of a measure with respect to a family of bounded holomorphic functions. The density criterion is obtained in § 7. Our approximation problem was suggested by, and is in fact equivalent to, a certain question about the invariant subspaces of normal operators on Hubert spaces. That question is discussed in § 8.

On the existence and characterization of minimal projections.

Abstract: : The paper is devoted to the characterization problem of minimal projections. (Author)

Continued fractions for some algebraic numbers.

Abstract: where <*! = 2 cos — =is a root of x -\\x — 2x — l, #2 is a root of x — χ — l, OCB = j/2~+ j/3\"is a root of X — 9z — 4# + 27z — 36z — 23. The first numbers were chosen s cubic irrationalities, oc1 s being totally real, and we picked a2, OCB at random to be non cubic. The original motivation was to see if computations for algebraic numbers would be in line with some conjectures made in [3] and [4], and to observe whatever eise might come up.

Closed ideals in locally convex algebras of analytic functions.

Abstract: Given a region Ω < C and a function p ^ 0 on , denote by Ap = Αρ(Ω) the ring of all functions f : fi-> C analytic on Ω for which there exist constants C1 = C^f) Ξ> 0 and C2 = C 2 ( / )^0 such that | f(z) \\ ̂ C1 exp (ί\\Ρ())> z € . These rings carry a natural locally convex topology, and in this paper we consider the problem of describing the closed ideals of Ap. In particular, we ask if every closed ideal is determined by its local ideals. This is the case, for example, in the ring of all functions analytic on a Stein manifold, and various authors have proved this when Ω = C, particularly under radial growth restrictions (e. g., p (z) = p(\\ z |), z € C). However, it is not in general true that every closed ideal of Ap is determined by its local ideals unless some conditions are imposed on the growth function p. Our principal result is an approximation theorem which shows that the closure of an ideal in Ap contains all functions that are sufficiently small\" wherever all the functions in the ideal are small. This theorem essentially reduces the problem of describing the closure of an ideal to an extremal problem for plurisubharmonic functions, and we use results from potential theory to solve the extremal problem in several cases when Ω = C. Thus, when Ω = C, the method yields a wide class of growth functions for which the closed ideals are determined by their local ideals. Finally, we briefly describe how the methods apply to describing the closed submodules of the topological vlp-modules Jt = (Ap), m Ξ> 1. I. Introduktion Given a region Ω < C and a function p Ξ> 0 on , denote by Ap = Αρ(Ω) the ring of all complex-valued analytic functions f on Ω for which there exist C± = C^f) ^ 0 and C2 = Ct(f) Ξ> 0 such that | f(z) | <Ξ C± exp (C2jp(z)) for all z € Ω. These rings carry a natural locally convex topology (see Section 2), and we consider here the problem of describing the closed ideals in the rings Ap. In particular, when is every closed ideal determined by its local ideals ? This is the case, for example, in the ring A (Ω) of all analytic functions on a Stein manifold , s follows directly from H. Cartan's Theorem B. In addition, various authors (see e. g. L. Ehrenpreis [8], I. F. Krasickov [18], [19], or Taylor [25], p. 455) have proved this when Ω = C and the growth restrictions are radial (i. e. p (z) = p(\\ z D), and a celebrated paper of L. Schwartz [24], p. 879 contains this result for Ω = C and p(z) = | Re z \\ + log (l + l z |). Kelleher and Taylor, Ciosed ideals in loeally convex algebras 191 However, it is not true in general that every closed ideal is determined by its local ideals unless some restrietions are made on the weight function p (see Example 6. 1), Our main results are given in Section 6 where we present some criteria for the case Ω = C which are sufficient to ensure that every closed ideal of Ap is determined by its local ideals. These criteria can be applied to various nonradial growth conditions, including the examples p (z) = | Re z |* + | Im z \\ (α, β ̂ 1), and p(z) = | R e z They may also be applied in the case of radial growth conditions, where they reduce the problem to a well-known classical minimum modulus theorem. The method we use is based on a slight extension of a theorem of L. H rmander [12]. Our principal result (Theorem 3. 3) shows basically that the closure of an ideal in Ap contains all functions that are \"sufficiently small\" whenever all the functions in the ideal are small and reduces the problem to what is essentially an extremal problem for plurisubharmonic functions. In particular, Theorem 3. 3 provides a criterion for deciding when an ideal of Ap is dense in Ap. In Section 4 it is shown that in the case of one complex variable the general problem reduces to that characterizing the dense ideals. However, in more than one variable we have been unable to make this reduction except in some very special cases (see Theorem 4. 