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Restricted range simultaneous approximation and interpolation with preservation of the norm

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Restricted range simultaneous approximation and interpolation with preservation of the norm ANNALES MATHÉMATIQUES BLAISE PASCAL J.B. PROLLA S. NAVARRO Restricted range simultaneous approximation and interpolation with preservation of the norm Annales mathématiques Blaise Pascal, tome 2, no 1 (1995), p. 225-235. <http://www.numdam.org/item?id=AMBP_1995__2_1_225_0> © Annales mathématiques Blaise Pascal, 1995, tous droits réservés. L’accès aux archives de la revue « Annales mathématiques Blaise Pascal » (http:// math.univ-bpclermont.fr/ambp/), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ RESTRICTED RANGE SIMULTANEOUS APPROXIMATION AND INTERPOLATION WITH PRESERVATION OF THE NORM J.B. Prolla and S. Navarro Ann. Math. Blaise Pascal, Vol. 2, N° 1, 1995, pp.225-235 Abstract. Let (F. ; ~ ~ ) be a complete non-archimedean non-trivially valued division ring. w ith valuation ring V. Let X be a compact 0-dimensional Hausdorff space. and let D(X) be the ring of all continuous functions f from X into V equipped with the supremum norm. Let A C D(X). Assume that for every ordered pair (s. t) of distinct elements of X. there is some multiplier of A; say ~;. such that = 1 and = 0. Assume that A contains the constants. We show that A is uniformly dense in D(X ). and when A is an interpolating family then simultaneous approximation and interpolation, with preservation of the norm, by elements of .4 is always possible. We apply this to the case of von Neumann subsets and to the case of restricted range polynomial algebras. 1991 Mathematics subject classification: 46S10. 1. Introduction Throughout this paper X is a compact Hausdorf

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