Interpolation and approximation in Taylor spaces

Barbara Zwi knagl1 and Robert S haba k2 Abstra t: The univariate Taylor formula without remainder allows to reprodu e a fun tion ompletely from ertain derivative values. Thus one an look for Hilbert spa es in whi h the Taylor formula a ts as a reprodu tion formula. It turns out that there are many Hilbert spa es whi h allow this, and they should be alled Taylor spa es. They have ertain reprodu ing kernels whi h are either polynomials or power series with nonnegative oe ients. Consequently, Taylor spa es an be spanned by translates of various lassi al spe ial fun tions su h as exponentials, rationals, hyperboli osines, logarithms, and Bessel fun tions. Sin e the theory of kernel based interpolation and approximation is well established, this leads to a variety of results. In parti ular, interpolation by shifted exponentials, rationals, hyperboli osines, logarithms, and Bessel fun tions provides exponentially onvergent approximations to analyti fun tions, generalizung the lassi al Bernstein theorem for polynomial approximation to analyti fun tions.