Interpolation and approximation in Taylor spaces

The univariate Taylor formula without remainder allows to reproduce a function completely from certain derivative values. Thus one can look for Hilbert spaces in which the Taylor formula acts as a reproduction formula. It turns out that there are many Hilbert spaces which allow this, and they should be called Taylor spaces. They have certain reproducing kernels which are either polynomials or power series with nonnegative coefficients. Consequently, Taylor spaces can be spanned by translates of various classical special functions such as exponentials, rationals, hyperbolic cosines, logarithms, and Bessel functions. Since the theory of kernel-based interpolation and approximation is well-established, this leads to a variety of results. In particular, interpolation by shifted exponentials, rationals, hyperbolic cosines, logarithms, and Bessel functions provides exponentially convergent approximations to analytic functions, generalizing the classical Bernstein theorem for polynomial approximation to analytic functions. Finally, we prove sampling inequalities in Taylor spaces that allow to derive similar convergence rates for non-interpolatory approximations.