D.2 The finite-dimensional reference measure🔗

For 0<\alpha<m, the mixture of \mathcal N(0,2sI) with weight s^{\alpha/2-1}\,\mathrm ds has density c_{m,\alpha}|\xi|^{\alpha-m}. The constant is c_{m,\alpha}=2^{-\alpha}\pi^{-m/2}\Gamma((m-\alpha)/2). The Lebesgue-direction case must be treated separately; it is not obtained by substituting \alpha=m into this mixture formula.

Remark D.1 (Comparison with familiar spaces).

For 0<\alpha<m, the squared norm is c_{m,\alpha}\int_{\mathbb R^m}|\widehat{fp_Q}(\xi)|^2|\xi|^{\alpha-m}\,\mathrm d\xi: the homogeneous Sobolev norm of order (\alpha-m)/2 of the weighted density fp_Q. In infinite dimension there is no Lebesgue measure with which to remove the weight, and no identification of \mathcal E_\alpha with a Sobolev space is asserted. The closed space \mathcal K_\alpha\subset L^2(\nu_\alpha) is not asserted to be a reproducing kernel Hilbert space.