derive equation 3_7 its wrong
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A brief Word-document note in the Jim Ball GQ Binder, apparently by Phil, working out a derivation of equation (3.7) in a source paper. It compares the finite-N matrix equation with the infinite-N one using the recursion-coefficient (tridiagonal) matrix T_N and a remainder term r_N. Subtracting the two and dotting with a vector yields the eigenvalue relation. The extracted text lost most symbols, so details are approximate.
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Derivation of (3.7)
Look at (2.4). The functions of the vector are the ortho polys defined by interval (a.b) and weight w(x). As we increase N, the don't change, there are just more of them in the matrix equation.
So the matrix equation at large but finite N says this:
NN = TNN + rN (1)
The same equation at infinite N says this
= T
and r = 0. This last is an x matrix. If we look at the upper left (N+1)2 corner of this matrix we
see
N = TNN (2)
Notice that the components of don't change as noted above, nor does the matrix TN since it is composed of the recursion coefficients which do not in turn change.
So we can now compute (1) - (2) to get
(N - )N = rN
and now dot both sides with N to get
(N - )NN = NrN
which says
(N - )n=0N n2 = NN+1N
where we use (2.7) for r, knowing that -1 = 0 (in 3.5, we see that saying the xi are zeros of J(xi) causes the -1 term to be 0). Thus we get
N = + NN+1N / (n=0N n2) QED
which is (3.7). Notice that he says that i are the eigenvalues of TN, so he identifies i = . Then the finite but large eigenvalues are N = 1/xi2 .