Analytic extension formulas, integral transforms and reproducing kernels (Q2749824)

This is a survey paper on the recent results of the reproducing kernels, linear transforms, analytic extension formulas and representations of analytic functions. As for the reproducing kernels, the author states that the extensibility and representation of \(f\) in the Hilbert space \(H_K\) are established by the reproducing kernel \(K(p,q)\) in terms of \(f(q_j)\), \(q_j\in S\), where \(S\) denotes a countable set such that \(\{K(\cdot, q_j)\); \(q_j\in S\}\) is a basis for \(H_K\). As for the representations of analytic functions in terms of local values by the Riemann mapping functions, the author gives an example of the expansion of the analytic function \(f(Z)=\log(1+Z)\) on \(|Z|<1\) by using the Riemann mapping function \(z=\varphi(Z)=Z/(Z+2)\) from \(\{\operatorname {Re} Z>-1\}\) onto \(\{|z|<1\}\), that is, \(f(Z)=2(\frac{Z}{Z+2})+\frac{2}{3}(\frac{Z}{Z+2})^3+\frac{2}{5}(\frac{Z}{Z+2 })^5+\cdots\) on \(\{\operatorname {Re} Z>-1\}\). Throughout the paper, the author discusses a lot of topics with respect to the analytic extensions.NEWLINENEWLINEFor the entire collection see [Zbl 0963.00016].