Other Functional Series with Orthogonal Basis Sets

Other Functional Series with Orthogonal Basis Sets... 

OTHER FUNCTIONAL SERIES WITH ORTHOGONAL BASIS SETS... 

The orthogonal characteristic polynomials or eigenpolynomials Qn(u) play one of the central roles in spectral analysis since they form a basis due to the completeness relation (163). They can be computed either via the Lanczos recursion (84) or from the power series representation (114). The latter method generates the expansion coefficients q , -r through the recursion (117). Alternatively, these coefficients can be deduced from the Lanczos recursion (97) for the rth derivative Q /r(0) since we have qni r = (l/r )Q r(0) as in Eq. (122). The polynomial set Qn(u) is the basis comprised of scalar functions in the Lanczos vector space C from Eq. (135). In Eq. (135), the definition (142) of the inner product implies that the polynomials Qn(u) and Qm(u) are orthogonal to each other (for n= m) with respect to the complex weight function dk, as per (166). The completeness (163) of the set Q (u) enables expansion of every function f(u) e C in a series in terms of the... 

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