Wiener-type theorems for generalized functions and the Stieltjes transform (Q1594194)

Let \({\mathcal S}_+\) be the space of \(\mathbb{C}^\infty\)-functions on \([0,\infty)\) rapidly decrasing together with all their derivatives, \({\mathcal S}^{a,M}_{b,N,\delta}\) the completion of \({\mathcal S}_+\) by a norm, and \(\rho(k)> 0\) the regularly varying function of order \(\alpha\) satisfying \(\lim_{k\to\infty} \rho(kt)/\rho(t)= t^\alpha\). Theorem 1. The tempered generalized function \(f(t)\in ({\mathcal S}^{a,M}_{b,N,\delta})'\) has a quasi-asymptotics with respect to \(\rho(k)\); i.e., \((f(kt),\phi(t))/\rho(k)\to c_\phi\) as \(k\to\infty\) for \(\forall\phi\in{\mathcal S}_+\), if \(f\) has the bounded quasi-asymptotic with respect to the countable functions \(\phi^\beta_0\) for \(\rho(k)\). Let \[ F(z)= (f(t), \ln^n(c- (t/z))/(c- (t/z))^s)\quad\text{for }z\in\mathbb{C}\setminus [0,\infty). \] Theorem 2. If \(f(t)\in({\mathcal S}^a_b)'\), \(b< a< s-1\), has a quasi-asymptotic with respect to \(\rho(k)\), then \(F(re^{i\beta})/(r\rho(r))\to c^\beta\) as \(r\to\infty\) for \(\forall\beta\in (0,2\pi)\), and there exist \(A\), \(m\) and \(r_0\) such that \[ |F(re^{i\beta})|/(r\rho(r))\leq A/\text{sin}^m(\beta/2) \] hold for \(r> r_0\) and \(\beta\in (0,2\pi)\). The authors formulate the inverse of the Theorem 2 for the case \(n= 1\) and \(s>0\).