A new kernel method for the uniform approximation in reproducing kernel Hilbert spaces

We are concerned with the uniform approximation of functions of a generic reproducing kernel Hilbert space (RKHS). In this general context, classical approximations are given by Fourier orthogonal projections (if we know the Fourier coefficients) and their discrete versions (if we know the function values on well-distributed nodes). In case such approximations are not satisfactory, we propose to improve the approximation using the same data but combined with a new kernel function. For the resulting (both continuous and discrete) new approximations, theoretical estimates and concrete examples are given.