Download Analyse mathématique I: Convergence, fonctions élémentaires by Roger Godement PDF

By Roger Godement

Les deux premiers volumes de cet ouvrage sont consacrés aux fonctions dans R ou C, y compris los angeles théorie élémentaire des séries et intégrales de Fourier et une partie de celle des fonctions holomorphes. L'exposé, non strictement linéaire, mix symptoms historiques et raisonnements rigoureux. Il montre los angeles diversité des voies d'accès aux principaux résultats afin de familiariser le lecteur avec les méthodes de raisonnement et idées fondamentales plutôt qu'avec les suggestions de calcul, aspect de vue utile aussi aux personnes travaillant seules.
Les volumes three et four traiteront principalement des fonctions analytiques (théorie de Cauchy, théorie analytique des nombres et fonctions modulaires), ainsi que du calcul différentiel sur les variétés, avec un court docket exposé de l'intégrale de Lebesgue, en suivant d'assez près le célèbre cours donné longtemps par l'auteur à l'Université Paris 7.
On reconnaîtra dans ce nouvel ouvrage le kind inimitable de l'auteur, et pas seulement par son refus de l'écriture condensée en utilization dans de nombreux manuels.

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Additional info for Analyse mathématique I: Convergence, fonctions élémentaires

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Next we show that sense. an! 2. Let p E (0, 1) and f E 1l with f(z) = L::=o anzn. Given a sequence {Cn}, Cn ~ 0, one can choose coefficients an > 0 such that L::=o n 1-Pa;cn < oo but fw ¢. QP for any choice of w. Proof. Let {cn} be a sequence of positive constants decreasing monotonically to 0. Choose integers nk which satisfy: (i) no= 1, (ii) nk > 2nk-b kEN, 00 (iii) I: c~ 2 < oo. cn -1 -1/2 nk cnk-1 Cnk-1 ::; L c~~l < oo. k=l Notes 33 On the other hand, thus fw ~ VP and hence fw ~ QP for any choice of w.

Of course, it becomes a natural topic to compare QP with Bp. 1. Let p E (1, oo) and let f(z) = f E Bp if and only if E%:o nklakiP < oo. E%:o akznk belong to HG. Then Proof. 1 (i), so we give the key steps of the argument. In fact, if tn = En 3 Eln n]lail 2 and In = {j EN: 2n :::; j < 2n+l }, then one has 00 IIJII~P ~ L 2-n{p-l}t~2. n=O Since the number of the Taylor coefficients ai is at most [loge 2] + 1 when ni E In, t~/2 ~ 2pn L laj IP. n3Eln The above two estimates lead to 00 llfii~P ~ L nklakiP.

1 has its root in [33) where Bergh dealt with those BMO and B functions with nonnegative Taylor coefficients in an approach based on the Fourier series. 1 which displays a close relation among B, BMOA and QP, p E (0, 1) through the value distribution of holomorphic functions under consideration. 1 should be compared with a constructive example of HoilandTwomey [75). For the reader's convenience, we give the construction of the example in the sequel. G- 2v'k-1, put Fk = {2k + j : j = 0, 1, · · ·, m(k)}, where Fo = {0}, and set Ek = {2k + j : j E Uo::;z::;k-lFz}, 34 3.

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