5). The next two sections consider only the case Ω = C, where we are able to use results from potential theory to solve the extremal problem in several cases. The case of radial growth is discussed in Section 5, but the Situation for non-radial growth restrietions is more complicated, and our discussion of this in Section 6 is more technical. We introduce (Definition 6. 3) the notion of what it means for a function f € A p to be Singular with respect to the weight function JP, and our first result (Theorem 6. 6) shows that if a closed ideal of Ap is not determined by its local ideals, then every function in the ideal must be singular in this sense. The remainder of the section is then devoted to obtaining conditions on p which guarantee that no elements of Ap have such singularities, and we then apply these conditions to obtain, in particular, the examples referred to above. Finally, in the last section we describe how our method may be applied to the topological ^Ip-modules jfl = (^4p)7 m 5: l, to describe their closed submodules. Here the results of [16] allow us to reduce the case m = i. II. Preliminary results The method we use requires some results from [12] and [16], which are summarized in part here. Given Ω and p s in the introduction, let f : ί2-> Cbe a Lebesgue-measurable function. We shall say that: (i) f 6 Wp if there exists C = C(f) Ξ> 0 such that / | f \\e^dm < + oo, ' and (ii) f € Bp if there exists C = C(f) ^ 0 such that ess^sup (| f(z) \\ 2 subsets of Ω with Dl c int (D2) such that for some C > 0 (1) f l g l I I F \\\\~~@+)e~*dm < + oo. (2) There exist A 1 ? . . . , hN analytic on a neighborhood of D2 such that g = h.F, + · · · + hNFN inside D2 while DlD\\i\\\\\\ Then gtl. Kelleher and Taylor> Ciosed ideals in kcally convex algebras 193 Finally, we comment on the topology of Ap. There are two natural ways to put a topology on Ap. First, for each C > 0 let Apt0 denote the set of all /*€ Ap for which / | / | 2 e x p ( — Cp)dm < + oo, so Ap — U AptC. Since each Apt0 has the structure OO of a Hubert space, Ap can be given a locally convex topology δs the inductive limit of these spaces. Secondly, Ap can be given a topology by the family of seminorms defined δs follows: Let Jf = Jf(p) denote the family of all continuous functions k > 0 defined on for which k(z) exp ( — Cp(z)^)-> + oo δs z~> SO for all C > 0. Then for each such A, defines a seminorm on Ap. Under the conditions on p imposed above, it is a consequence of Theorem l of [26] that these two topologies coincide. (Actually, the proof in [26] applies only to the case = C, but it can be modified so δs to include the rings Ap considered here.) Note that the topology on Ap is never a metric topology. However, Ap is the dual of a separable Frechet space, so a corollary of the Banach-Dieudonno Theorem ([17], p. 273) implies that a convex set in Ap is closed if it is sequentially closed. Subsets S of Ap are bounded if and only if there exist constants C^ C2 > 0 such that for all /\"€ S we have z € . In the following we shall consider only convergence of sequences of functions of Ap. It is not difficult to show that a sequence {hj}i^1 of elements of Ap converges if and only if : (1) The sequence converges uniformly on compact subsets of ί, and (2) The elements of the sequence lie in some bounded subset of Ap. Evidently the limit of a convergent sequence is again an element of Ap. The ring multiplication is jointly continuous in the topology on Ap. HL Dense ideals in Ap We first consider the question of density of ideals and ask when every element of Ap can be sequentially approximated by elements of a given ideal /. Evidently, this is the case if and only if g = l can be so approximated. In order that this be possible it is clearly necessary that / be a free ideal (that is, the elements of / have no

Minimally n-line connected graphs.

Abstract: The following result is proved. For n ̂ 2, every n-line connected graph contains a line whose removal results in a graph which is also n-line connected or a point of degree n.

On the Brun-Titchmarsh theorem.

Abstract: where π(χ\\ α, k) denotes s is customary the number of primes not exceeding χ that are eongruent to a, modulo k. This theorem, which is now commonly known s the BrunTitchmarsh theorem, has subsequently had many other applications in the theory of numbers mainly because, s Titchmarsh himself remarked in his paper, the r nge of validity of the result in terms of k is much wider than that of the corresponding asymptotic formulae that can be obtained by analytic methods. Indeed, despite recent advances in the theory of prime numbers associated with Bombieri's theorem [1], no other known 1 results give effective Information for k 3g x even if one include the formula

Some criteria for weak compactness.

