Analyse mathématique IV: Intégration et théorie spectrale, by Roger Godement PDF

By Roger Godement

ISBN-10: 3540438416

ISBN-13: 9783540438410

Ce 4?me quantity de l'ouvrage Analyse math?matique initiera le lecteur ? l'analyse fonctionnelle (int?gration, espaces de Hilbert, examine harmonique en th?orie des groupes) et aux m?thodes de l. a. th?orie des fonctions modulaires (s?ries L et theta, fonctions elliptiques, utilization de l'alg?bre de Lie de SL2). Tout comme pour les volumes 1 ? three, on reconna?tra ici encore, le type inimitable de l'auteur et pas seulement par son refus de l'ecriture condens?e en utilization dans de nombreux manuels. Mariant judicieusement les math?matiques dites 'modernes' et' classiques', l. a. premi?re partie (Int?gration) est d'utilit? universelle tandis que los angeles seconde oriente le lecteur vers un domaine de recherche sp?cialis? et tr?s actif, avec de vastes g?n?ralisations possibles.

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Additional info for Analyse mathématique IV: Intégration et théorie spectrale, analyse harmonique, le jardin des délices modulaires

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Bn ) be two sequences of positive numbers f, g : [0, ∞) × [0, ∞) → (0, ∞) . 94) for any a, b, k > 0. PROOF We will follow the proof in Mitrinovi´c, Peˇcari´c, and Fink [141, pp. 88–89]. 92). 95) a22 b21 for any ai , bi > 0, i ∈ {1, 2} . 96) for any ai , bi > 0, i ∈ {1, 2} . 96) a, b for a1 , b1 and ka, kb for a2 , b2 (k > 0) , we get 2≤ f (a, b) 2 f (ka, kb) −2 k + k ≤ 2. 93). 96) for a1 = a, b1 = 1, a2 = b, and b2 = 1, we have 2≤ f (a,1) a f (b,1) b + f (b,1) b f (a,1) a ≤ b a + . 94). 92) holds.

78) we choose p = 1, q = n, di = Pi An (p) − Pn Ai (p) and vi = © 2011 by Taylor and Francis Group, LLC 26 Mathematical Inequalities: A Perspective bi (i ∈ {1, . . 77) . The second and third identities are obvious; and we omit the details. Before proving the second result, we need the following interesting identity (see Dragomir [55]): Let p = (p1 , . . , pn ) and a = (a1 , . . , an ) be n-tuples of real numbers. Then we have the result det Pi Pn Ai (p) An (p) n−1 Pmin{i,j} P¯max{i,j} · ∆aj , = j=1 for each i ∈ {1, .

152) can be seen as both a refinement of the arithmetic mean– geometric mean inequality An (p, x) ≥ Gn (p, x) and a counterpart of it. 152) we can also obtain a refinement and a counterpart of the geometric mean–harmonic mean inequality Gn (p, x) ≥ Hn (p, x) as follows. 152) x1i instead of xi and note that x1i ∈ 1b , a1 . 154) for all xi ∈ [a, b], pi ≥ 0 and Pn > 0. 23 Slater’s Inequality for Convex Functions ◦ Suppose that I is an interval of real numbers with interior I and f : I → R ◦ is a convex function on I.

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Analyse mathématique IV: Intégration et théorie spectrale, analyse harmonique, le jardin des délices modulaires by Roger Godement


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