theorem holds if (ithe kernel is continuous on (iithe function ( : (xxdefined for is integrable extensions of theorem to more general spaces and measures holdseee [ or [ the key importance of theorem lies in the fact that the series representation ( converges absolutely and uniformly on xxx the uniform convergence is much stronger condition than pointwise convergenceand means for instance that properties of the sequence of partial sumssuch as continuity and integrabilityare transferred to the limit example (mercersuppose [- and the kernel is (xx xx which corresponds to the rkhs of affine functions from to find the (eigenvalueeigenfunctionpairs for the integral operator appearing in theorem we need to find numbers {land orthonormal functions { ( )that solve - ( xx ( dx lx(xfor all [- consider first constant function (xc thenr for all [- ]we have that cand the normalization condition requires that - dx togetherthese give and +- nextconsider an affine function (xa bx orthogonality requires that - ( bxdx which implies (since moreoverthe normalization condition then requires - dx orequivalently / implying + / finallythe integral equation reads - ( xx bx dx bx = bx bx implying that / we take the positive solutions ( and )and note that (xx ( (xx ( xx (xx ) and so we have found the decomposition appearing in ( as an asideobserve that and are orthonormal versions of the first two legendre polynomials the corresponding feature map can be explicitly identified as ph (xl ( and ph (xl (xx kernels from kernels the following theorem lists some useful properties for constructing reproducing kernels from existing reproducing kernels