theorem rules for constructing kernels from other kernels if is reproducing kernel and ph is functionthen (ph( )ph( )is reproducing kernel from if is reproducing kernel and ris functionthen ( ) (xx ( is also reproducing kernel from if and are reproducing kernels from rthen so is their sum if and are reproducing kernels from rthen so is their product if and are reproducing kernels from and respectivelythen ((xy)( ): (xx (yy and kx ((xy)( ): (xx ) (yy are reproducing kernels from ( yx ( yr prooffor rules and it is easy to verify that the resulting function is finitesymmetricand positive semidefiniteand so is valid reproducing kernel by theorem for examplefor rule we have ni= nj= ai (yi ) for every choice of {ai }ni= and {yi }ni= since is reproducing kernel in particularit holds true for yi ph(xi ) rule is easy to show for kernels that admit representation of the form ( )since ( ( ( ( (xx (xx phi (xphi ( ph (xph ( > > ( ( ( phi (xph (xphi ( ph( ( ij> phk (xphk ( = (xx ) > showing that also admits representation of the form ( )where the new (possibly infinitesequence of features (phk is identified in one-to-one way with the sequence ( (ph( ph we leave the proof of rule as an exercise (exercise example (polynomial kernelconsider xx with (xx ( hxx ) polynomial kernel where hxx xx this is an example of polynomial kernel combining the fact that sums and products of kernels are again kernels (rules and of theorem )we find thatsince hxx and the constant function are kernelsso are hxx and ( hxx ) by writing (xx ( ) ( ) ( )