Holomorphic $$L^2$$ Signals of Several Complex Variables

Following the line of thought by P. Bouboulis & S. Theodoridis, [6], we take up a program of recovering kernel methods (as employed in signal analysis and machine learning theory) from real RKHS and kernels, to the complex domain. We solve the maximum problem sup{∑j=1p|f(zj)|2:‖f‖2≤E} in the complex RKHS of holomorphic L2 functions f:Ω→C, for any bounded domain Ω⊂Cn and any finite set of points z1,⋯,zp∈Ω, and apply the result to the space L2H(Bn) of holomorphic L2 functions on the unit ball Bn⊂Cn. We produce sampling expansions of functions f∈L2H(Ω) associated to infinite sequences {ζk}k≥0⊂Ω, by starting from complete orthonormal systems {ϕν}ν≥0⊂L2H(Ω) and approximating each ϕν uniformly on Ω by a linear combination of reproducing kernels. The means to said approximation are provided by the Faber-Kaczmarz-Mycielski algorithm A(h) learning (cf. [23]) from the data {(ζk,ϕν(ζk))}k≥0 and producing an approximating sequence {Jkϕν}k≥0⊂L2H(Ω).