Abstract: In [2] W. F. Eberlein has given a criterion for weak eompactness in Banach spaces, which has been extended by A. Grothendieck for locally convex spaces, quasi-complete for the topology of Mackey [3]. V. Smulian proves in [5] that a closed convex set A, of a normed space, is weakly compact if and only if every decreasing sequence of non-empty closed convex sets of A has non-empty intersection. J. Dieudonne proves that the theorem of Smulian is also true for separated, locally convex spaces, which are quasi-complete for the topology of Mackey [1]. In this paper we prove that the criteria of W. F. Eberlein and V. Smulian can be used in a wide class of locally convex spaces, more extensive than the class of the spaces which are quasi-complete for the topology of Mackey. The vector spaces used here are of non-zero dimension and they are defined over the field K of real or complex numbers. The topologies on them are separated. If £ is a locally convex space we denote, s usual, by E' and E* its topological and algebraic dual respectively.

A structure theory for isometric representations of a class of semigroups.

Abstract: Using Mackey's theory of Systems of imprimitivity, we develop a structure theory for isometric representations of subsemigroups of the nonnegative real numbers. This structure theory extends the well known results of Helson and Lowdenslager concerning those isometric representations of the nonnegative real numbers which arise naturally in the study of stationary stochastic processes and prediction theory.

On units and fundamental units.

Abstract: Next to the ingenious revelation by Dedekind about the exact number of members in a System of fundamental units in an algebraic number field, very little is yet known about these bricklayers of higher arithmetics but some numeric examples, äs were so laborously construcjed by previous authors like Voronoi [10], Daus [4], Billevic [3] and more recently by Wada [11]. It was not before some ten years ago when the author [1] revived the Algorithm of Jacobi [6] and its generalization by Perron [8] that substantial progress was made jointly by Helmut Hasse and the author in calculating units of algebraic number fields of any degree. The Generalized Jacobi-Perron Algorithm, äs it was called by the author, offered a convenient method to solve this challenging problem which proved to be expecially effective for cubic fields, äs was shown recently by the author [1]. In [2] Hasse and the author proved the theorem: Let

Die 2-Klassenzahlen spezieller quadratischer Zahlkörper.

Abstract: Im folgenden wird eine Tafel der engeren 2-Klassenzahlen h solcher quadratischen Zahlkörper k = J2(]/<^) vorgelegt, deren Diskriminante d mit | d \\ < 8000 genau zwei verschiedene Primteiler besitzt. In [1] entwickelte ich im Anschluß an einige Arbeiten von Hasse [2], [3], [4], [5] ein Kriterium, nach dem mittels der ganzrationalen Lösung einer indefiniten quadratischen Form entschieden werden kann, ob für n ̂ 3 die Klassenzahl h von k durch 2 teilbar ist, falls die Teilbarkeit durch 2\"\" bekannt ist. Dieses Kriterium bildet die Grundlage für ein ALGOL-Programm, mit welchem ich — abgesehen von einigen von Hand bearbeiteten Sonderfällen — die vorliegenden Ergebnisse auf der EL-X8 des Rechenzentrums der Universität Karlsruhe errechnete. Zum Programm selbst ist folgendes zu bemerken:

Einheiten für eine allgemeine Klasse total reeller algebraischer Zahlkörper.

Abstract: Fällt die Entwicklung eines Systems algebraischer Irrationalitäten vom Grade n nach dem Jacobi-Perron-Algorithmus (JPA) oder dem Modifizierten Jacobi-PerronAlgorithmus (M JPA) periodisch aus, so lassen sich explizit algebraische Einheiten berechnen. L. Bernstein und H. Hasse haben in mehreren Arbeiten [5], [7], [8], [17]) solche Einheiten bestimmt. Offen bleibt jedoch die Frage, ob die gefundenen Einheiten in einer mit dem Ausgangssystem zusammenhängenden Ordnung des zugehörigen algebraischen Zahlkörpers Grundeinheiten sind. Schon in einer früheren Arbeit [19] ist der Verfasser dieser Frage nachgegangen und hat in einer vom Algorithmus unabhängigen Untersuchung für einige unendliche Klassen reiner kubischer Zahlkörper eine Vermutung von Bernstein und Hasse bestätigen können: Von zwei Ausnahmen abgesehen stimmte die Einheit aus dem Algorithmus mit der Grundeinheit des Körpers überein. Ausgangspunkt der Betrachtungen in der vorliegenden Arbeit) ist das Polynom

Über die Fundamentalgruppe von Körpern mit Divisorentheorie.

