Ce mémoire contient quelques résultats sur l'intégration numérique. Ils sont liés à la célèbre formule de quadrature de K. F. Gauss. Une généralisation très intéressante de la formule de Gauss a été obtenue par P. Turán. Elle est contenue dans son article publié en 1948, seulement quelques années après la seconde guerre mondiale. Étant données les circonstances défavorables dans lesquelles il se trouvait à l'époque, l'auteur (Turán) a laissé beaucoup de détails à remplir par le lecteur. Par ailleurs, l'article de Turán a inspiré une multitude de recherches; sa formule a été étendue de di érentes manières et plusieurs articles ont été publiés sur ce sujet. Toutefois, il n'existe aucun livre ni article qui contiennent un compte-rendu détaillé des résultats de base, relatifs à la formule de Turán. Je voudrais donc que mon mémoire comporte su samment de détails qui puissent éclairer le lecteur tout en présentant un exposé de ce qui a été fait sur ce sujet. Voici comment nous avons organisé le contenu de ce mémoire. 1-a. La formule de Gauss originale pour les polynômes - L'énoncé ainsi qu'une preuve. 1-b. Le point de vue de Turán - Compte-rendu détaillé des résultats de son article. 2-a. Une formule pour les polynômes trigonométriques analogue à celle de Gauss. 2-b. Une formule pour les polynômes trigonométriques analogue à celle de Turán. 3-a. Deux formules pour les fonctions entières de type exponentiel, analogues à celle de Gauss pour les polynômes. 3-b. Une formule pour les fonctions entières de type exponentiel, analogue à celle de Turán. 4-a. Annexe A - Notions de base sur les polynômes de Legendre. 4-b. Annexe B - Interpolation polynomiale. 4-c. Annexe C - Notions de base sur les fonctions entières de type exponentiel. 4-d. Annexe D - L'article de P. Turán. / This mémoire contains some results about numerical integration. They are related to the famous quadrature formula of K. F. Gauss. A very interesting generalization of the formula of Gauss was obtained by P.Turán. It is contained in a paper that was published in 1948, only a few years after the second world war. Due to adverse circunstances he was in at the time, the author (Turán) left many details for the reader to fill in. Otherwise, the article of Turán inspired a multitude of research, and his formula has been extended in many ways and several papers have been written on this subject. However, there is no single book or paper where one can nd a clear and comprehensive account of the basic results pertaining to Turán's formula. Thus, I would like my Master's mémoire to contain enough details that can enlighten the reader and present an exposition of much that has been done on this subject. Here is how we have arranged the contents of the mémoire. 1-a. The original formula of Gauss for polynomials - statement along with a proof. 1-b. Turán's point of view - detailed account of the results contained in his paper. 2-a. A formula for trigonometric polynomials analogous to that of Gauss. 2-b. A formula for trigonometric polynomials analogous to that of Turán. 3-a. Two formulae for entire functions of exponential type, analogous to the one of Gauss for polynomials. 3-b. A formula for entire functions of exponential type, analogous to that of Turán. 4-a. Annexe A - Basic facts about Legendre polynomials. 4-b. Annexe B - Polynomial interpolation. 4-c. Annexe C - Basic facts about entire functions of exponential type. 4-d. Annexe D - Paper of P. Turán.

In this thesis, we examined some Dirichlet type problems of the form: \begin{eqnarray*} \triangle u & = & 0\ {\rm in\ } \mathbb{R}^n \\ u & = & f\ {\rm on\ } \psi = 0, \end{eqnarray*} and we were particularly interested in finding entire solutions when entire data was prescribed. This is an extension of the work of D. Siegel, M. Mouratidis, and M. Chamberland, who were interested in finding polynomial solutions when polynomial data was prescribed. In the cases where they found that polynomial solutions always existed for any polynomial data, we tried to show that entire solutions always existed given any entire data. For half space problems we were successful, but when we compared this to the heat equation, we found that we needed to impose restrictions on the type of data allowed. For problems where data is prescribed on a pair of intersecting lines in the plane, we found a surprising dependence between the existence of an entire solution and the number theoretic properties of the angle between the lines. We were able to show that for numbers $\alpha$ with $\omega_1$ finite according to Mahler's classification of transcendental numbers, there will always be an entire solution given entire data for the angle $2\alpha\pi$ between the lines. We were also able to construct an uncountable, dense set of angles of measure 0, much in the spirit of Liouville's number, for which there will not always be an entire solution for all entire data. Finally, we investigated a problem where data is given on the boundary of an infinite strip in the plane. We were unable to settle this problem, but we were able to reduce it to other {\it a priori} more tractable problems.

