Example 1 Regression Function:Consider the first order autoregression function r1(x)=E(X:Xi-1=x). We can write Xt=ri(Xi-1)+t, where E(etX:-1)=0 by construction.We assume E(X?)<oo. Suppose a sequence of bases (r)}constitutes a complete orthonormal basis for the space of square-integrable functions.Then we can always decompose the function where the Fourier coefficient rn(e),(e), which is the projection of ri(r)on the base i(). Suppose there is a quadratic function ri(z)-x2forx∈【-元，x.Then r1(x)= 2 34 cos(ar)-o s(2y）+os(3d- 32 π2 4∑(-1p-1os0四 j=1 For another example,suppose the regression function is a step function,namely -1if-π<x<0, r(x) 0 if =0, 1if0<x<π. Then we can still expand it as an infinite sum of periodic series, n()= 4 sin(e)sin)sin( 3 5 4 户m2j+1网 (2j+1) 5Example 1 [Regression Function]: Consider the Örst order autoregression function r1(x) = E(Xt jXt￾1 = x): We can write Xt = r1(Xt￾1) + "t ; where E("t jXt￾1) = 0 by construction. We assume E(X2 t ) < 1: Suppose a sequence of bases f j (x)g constitutes a complete orthonormal basis for the space of square-integrable functions. Then we can always decompose the function r1(x) = X1 j=0 j j (x); where the Fourier coe¢ cient j = Z 1 ￾1 r1(x) j (x)dx; which is the projection of r1(x) on the base j (x): Suppose there is a quadratic function r1(x) = x 2 for x 2 [￾; ]: Then r1(x) =  2 3 ￾ 4  cos(x) ￾ cos(2x) 2 2 + cos(3x) 3 2 ￾


 =  2 3 ￾ 4 X1 j=1 (￾1)j￾1 cos(jx) j 2 : For another example, suppose the regression function is a step function, namely r1(x) = 8 >>< >>: ￾1 if ￾  < x < 0; 0 if x = 0; 1 if 0 < x < : Then we can still expand it as an inÖnite sum of periodic series, r1(x) = 4   sin(x) + sin(3x) 3 + sin(5x) 5 +


 = 4  X1 j=0 sin[(2j + 1)x] (2j + 1) : 5