Abstract: In [5] hat Ishida gezeigt, wie man durch Betrachten der Diskriminante eines algebraischen Zahlkörpers Teilbarkeitsaussagen für seine Klassenzahl gewinnt und wie man dadurch (ganz elementar) Zahlkörper konstruieren kann, deren Klassenzahl durch eine vorgegebene Zahl teilbar ist. In [6] hat Madan gezeigt, wie man Zahlkörper konstruiert, deren Klassengruppe eine vorgegebene abelsche Gruppe enthält. Beide Arbeiten benutzen dieselbe Methode, sie betrachten zahm verzweigte Kreiskörper über Q und sehen nach, wann diese über einem Zahlkörper unverzweigt werden. Mit etwas mehr Aufwand hat Madan sein Resultat auch auf Funktionenkörper einer Variablen über endlichem Konstantenkörper übertragen, wobei allerdings Gharakteristikschwierigkeiten nicht ganz aus dem Weg geräumt wurden. In der vorliegenden Arbeit sollen die geschilderten Ansätze weitergeführt und durch Axiomatisierung der elementare Charakter stärker betont werden. Zunächst wird die Klassengruppe eines globalen Körpers als seine abelsche Fundamentalgruppe aufgefaßt. Dadurch erhalten wir auch Verbindung zur algebraischen Geometrie, in der die Fundamentalgruppe ein wichtiger Begriff ist. In § l definieren wir, was wir unter einer Divisorentheorie eines Körpers verstehen, und bilden unter diesen Minimalvoraussetzungen die wichtigsten Begriffe, u. a. den der Fundamentalgruppe. In § 2 wird die Fragestellung von Madan verallgemeinert: Es werden Körper konstruiert, deren Fundamentalgruppe eine vorgegebene Faktorgruppe hat. Im Fall von p-Gruppen bei Ghar. p und im abelschen Fall werden gesonderte Konstruktionen angegeben, die zu besseren Resultaten führen. Der § 3 schließt sich enger an die Arbeiten [5] und [6] an, es wird untersucht, wann Erweiterungen von K über einem festen Oberkörper unverzweigt werden. Insbesondere wird Satz l aus [5] elementarer bewiesen, und eine kürzere (allerdings Klassenkörpertheorie benutzende) Konstruktion für die Ergebnisse aus [6] im Funktionenkörperfall gegeben. In § 4 schließlich wird die abelsche Fundamentalgruppe im geometrischen Fall von unserem axiomatischen Standpunkt aus behandelt. Dies bringt keine eigentlich neuen Ergebnisse, man stößt aber auf ein unserer Kenntnis nach ungelöstes Problem über Kurven bei Charakteristik p.

Die Primteilerdichte von ganzzahligen Polynomen. I.

Abstract: 1. Problemstellung ....................................... 176 2. Zusammenhang zwischen Primteilermengen ganzzahliger und irreduzibler normierter Polynome . . . .176 3. Dichte und Durchschnitt von Tschebotareff -Mengen ....................... 177 4. Die Primteilermenge irreduzibler normierter Polynome ...................... 181 5. Die Primteilerdichte ganzzahliger Polynome ........................... 181 6. Durchschnitte von Primteilermengen ganzzahliger Polynome .................... 182 7. Beispiele .......................................... 182 Literatur .......................................... 185

The lowest zero of sections of the zeta function.

Abstract: N Lei fjv(s) = Σ ~~*> s = a-\\-it.lt is shown that for N sufficiently large ζ N (s) has a simple zero with n=l least positive t. Close asymptotic bounds are given for the location of this zero.

Homomorphismen projektiver Räume und Hjelmslevsche Geometrie.

