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Approximation de fonctions holomorphes d’un nombre infini de variables. (Approximation of holomorphic functions of infinitely many variables.). (French) Zbl 0944.46046
The author’s motivation comes from the following open problem:
Let \(B(R)\subset X\) be the \(R\)-ball centered at \(0\) in a complex Banach space \(X\), let \(f:B(R)\to \mathbb{C}\) be a holomorphic function, and let \(0< r< R\). Given \(\varepsilon> 0\), does there exist an entire function \(g\) on \(X\) such that \(\|f- g\|_{B(r)}< \varepsilon\)? (Note that \(f\) may well be unbounded on \(B(R)\), so that \(f\) is not in general uniformly approximable by partial sums of its Taylor series.)
In the first part of this note, the author solves a special case of this problem: Let \(M\) be a Stein (finite-dimensional) variety, let \(K\subset M\) be a holomorphically convex compact set, and let \(G\subset M\) be an open precompact neighborhood of \(K\). Further, let \(X\) and \(V\) be locally convex spaces, and let \(f\in{\mathcal O}(G\times X,V)\), that is a \(V\)-valued holomorphic mapping on \(G\times X\).
Theorem: For any bounded subset \(B\subset X\), any continuous seminorm \(p\) on \(V\), and any \(\varepsilon> 0\), there is \(g\in{\mathcal O}(M\times X,V)\) such that \(\sup_{K\times B}p(f- g)< \varepsilon\).
The proof makes use of work of L. Bungart [Trans. Am. Math. Soc. 111, 317-344 (1964) and ibid. 113, 547 (1964; both Zbl 0142.33902)]. The major part of this paper deals with the case \(X= \ell^1(\Gamma)\).
Let \(M\) be a Stein variety, let \(R: M\to (0,\infty)\) be continuous, and let \(\Omega= \{(\zeta, z)\in M\times \ell^1(\Gamma):\|z\|< R(\zeta)\}\). Let \({\mathcal C}_{\mathcal O}(\Omega, V)\) be the space of continuous functions \(f:\Omega\to V\) such that for each \(\zeta\in M\), \(f(\zeta,\cdot)\) is holomorphic. After describing the monomial development of elements of \({\mathcal C}_{\mathcal O}(\Omega, V)\) [cf. R. A. Ryan, Trans. Am. Math. Soc. 302, 797-811 (1987; Zbl 0637.46045) and the author’s paper, J. Am. Math. Soc. 12, No. 3, 775-793 (1999; Zbl 0926.32048)], the author proves the following approximation theorem.
Let \(K\) be a compact subset of \(M\) which is \({\mathcal O}(M)\)-convex, let \(M'\subset M\) be an open neighborhood of \(K\), and let \(r,R: M'\to (0,\infty)\) be continuous functions such that \(r< R\). Let \(\Omega= \{(\zeta, z)\in M'\times \ell^1(\Gamma):\|z\|< R(\zeta)\}\) and \(A= \{(\zeta, z)\in \Omega:\|z\|\leq r(\zeta)\}\).
Theorem. Let \(V\) be a locally convex space and let \(p\) be a continuous seminorm on \(V\). For any \(f\in{\mathcal O}(\Omega, V)\) and any \(\varepsilon> 0\), there is \(g\in{\mathcal O}(M\times \ell^1(\Gamma), V)\) such that \(\sup_Ap(f- g)<\varepsilon\).

MSC:
46G20 Infinite-dimensional holomorphy
32A05 Power series, series of functions of several complex variables
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
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References:
[1] L. BUNGART, Holomorphic functions with values in locally convex spaces and applications to integral formulas, Trans. Amer. Math. Soc., 111 (1964), 317-344. · Zbl 0142.33902
[2] L. BUNGART, Errata to volume 111, Trans. Amer. Math. Soc., 113 (1964), 547. · Zbl 0142.33902
[3] S. DINEEN, Complex analysis in locally convex spaces, North Holland, Amsterdam, 1981. · Zbl 0484.46044
[4] S. DINEEN, Complex analysis on infinite dimensional spaces, Springer, Berlin, 1999. · Zbl 1034.46504
[5] N. DUNFORD, T. SCHWARTZ, Linear operators I., John Wiley & Sons, New York, 1988.
[6] L. LEMPERT, The Dolbeault complex in infinite dimensions, II, à paraître, J. Amer. Math. Soc. · Zbl 0926.32048
[7] P. NOVERRAZ, Pseudo-convexité, convexité polynomiale et domaines d’holomorphie en dimension infinie, North Holland, Amsterdam, 1973. · Zbl 0251.46049
[8] R.A. RYAN, Holomorphic mappings in l1, Trans. Amer. Math. Soc., 302 (1987), 797-811. · Zbl 0637.46045
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