On the construction of biorthogonal systems for subspaces generated by integral shifts of a single function (Q2343882)

A method is proposed to construct a base dual function for a biorthogonal system by solving an equation of convolution type. This construction is carried out for two base functions: the Gaussian function \(\varphi (t) = \exp (- t^2 / (2\sigma^2))\) and the Lorentz function \(\varphi (t) = \sigma^2 / (\sigma^2 + t^2)\). Explicit formulae are obtained for the Fourier transform of the resulting families of base dual functions, which can be used in computational algorithms.