  We analyze the Nystr\"om approximation of a positive definite kernel
associated with a probability measure. We first prove an improved error bound
for the conventional Nystr\"om approximation with i.i.d. sampling and
singular-value decomposition in the continuous regime; the proof techniques are
borrowed from statistical learning theory. We further introduce a refined
selection of subspaces in Nystr\"om approximation with theoretical guarantees
that is applicable to non-i.i.d. landmark points. Finally, we discuss their
application to convex kernel quadrature and give novel theoretical guarantees
as well as numerical observations.
