In the sense of kernel which one can use when weighing coefficients together in Discrete Fourier Transform (Dirichlet and Fejér kernels maybe most well known). Here are some examples of such kernels.
Say we have a Taylor expansion for $$\text{ }x \to f(x) \text{ around } x=x_0, \text{ : } p(x) = \sum_{k=0}^{N}c_k(x-x_0)^k$$
When evaluating the Taylor polynomials for this question I was amazed how very localized they can be sometimes. Does there exist any similar concepts of kernel for Taylor series so that we can trade away some local moments for achieving a better fit further away i.e. larger $|x-x_0|$.
Can we derive properties for re-weighting the Taylor coefficients $\{c_k\}$ prior to "inverse transformation" to dampen the sometimes extremely localized behaviour that they posess?
Yes, this kernel is know as the Cauchy kernel and we have the celebrated Cauchy integral formula $$ f(z) =\frac{1}{2\pi i}\int_C \frac{f(w)}{w-z}dw$$ where $C$ is a circle in the complex plane, $\mathbb{C}$, surrounding the point $z$ (traversed in the counter clockwise direction). The formula holds for analytic functions, that is functions that can be expressed as an infinite Taylor expansion.