3.14 ( ) In this exercise, we explore in more detail the properties of the equivalent kernel defined by (3.62), where SN is defined by (3.54). Suppose that the basis functions φj(x) are linearly independent and that the number N of data points is greater than the number M of basis functions. Furthermore, let one of the basis functions be constant, say φ0(x) = 1. By taking suitable linear combinations of these basis functions, we can construct a new basis set ψj(x) spanning the same space but that are orthonormal, so that

N n=1

ψj(xn)ψk(xn) = Ijk (3.115)

where Ijk is defined to be 1 if j = k and 0 otherwise, and we take ψ0(x) = 1. Show that for α = 0, the equivalent kernel can be written as k(x, x) = ψ(x)Tψ(x)

where ψ = (ψ1, . . . , ψM)T. Use this result to show that the kernel satisfies the summation constraint

N n=1 k(x, xn) = 1. (3.116)