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We analyze the Nystr\"om approximation of a positive definite kernelassociated with a probability measure. We first prove an improved error boundfor the conventional Nystr\"om approximation with i.i.d. sampling andsingular-value decomposition in the continuous regime; the proof techniques areborrowed from statistical learning theory. We further introduce a refinedselection of subspaces in Nystr\"om approximation with theoretical guaranteesthat is applicable to non-i.i.d. landmark points. Finally, we discuss theirapplication to convex kernel quadrature and give novel theoretical guaranteesas well as numerical observations.

Sampling-based Nystr\"om Approximation and Kernel Quadrature