Theorems in the additive theory of numbers

Summary This paper extends some earlier results on difference sets andB ₂ sequences bySinger, Bose, Erd?s andTuran, andChowla. This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract No. AF 49 (638)-213. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Problems and methods in partial differential equations

Summary Finite part and logarithmic part of some divergent integrals with applications to theCauchy problem. A Giovanni Sansone nel suo 70^mo compleanno. The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research of the Air Research and Development Command, United States Air Force, through its European Office under Contract AF 61 (056)-86.     

On a cauchy problem with subharmonic initial values

Summary Investigation of the Euler-Poisson-Darboux equation for special initial values. Generalized transformation ofTricomi-Cibrario. Extension of classical theorems ofF. Riesz andP. Montel on mean values of subharmonic functions. This research was supported in part by the United States Air Force under Contract AF 18(600)573 — monitored by the Office of Scientific Research, Air Research and Development Command.     

Some singular cauchy problems

Summary Existence and uniqueness theorems for some generalizedEuler-Poisson-Darboux equations are proved and growth and convexity properties of the solutions are studied for multiply subharmonic initial values. This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. AF 49 (638)-228, at the University of Maryland, and in part by the National Science Foundation, under a fellowship grant.     

A generalization of Riemann's method for partial differential equations

Summary A bilinear divergence identity is obtained, which differs from the usualLagrange divergence identity employed byRiemann. In the case of two independent variables, this new identity is used to unify the treatment ofCauchy's problem for hyperbolic equations, the initial value problem for parabolic equations, and theDirichlet problem for elliptic equations. This research was supported in whole or in part by the United States Air Force under Contract N^o. AF18(600)-573 monitored by the Office of Scientific Research, Air Research and Development Command.     

Über das Riemann-Privalovsche Randwertproblem

Ohne ZusammenfassungHerrn Professor Dr.H. Petersson zum 60. Geburtstag gewidmetThis research was supported by the Air Force Office of Scientific Research.     

Reflection principles for linear elliptic second order partial differential equations with constant coefflcients

Summary Reflection principles, analogous to the classicalSchwarz reflection principle for harmonic functions, are obtained for solutions of linear elliptic second order partial differential equations with constant coefficients. The boundary conditions employed are supposed to be satisfied in a limiting sense only, and do not require (a priori) the existence of derivatives on the boundary. To Mauro Picone on his 70^th birth day. This research was supported in part by the United States Air Force under Contract N^o. AF(600)-573 — monitored by the Office of Scientific Research, Air Research and Development Command. The work of this author was sponsored by the Office of Ordnance Research, U.S. Army, under the Contract DA-36-034-ORD-1486.     

Über die Transformation einer Reihe nachBesselschen Funktionen

Ohne ZusammenfassungHerrn Professor W.Süss zum 60. Geburtstag gewidmetThe preparation of this note was sponsored by the European Office Air Research and Development Command, U.S. Air Force, Project No. AF 61(514)-442.     

Lokalisation bei absoluter Cesàro-Summierbarkeit von Potenzreihen und trigonometrischen Reihen. I

Ohne ZusammenfassungThis research was supported in part (as far as the work ofW. Jurkat is concerned) by the United States Air Force, through the Office of Scientific Research of the Air Research and Development Command.     

Die geometrische Realisierung eines schiefen kartesischen Produktes

Ohne ZusammenfassungDiese Arbeit wurde zum Teil durch Unterstützung durch die United States Air Force ermöglicht (Contract No. AF 18(600)-1383 vom AF Office of Scientific Research, Air Research and Development Command).Herrn Professor Dr.Hellmuth Kneser zum 60. Geburtstag gewidmet     

Nörlund-Verfahren für Funktionen

Ohne ZusammenfassungDem Andenken anIssai Schur gewidmet.This research was supported by the United States Air Force through the Office of Scientific Research, of the Air Research and Development Command, under Contract Number AF 18 (600)–691.     

On a regular cauchy problem for the Euler-Poisson-Darboux equation

Summary The explicit solution of a particularCauchy problem for the n-dimensionalEuler-Poisson-Darboux equation is found. To obtain the solution the method ofM. Riesz is extended to include non self-adjoint equations. Existence and uniqueness are shown. This research was supported in part by the United States Air Force under Contract No. AF18(600)-573 — monitored by the Office of Scientific Research, Air Research and Development Command.     

