On extension to Fourier transforms (Q6115667)

Let \(G\) be a locally compact abelian group and let \(\Gamma\) be its dual group. Denote by \(A(\Gamma)\) the space of Fourier transforms of functions in \(L^1(G)\). For a closed subset \(K\) of \(\Gamma\), we shall write \(C_0(K)\) for the space of complex-valued continuous functions on \(K\) that vanish at infinity. It is known that \(A(\Gamma) \subseteq C_0(\Gamma)\). Suppose \(K\) is a closed subset of \(\Gamma\) and let \(f \in C_0(K).\) An interesting question is: does there exist \(F \in A(\Gamma)\) that coincides with \(f\) on \(K\)? If such a function \(F\) exists, then it is known as an extension of \(f\). Let \(\mathcal{E} \colon C_0(K) \rightarrow A(\Gamma)\) be an extension operator. In [\textit{C. C. Graham}, Stud. Math. 43, 57--60 (1972; Zbl 0214.38101)] it was shown that if \(K\) is an infinite compact subset of \(\Gamma\), then \(\mathcal{E}\) can never be a bounded linear operator. In the present paper the author shows that this result is also true if \(K\) is an infinite closed subset of \(\Gamma\). We briefly describe the proof of this result. For a finite set \(K \subseteq \Gamma\), define \[ \alpha_{\Gamma}(K) := \inf\{ \Vert \mathcal{E} \Vert \colon \mathcal{E} \text{ is a linear extension operator from } \ell^{\infty}(K) \text{ to } A(\Gamma)\}. \] The following result about \(\alpha_{\Gamma}(K)\) is proved in the paper: Let \(\Gamma\) be a locally compact abelian group. If \(K\) is an \(n\)-point subset of \(\Gamma\), then \(\sqrt{n/2} \leq \alpha_{\Gamma}(K) \leq \sqrt{n}\). Now suppose \(K\) is an infinite closed subset of \(\Gamma\) and let \(K_n\) be an \(n\)-point subset of \(K\). It is shown that if there exists a bounded linear extension operator \(\mathcal{E} \colon C_0(K) \rightarrow A(\Gamma)\), then there is a linear extension operator from \(\ell^{\infty}(K_n)\) to \(A(\Gamma)\) whose norm is at most \(\Vert \mathcal{E} \Vert\). However, this contradicts the result given above concerning the growth of \(\alpha_{\Gamma}(K_n)\) as \(n \rightarrow \infty\). This establishes the main result of the paper: Let \(\Gamma\) be a locally compact abelian group. If \(K\) is an infinite closed subset of \(\Gamma\), then there does not exist a bounded linear extension operator from \(C_0(K)\) to \(A(\Gamma)\).