construction of reproducing kernels in this section we describe various ways to construct reproducing kernel for some feature space recall that needs to be finitesymmetricand positive semidefinite function (that isit satisfies ( )in view of theorem specifying the space and reproducing kernel corresponds to uniquely specifying an rkhs reproducing kernels via feature mapping perhaps the most fundamental way to construct reproducing kernel is via feature map ph we define (xx :hph( )ph( )iwhere denotes the euclidean inner product the function is clearly finite and symmetric to verify that is positive semidefinitelet ph be the matrix with rows ph( )ph(xn )and let [ an ]rn thenn = = ai (xi = = ai ph(xi ph( aphpha kphak example (linear kerneltaking the identity feature map ph(xx on gives the linear kernel (xx hxx xx linear kernel as can be seen from the proof of theorem the rkhs of functions corresponding to the linear kernel is the space of linear functions on this space is isomorphic to itselfas discussed in the introduction (see also exercise it is natural to wonder whether given kernel function corresponds uniquely to feature map the answer is noas we shall see by way of example example (feature maps and kernel functionslet andconsider feature maps ph and ph with ph ( : and ph ( :[xx] then kph (xx hph ( )ph ( ) xx but also kph (xx hph ( )ph ( ) xx thuswe arrive at the same kernel function defined for the same underlying set via two different feature maps kernels from characteristic functions another way to construct reproducing kernels on makes use of the properties of characteristic functions in particularwe have the following result we leave its proof as exercise