Mortini, Raymond; Rupp, Rudolf A constructive proof of the Nullstellensatz for subalgebras of \(A(K)\). (English) Zbl 0735.30040 Séminaire de mathématique de Luxembourg, Trav. Math. 3, 45-49 (1991). [For the entire collection see Zbl 0728.00002.]Let \(A(K)\) be the Banach algebra of all continuous complex valued functions on a compact set \(K\subset\mathbb{C}\), which are analytic in the interior \(K^ 0\) of \(K\). The authors prove, in an elementary and constructive way, the following version of the Nullstellensatz for subalgebras \(A\) of \(A(K)\) (it can also be considered as a form of the Corona theorem): Assume that \(A\) satisfies the conditions: (1) Each continuous function on \(\partial K\) can be approximated uniformly by rational functions with poles off \(\partial K\), (2) A contains the polynomials, (3) \((f-f(z_ 0))/(z-z_ 0)\in A\) whenever \(f\in A\) and \(z_ 0\in K^ 0\). Then, if \(f_ 1,\dots,f_ N\in A\) have no common zero in \(K\) then there exist functions \(g_ 1,\dots,g_ N\in A\) such that \(\sum_{i=1}^ N g_ i f_ i\) has no zero in \(K\). Reviewer: T.Krasiński Cited in 1 Document MSC: 30H05 Spaces of bounded analytic functions of one complex variable 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces Keywords:Corona theorem; Nullstellensatz for Banach algebra Citations:Zbl 0728.00002 PDFBibTeX XMLCite \textit{R. Mortini} and \textit{R. Rupp}, in: Fourier coefficients of functions with small supports on 0-dimensional compact Abelian groups. . 45--49 (1991; Zbl 0735.30040)