  In this article, we solve a deterministically generalized interpolation
problem by a stochastic approach. We introduce a kernel-based probability
measure on a Banach space by a covariance kernel which is defined on the dual
space of the Banach space. The kernel-based probability measure provides a
numerical tool to construct and analyze the kernel-based estimators conditioned
on non-noise data or noisy data including algorithms and error analysis. Same
as meshfree methods, we can also obtain the kernel-based approximate solutions
of elliptic partial differential equations by the kernel-based probability
measure.
