the eigenvalues of are { / / / / / {- and so by theorem is not positive semidefinite matrixsince it has negative eigenvalue consequentlyk is not valid reproducing kernel one of the reasons why the gaussian kernel ( is popular is that it enjoys the universal approximation property [ ]the space of functions spanned by the gaussian kernel is dense in the space of continuous functions with support naturallythis is desirable property especially if there is little prior knowledge about the properties of ghowevernote that every function in the rkhs associated with gaussian kernel is infinitely differentiable moreovera gaussian rkhs does not contain non-zero constant functions indeedif is non-empty and openthen the only function of the form (xc { acontained in is the zero function ( consequentlyif it is known that is differentiable only to certain orderone may prefer the matern kernel with parameters ns kn (xx  - kx / kn kx / (nmatern kernel reproducing kernels using orthonormal features we have seen in sections and how to construct reproducing kernels from feature maps and characteristic functions another way to construct kernels on space is to work directly from the function class (xu)that isthe set of square-integrable functions on with respect to usee also definition for simplicityin what followswe will consider to be the lebesgue measureand will simply write (xrather than (xuwe will also assume that let { be an orthonormal basis of (xand let be sequence of positive numbers as discussed in section the kernel corresponding to feature map pp ph is (xx ph( )ph( = phi (xphi ( now consider (possibly infinitesequence of feature functions phi ci xi and define (xx :phi (xphi ( li xi (xxi ( )( > universal approximation property ( which gives functions that are (weaklydifferentiable to order bnc (but not necessarily to order dneherekn denotes the modified bessel function of the second kindsee ( the particular form of the matern kernel appearing in ( ensures that limnkn (xx (xx )where is the gaussian kernel appearing in ( we remark that sobolev spaces are closely related to the matern kernel up to constants (which scale the unit ball in the space)in dimension and for parameter / these - spaces can be identified with ps( (sktk - / / - (ktk)which in turn can be viewed as the characteristic function corresponding to the (radially symmetricmultivariate student' distribution with degrees of freedomthat iswith pdf ( ( kxk )- > the term radial basis function is sometimes used more generally to mean kernels of the form (xx (kx kfor some function function is said to be square-integrable if (xu(dxwhere is measure on