Series expansions of solutions of the heat equation inn dimensions

Summary This paper is an extensions of a recent result byD. V. Widder andP. C. Rosenbloom concerning the expansion of a solution u(x, t) of the heat equation, u_x ^x=u_t, in series of polyonomial solutions. Whereas that result involved two variables only, the present one permits an arbitrary finite number of variables. It is found that the polynomial solutions of the heat equation in n dimensions are factorable into polynomials in two variables. The expansion problem may be considered as an investigation of the space spanned by linear combinations of these factorable solutions To Enrico Bompiani on his scientific Jubilee This research was supported (in part) by the U. S. Air Force, through the Air Force Office of Scientific Research, Air Research and Development Command, under Contract AF 49(638)-574. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Problems and methods in partial differential equations

Summary The method of singularities is used to solve theCauchy problem for simple hyperbolic partial differential equations, namely, the wave equation and the damped wave equation. The representation formula for the solution of theCauchy problem is written in terms of finite parts and logarithmic parts of certain divergent integrals. A process of analytic continuation is also used to solve theCauchy problems under consideration. However, to obtain explicitly the representation formulas for the solutions, one must actually perform the analytic continuation. It is shown that this is best achieved by making use of finite and logarithmic parts. Simple examples were purposely chosen so as to show that consideration of finite and logarithmic parts is naturally unavoidable and ? in the very nature of things ?. To Enrico Bompiani on his scientific Jubilee. This work was sponsored in part by the Air Force Office of Scientific Research of the Air Research and Development Command, United States Air Force, through its European Office.     

On banach spaces whose duals areL 1 spaces

A structure theorem for Banach spaces whose duals areL ₁ spaces, is proved. The research of the second named author has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18 through the European Office of Aerospace Research (OAR) United States Air Force.     

Geometric convergence of rational approximations toe −z in infinite sectors

Summary In this paper, we study the geometric convergence of rational approximations toe ^–z in infinite sectors symmetric about the positive real axis.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Florida Research CouncilResearch supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Atomic Energy Commission under Grant AT (11-1)-2075     

The Dirichlet problem for the wave equation

Summary We consider a vibrating string fixed at the ends for which the position is known at two different times. This corrisponds to a classical not well posed problem, theDirichlet problem for the wave equation, which we reconsider here in order to determine under what conditions it is possible to obtain useful information about the physical phenomenon. This problem is related to a functional equation from which the principal results can be deduced.

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Sunto Si riesamina un classico esempio di problema ? non ben posto ?, il problema diDirichlet corrispondente al modello fisico di una corda vibrante, fissa agli estremi e assumente posizioni note in due diversi istanti. Lo studio relativo all'esistenza ed unicità della soluzione è ricondotto a quello di una equazione funzionale ed è strettamente connesso a questioni di teoria dei numeri. Si esamina poi la dipendenza della soluzione dai dati e si discute il problema dal punto di vista delle sue applicazioni.

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This research was supported in part by the United States Air Force under contract No. AF 49 (638) 228 monitored by the Office of Scientific Research, Air Research and Development Command.     

Sur l'approximation par fonctions rationnelles et par fonctions holomorphes bornées

Résumé Nous définissons et étudions (I) une série d'interpolation de fonctions rationnelles, une toute naturelle généralisation de la série deTaylor, qui concerne un domaine multiplement connexe d'un type assez général. Moyennant une représentation conforme nous obtenons ensuite (II) des résultats sur l'approximation dans uu domaine arbitraire multiplement connexe par fonctions holomorphes bornées. Ce dernier sujet n'est pas nouveau [comparer1, 2], mais ces méthodes de la représentation conforme sont plus puissantes que les méthodes antérieures. A Mauro Picone pour son 70^me anniversaire. Recherche subventionnée par U. S. Air Force, Office of Scientific Research of the Air Research and Development Command.     

Ein Einschließungssatz für Minimalflächen

Ohne ZusammenfassungMeinem LehrerErnst Hölder zu seinem 65. Geburtstag in Verehrung gewidmetResearch sponsored by the Air Force Office of Scientific Research under AFOSR Grant Nr. 883–65.     

On extreme points inl 1

It is proved that every bounded closed and convex subset ofl ₁ is the closed convex hull of its extreme points. The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18 through the European Office of Aerospace Research (OAR) United States Air Force.