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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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