In this thesis, we examined some Dirichlet type problems of the form: \begin{eqnarray*} \triangle u & = & 0\ {\rm in\ } \mathbb{R}^n \\ u & = & f\ {\rm on\ } \psi = 0, \end{eqnarray*} and we were particularly interested in finding entire solutions when entire data was prescribed. This is an extension of the work of D. Siegel, M. Mouratidis, and M. Chamberland, who were interested in finding polynomial solutions when polynomial data was prescribed. In the cases where they found that polynomial solutions always existed for any polynomial data, we tried to show that entire solutions always existed given any entire data. For half space problems we were successful, but when we compared this to the heat equation, we found that we needed to impose restrictions on the type of data allowed. For problems where data is prescribed on a pair of intersecting lines in the plane, we found a surprising dependence between the existence of an entire solution and the number theoretic properties of the angle between the lines. We were able to show that for numbers $\alpha$ with $\omega_1$ finite according to Mahler's classification of transcendental numbers, there will always be an entire solution given entire data for the angle $2\alpha\pi$ between the lines. We were also able to construct an uncountable, dense set of angles of measure 0, much in the spirit of Liouville's number, for which there will not always be an entire solution for all entire data. Finally, we investigated a problem where data is given on the boundary of an infinite strip in the plane. We were unable to settle this problem, but we were able to reduce it to other {\it a priori} more tractable problems.

We prove several inequalities involving the Fourier transform of functions which are compactly supported. The constraint that the functions have compact support is a simplifying feature which is desirable in applications, but there is a trade-off in control of other relevant quantities-- such as the mass of the function. With applications in mind, we prove inequalities which quantify these types of trade-offs. / text

Ce mémoire contient quelques résultats sur l'intégration numérique. Ils sont liés à la célèbre formule de quadrature de K. F. Gauss. Une généralisation très intéressante de la formule de Gauss a été obtenue par P. Turán. Elle est contenue dans son article publié en 1948, seulement quelques années après la seconde guerre mondiale. Étant données les circonstances défavorables dans lesquelles il se trouvait à l'époque, l'auteur (Turán) a laissé beaucoup de détails à remplir par le lecteur. Par ailleurs, l'article de Turán a inspiré une multitude de recherches; sa formule a été étendue de di érentes manières et plusieurs articles ont été publiés sur ce sujet. Toutefois, il n'existe aucun livre ni article qui contiennent un compte-rendu détaillé des résultats de base, relatifs à la formule de Turán. Je voudrais donc que mon mémoire comporte su samment de détails qui puissent éclairer le lecteur tout en présentant un exposé de ce qui a été fait sur ce sujet. Voici comment nous avons organisé le contenu de ce mémoire. 1-a. La formule de Gauss originale pour les polynômes - L'énoncé ainsi qu'une preuve. 1-b. Le point de vue de Turán - Compte-rendu détaillé des résultats de son article. 2-a. Une formule pour les polynômes trigonométriques analogue à celle de Gauss. 2-b. Une formule pour les polynômes trigonométriques analogue à celle de Turán. 3-a. Deux formules pour les fonctions entières de type exponentiel, analogues à celle de Gauss pour les polynômes. 3-b. Une formule pour les fonctions entières de type exponentiel, analogue à celle de Turán. 4-a. Annexe A - Notions de base sur les polynômes de Legendre. 4-b. Annexe B - Interpolation polynomiale. 4-c. Annexe C - Notions de base sur les fonctions entières de type exponentiel. 4-d. Annexe D - L'article de P. Turán. / This mémoire contains some results about numerical integration. They are related to the famous quadrature formula of K. F. Gauss. A very interesting generalization of the formula of Gauss was obtained by P.Turán. It is contained in a paper that was published in 1948, only a few years after the second world war. Due to adverse circunstances he was in at the time, the author (Turán) left many details for the reader to fill in. Otherwise, the article of Turán inspired a multitude of research, and his formula has been extended in many ways and several papers have been written on this subject. However, there is no single book or paper where one can nd a clear and comprehensive account of the basic results pertaining to Turán's formula. Thus, I would like my Master's mémoire to contain enough details that can enlighten the reader and present an exposition of much that has been done on this subject. Here is how we have arranged the contents of the mémoire. 1-a. The original formula of Gauss for polynomials - statement along with a proof. 1-b. Turán's point of view - detailed account of the results contained in his paper. 2-a. A formula for trigonometric polynomials analogous to that of Gauss. 2-b. A formula for trigonometric polynomials analogous to that of Turán. 3-a. Two formulae for entire functions of exponential type, analogous to the one of Gauss for polynomials. 3-b. A formula for entire functions of exponential type, analogous to that of Turán. 4-a. Annexe A - Basic facts about Legendre polynomials. 4-b. Annexe B - Polynomial interpolation. 4-c. Annexe C - Basic facts about entire functions of exponential type. 4-d. Annexe D - Paper of P. Turán.

