Fourier-Laplace transformation of functionals on a weighted space of infinitely smooth functions (Q2710736)

The author of this note states: the dual to a weighted space \(G\) of infinitely smooth functions on the real axis is described by means of the Fourier-Laplace transformation. Let \(M_0= 1,M_1,M_2,\dots\) be an increasing sequence of positive numbers satisfying the following conditions:NEWLINENEWLINENEWLINE(i) \(M^2_k\leq M_{k-1} M_{k+1}\) for each \(k\in\mathbb{N}\);NEWLINENEWLINENEWLINE(ii) there exist constants \(H_1> 0\) and \(H_2> 1\) such that \(M_{k+n}\leq H_1 H^{k+n}_2 M_kM_n\) for all \(k,n\in\mathbb{Z}_+\);NEWLINENEWLINENEWLINE(iii) there exist positive numbers \(Q_1\) and \(Q_2\) such that \(M_k\geq Q_1 Q^k_2 k!\) for each \(k\in\mathbb{Z}_+\).NEWLINENEWLINENEWLINESetting \(M_k= (k!)^s\) or \(M_k= k^{ks}\) for \(s\geq 1\) and \(k\in\mathbb{N}\) we obtain important examples of such sequences. We set \(w(r)= \sup_{k\in\mathbb{Z}_+} \ln{r^k\over M_k}\), \(r> 0\), \(w(0)= 0\). The function \(w\) is continuous on the nonnegative half-axis. Since \(w(r)= 0\) for \(r\in [0, M_1]\), it follows by (iii) that there exists a positive quantity \(A_w\) such that \(w(r)\leq A_wr\), \(r\geq 0\). Clearly, \(w(|z|)\), \(z\in\mathbb{C}\), is a subharmonic function in the complex domain. \(\alpha> 1\) is a fixed number and \(\psi: \mathbb{R}\to[0,\infty)\) is a convex function such thatNEWLINENEWLINENEWLINE(1) there exists a constant \(A_\psi> 0\) such that for all \(x_1,x_2\in\mathbb{R}\) we have NEWLINE\[NEWLINE|\psi(x_1)- \psi(x_2)|\leq A_\psi(1+|x_1|+|x_2|)^{\alpha- 1}|x_1- x_2|;NEWLINE\]NEWLINE (2) \(\lim_{x\to\infty} {\psi(x)\over|x|}= +\infty\).NEWLINENEWLINENEWLINEIf \(g\) is a convex function on the real axis such that \({g(x)\over|x|}\to +\infty\) as \(x\to\infty\), then \(g^*(x)= \sup_{y\in\mathbb{R}}(xy- g(y))\), \(x\in\mathbb{R}\), is the Young transform of \(g\). It is known that \(g^*\) is a convex function, \({g^*(x)\over|x|}\to +\infty\) as \(x\to\infty\), and the conversion formula for the Young transform \((g^*)^*= g\) holds.NEWLINENEWLINENEWLINELet \(\varphi= \psi^*\). As usual, \({\mathcal E}(\mathbb{R})\) is the space of infinitely smooth functions on the real axis with the topology of uniform convergence of all derivatives and \(C(\mathbb{R})\) is the space of continuous functions on the real axis. For an arbitrary domain \(\Omega\) in \(\mathbb{C}\) let \({\mathcal D}(\Omega)\) be the space of infinitely smooth functions with compact support in \(\Omega\), with the usual topology. We denote by \(H(\mathbb{C})\) the space of entire functions with the topology of uniform convergence on compact subsets of the complex plane. For an arbitrary locally convex space \(X\) we denote by \(X^*\) its strong dual.NEWLINENEWLINENEWLINEFor brevity we set \(\varphi_\varepsilon(x)= \varphi(x)- \varepsilon^{-1}\ln (1+|x|)\), \(x\in\mathbb{R}\), \(\varepsilon> 0\). Using the normed spaces NEWLINE\[NEWLINEG(\varphi_\varepsilon)= \Biggl\{f\in{\mathcal E}(\mathbb{R}):\|f\|_{\varphi_\varepsilon}= \sup_{x\in\mathbb{R}, k\in\mathbb{Z}_+} {|f^{(k)}(x)|\over \varepsilon^k M_k\exp(\varphi_\varepsilon(x))}< \infty\Biggr\},\quad \varepsilon> 0,NEWLINE\]NEWLINE we can define the space \(G= \text{lim pr}_{\varepsilon\to 0}G(\varphi_\varepsilon)\) (projective limit).NEWLINENEWLINENEWLINEConsider now the space \(P\) of entire functions defined as the inductive limit of the normed spaces NEWLINE\[NEWLINEP(\varepsilon)= \Biggl\{f\in H(\mathbb{C}): p_\varepsilon(f)= \sup_{z\in\mathbb{C}} {|f(z)|\over\exp(\psi(\text{Im }z)+ w(\varepsilon^{-1}|z|))}< \infty\Biggr\},\quad \varepsilon> 0.NEWLINE\]NEWLINE The Fourier-Laplace transform \(\widehat T\) of a functional \(T\in G^*\) is defined by the formula \(\widehat T(z)= T(e^{-ixz})\), \(z\in\mathbb{C}\). The central result of this paper is as follows.NEWLINENEWLINENEWLINETheorem. The Fourier-Laplace transformation establishes a topological isomorphism between the space \(G^*\) and \(P\).