  This paper presents a convergence analysis of kernel-based quadrature rules
in misspecified settings, focusing on deterministic quadrature in Sobolev
spaces. In particular, we deal with misspecified settings where a test
integrand is less smooth than a Sobolev RKHS based on which a quadrature rule
is constructed. We provide convergence guarantees based on two different
assumptions on a quadrature rule: one on quadrature weights, and the other on
design points. More precisely, we show that convergence rates can be derived
(i) if the sum of absolute weights remains constant (or does not increase
quickly), or (ii) if the minimum distance between design points does not
decrease very quickly. As a consequence of the latter result, we derive a rate
of convergence for Bayesian quadrature in misspecified settings. We reveal a
condition on design points to make Bayesian quadrature robust to
misspecification, and show that, under this condition, it may adaptively
achieve the optimal rate of convergence in the Sobolev space of a lesser order
(i.e., of the unknown smoothness of a test integrand), under a slightly
stronger regularity condition on the integrand.