Abstract: Die Untersuchung von Geometrien, in denen Verbindung und Schnitt zweier Elemente i. a. nicht eindeutig bestimmt sind, wurde von J. Hjelmslev angeregt [5]. In seinen Arbeiten zur allgemeinen Kongruenzlehre [6] schränkt er das Schnittpunktaxiom, das die Eindeutigkeit des Schnittpunktes zweier Geraden fordert, auf orthogonale Geraden ein. Durch den Verzicht auf metrische Begriffe konnte Klingenberg den Hjelmslevschen Ansatz wesentlich verallgemeinern. Die von ihm untersuchten ,,Projektiven Räume mit Homomorphismus\" sind Hjelmslevsche Geometrien, die sich analytisch als Geometrien über lokalen Ringen charakterisieren lassen [9]. Die vorliegende Arbeit stützt sich in vielen Punkten auf die Arbeiten von Klingenberg. Die hier konstruierten Geometrien gehören wie auch Klingenbergs „Projektive Räume mit Homomorphismus\" zur reinen Inzidenzgeometrie, Untersucht werden daher ausschließlich Inzidenzeigenschaften. Fragen der Metrik oder der Anordnung bleiben unberücksichtigt. Den Ausgangspunkt der Konstruktion bildet ein Homomorphismus : P -> P', x->x, der projektiven Räume P und P'. Mit Hilfe des zu gehörigen Stellenringes [10] kann in der Translationsgruppe T (zu einer fest gewählten Hyperebene h) .eine Filtration

On the Cartan-Thullen Theorem for some subalgebras of holomorphic functions in a locally convex space.

Abstract: In this paper we prove some theorems which contain the results stated in [1], [2] and [3] äs special cases. Let E be a complex Hausdorff locally convex space. If U is a non-empty open subset of E, H (U) denotes the algebra of all complex holomorphie functions in U. For the general theory of this space see [4] and [5], We use the notations of these references. Let ̂ be a family of closed subsets of U with the following properties: (i) If B € ̂ and K is a finite dimensional compact subset of E such that B + K < t/, then B + K 3&. (ii) For every B € 3$ there is an open balanced convex neighborhood VB of zero in E such that B + VB < U. Let S(&) be a subalgebra of H (U) such that:

Compactness of the inverse of the minimal operator for a class of ordinary differential expressions.

Abstract: Physically, a self-adjoint ordinary differential operator which has a purely discrete spectrum is interesting because it corresponds to a quantum mechanical observable which has a complete orthonormal set of eigenfunctions in the Hubert space. Also, for such an observable, its eigenvalues, the values the observable may have in states where it can be measured exactly, do not cluster. We shall work with a large class K(I), defined in section 2, of ordinary differential expressions which have this property. The harmonic oscillator is one such expression. One of our theorems, Theorem 3, is devoted to showing that the class K ( I ) is larger than it might at first appear. Theorem 3 contains many known results s special cases, but is not contained in any results known to this author. For a classically self-adjoint differential expression Γ, on an interval of type [0, σο), the existence of a self-adjoint operator H in L2(I) associated with T such that H has a purely discrete spectrum is equivalent to the complete continuity of the inverse of the minimal operator. (This inverse is not defined in all of L2(/).) If / is (—σο, σο), the minimal operator on each half interval must have a completely continuous inverse. We shall thus concentrate our attention on establishing the last result for our class. K ( I ) includes many non self-adjoint expressions, also, and the inverse of the minimal operator on each half line is completely continuous for these s well. This fact, while it does not have a quantum mechanical Interpretation, is nevertheless useful because operators associated with these expressions are very stable under perturbations. Special cases of some of our results can be found in Naimark [5] and Dunford and Schwartz [1]. Our methods of proof seem quite different from the methods employed in these books.

A summability integral.

Abstract: In [3] a weak characterization of the Wirkfelder of certain summability methods was obtained by using techniques of Integration theory, though no integral was developed. In this paper a summability integral is developed for positive convergence preserving methods and integral techniques are used in an effort to obtain more meaningful characterizations of the Wirkfelder. The general approach parallele the development in [2], eh. 3 sect. 2, but differs radically at certain critical points which will become clear in the sequel. In section 2 we define a summability field and an integral over this field. Due to the distinction between a summability field and a field we show that the usual integral approach to define a norm does not appear to be the proper one, and subsequently a norm which has the character of an integral norm is defined and used to define a Lebesguetype space of sequences. Several dominated convergence theorems are obtained. In section 3, this JLebesgue space is shown to be contained in the Wirkfeld. Also, a characterization of the Wirkfeld for Cl is given in terms of an extended integral and it is shown that this particular characterization does not extend significantly beyond CL. The final section of the paper contains a Riesz representation theorem, a general dominated convergence theorem for the Wirkfeld s opposed to the Lebesgue space, and assorted questions which appeared to the authors to be interesting and desirable to have answered.