Analyse mathématique III: Fonctions analytiques, by Roger Godement

By Roger Godement

Ce vol. III disclose l. a. théorie classique de Cauchy dans un esprit orienté bien davantage vers ses innombrables utilisations que vers une théorie plus ou moins complète des fonctions analytiques. On montre ensuite remark les intégrales curvilignes à l. a. Cauchy se généralisent à un nombre quelconque de variables réelles (formes différentielles, formules de variety Stokes). Les bases de l. a. théorie des variétés sont ensuite exposées, principalement pour fournir au lecteur le langage "canonique" et quelques théorèmes importants (changement de variables dans les intégrales, équations différentielles). Un dernier chapitre montre touch upon peut utiliser ces théories pour construire l. a. floor de Riemann compacte d'une fonction algébrique, sujet rarement traité dans los angeles littérature non spécialisée bien que n'éxigeant que des strategies élémentaires. Un quantity IV exposera, outre, l'intégrale de Lebesgue, un bloc de mathématiques spécialisées vers lequel convergera tout le contenu des volumes précédents: séries et produits infinis de Jacobi, Riemann, Dedekind, fonctions elliptiques, théorie classique des fonctions modulaires et l. a. model moderne utilisant los angeles constitution de groupe de Lie de SL (2, R).

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6. oo JRmg(x)ei(x'Y)dx = o. Proof. 2 with f = e i '2>jYj (lIflloo = 1) to each of the 2m octants eR+. Assuming, without loss of generality, that Y > 0, we obtain 11 yJ meas[O, [O ,y] f(x) dx = m /1- } ] Yj 1 0 Yj etCj Xjdxj < . / - 2 maxj IYj I • which tends to zero at infinity. 1. Prove the following lemma due to 1. Fejer. OO Hint. 2. 1 211' f(,Xt) g(t) dt = (211")-1 1211' f(t) dt 1211' g(t) dt. 2. 4 for ,X ---* +00. 3. 1. 2 by substituting K by its Taylor polynomial. This is just the way K. Weierstrass proved the theorem.

And = r~I-0 IIh - Irllp --t 0 as r --t 1 - O. If f(z) Proof. 13). Taking into account that Hp(D) C HI (D) for p > 1, we further assume that p E (0,1]. 9 we have h(z) = gP/2(z) E H2(D). e. e. on T provided f(O) "# O. Suppose that hI = 0 on a set of positive Lebesgue measure. 2), one can assume that hr as r --t 1 - 0, uniformly on this set. 5 yields f(O) = 0 which contradicts the assumption. e. e. The proof of the convergence in Lp-norm reduces to the case p > l. gp = II'ljJII}/:p = II fll Hp . i: Since Ir - h = (r - 1)'ljJr Ifr - hl Pdt ~ + ('ljJr - 'ljJdl' i: we have lr - II P l'ljJrl Pdt + l'ljJr - 'ljJII P ld Pdt , and applying the Cauchy-Schwarz-Bunyakovskii inequality yields lIr - 111~plI'ljJrll~p + 1III1~pll'IjJr - 'ljJl11~p < 1I/1I;:2P (lIr - 111~p + lI'ljJr - 'ljJll1~p)· FO URIER SERIES 48 .

5, see Stein and Weiss [M-1971], Ch II, §3. 9, see Stein [1961b]' where the result is proved for compact groups. 8 is due to A. P. 2, Ch. 22). The condition 1 ::; p ::; 2 is necessary, since the theorem fails for p > 2 (see Stein [1961b]). Chapter 2 FOURIER SERIES In this chapter trigonometric Fourier series are studied. Convergence and divergence problems, the Fejer and the Abel-Poisson summability methods, and the association of the Fourier series problems with those for the Hardy spaces in the circle are among the subjects.

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