If f(z) is an entire function of exponential type, hf(π/2) = 0 and f(z) 0 for Im(z) > 0 then according to a well-known result of R. P. Boas, for, we have. R. P. Boas proposed the problem of obtaining an inequality analogous to this if being real and the answer to this question in the case k < 0 was given by Govil and Rahman. In this paper we present generalizations of these results of Govil and Rahman.

If f(z) is an asymmetric entire function of exponential type t, Both of these inequalities are sharp. In this paper we generalize the above two inequalities of Boas by proving a sharp inequality which, besides giving as special cases the above two inequalities of Boas, yields some other results as well.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

We investigate the famous Laguerre–Pólya class of entire functions and its subclass, the Laguerre–Pólya class of type I. The functions from these classes can be expressed in terms of the Hadamard Canonical Factorization (see Chapter 1, Definition 1.2 and 1.3). The prominent theorem by E. Laguerre and G. Pólya gives a complete description of the Laguerre–Pólya class and the Laguerre–Pólya class of type I, showing that these classes are the respective closures in the topology of uniform convergence on compact sets of the set of real polynomials having only real zeros (that is, the set of so-called hyperbolic polynomials) and the set of real polynomials having only real negative zeros. Both the Laguerre–Pólya class and the Laguerre–Pólya class of type I play an essential role in complex analysis. For the properties and characterizations of these classes, see, for example, [31] by A. Eremenko, [40] by I.I. Hirschman and D.V. Widder, [43] by S. Karlin, [57] by B.Ja. Levin, [66, Chapter 2] by N. Obreschkov, and [74] by G. Pólya and G. Szegö. In the thesis, we study entire functions with positive coefficients and with the monotonic sequence of their second quotients of Taylor coefficients. We find necessary and sufficient conditions under which such functions belong to the Laguerre–Pólya class (or the Laguerre–Pólya class of type I).:List of symbols Introduction 1 Background of research 1 1.1 The Laguerre–Pólya class .................... 1 1.2 The quotients of Taylor coefficients ............... 3 1.3 Hutchinson’s constant ...................... 4 1.4 Multiplier sequences ....................... 4 1.5 Apolar polynomials........................ 8 1.6 The partial theta function .................... 10 1.7 Decreasing second quotients ................... 13 1.8 Increasing second quotients ................... 14 2 A necessary condition for an entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre–Pólya class 15 2.1 Proof of Theorem 2.1....................... 16 2.2 The q-Kummer function ..................... 29 2.3 Proof of Theorem 2.10 ...................... 31 2.4 Proof of Theorem 2.11 ...................... 43 3 Closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre–Pólya I class 49 3.1 Proof of Statement 3.1 ...................... 50 3.2 Proof of Theorem 3.2....................... 53 3.3 Proof of Theorem 3.4....................... 61 3.4 Proof of Theorem 3.6....................... 66 4 Entire functions from the Laguerre–Pólya I class having the increasing second quotients of Taylor coefficients 69 4.1 Proof of Theorem 4.1....................... 70 4.2 Proof of Theorem 4.3....................... 76 5 Number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients 81 5.1 Proof of Theorem 5.1....................... 82 5.2 Proof of Corollary 5.2....................... 88 5.3 Proof of Theorem 5.4....................... 88 6 Further questions 95 Acknowledgements 97 Selbständigkeitserklärung 101 Curriculum Vitae 103 Bibliography 107