Lanczos polynomials completeness

In Eq. (162), the result (156) is used to identify the residue dk. Note that the upper limit K — 1 in Eq. (163) could be replaced by K on account of the characteristic equation Qk(u>c) = 0. The expression (163) represents the local completeness relation or closure for the Lanczos polynomial Qn(uk) -Next, we are interested in considering the weighted products dkQn(uk)QM summed over all the eigenvalues uk k=1 for two arbitrary degrees n and m ... 

This duality enables switching from the work with the Lanczos state vectors fn) to the analysis with the Lanczos polynomials Q (m). A change from one representation to the other is readily accomplished along the lines indicated in this section, together with the basic relations from Sections 11 and 12, in particular, the definition (142) of the inner product in the Lanczos space CK, the completeness (163) and orthogonality (166) of the polynomial basis Qn,k -... 

The Lanczos vector space CM can be defined through its basis and the appropriate scalar product. A finite sequence of the Lanczos orthogonal polynomials of the first kind is complete, as will be shown in Section 12, and therefore, the set Q (u) with K elements represents a basis. Thus, the polynomial set Q (u) =0 will be our fixed choice for the basis in CK. Of particular importance is the set K, of the zeros uk %=1 of the Kth degree characteristic polynomial QK(u) ... 

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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