Orthogonal polynomials orthogonality relations

This is the orthogonality relation of the two Lanczos polynomials Q (m) and Qm(u) with the weight function, which is the residue dk [48]. We recall that the sequence Q = (Q ( z-) coincides with the set of eigenvectors of the Jacobi matrix (60). 

So if we consider the value of orthonormal polynomials P (E) at the energies Eg at which the first neglected polynomial vanishes, a new orthogonality relation is achieved from the orthogonality property of the matrix of eigenvectors ... 

Orthogonal polynomials are a very useful set of expansion functions on grids. The simplest case is to define gn(q) as w(q)Pn(q), where P (q) is a member of the set of orthogonal polynomials, and w(q) is a weight function. These functions obey the continuous orthogonal relation defined in the domain D ... 

The associated Legendre polynomials obey the following orthogonality relation ... 

The expansion coefficients S, can be found from the orthogonality relations of the Legendre polynomials ... 

Using the polynomial expansions (3.20a) and (3.20c) with appropriate orthogonality relations we then obtain (Hirschfelder et al, 1954)... 

The Hermite polynomials of degree n, H x), that appear in the HO functions in Cartesian form fulfil the orthogonality relation [11,12]... 

The first tenn is zero if j due to the orthogonality of the Hemiite polynomials. The recursion relation in equation (B 1,2.4 ) is rearranged... 

In other words, the unknown function is a perturbation of the (Banna distritxition such a perturbation is expressed in terms of orthogonal polynomials and unknown coefficients related to the... 

To obtain the orthogonality and normalization relations for the Hermite polynomials, we multiply together the generating functions g(, 5) and g( , t), both obtained from equation (D.l), and the factor e and then integrate over ... 

In order to obtain the orthogonality and normalization relations of the assoeiate Laguerre polynomials, we make use of the generating fimction (F. 10). We multiply together g(p, s j), g(p, t J), and the factor pj+ e-f and then integrate over p to give an integral that we abbreviate with the symbol /... 

The coefficients c in this probability distribution are referred to as the quasimoments of the distribution. Because of the orthogonality of Hermite polynomials, the quasimoments of a function are obtained by integration of the product of the function and the related Hermite polynomial over all space. For the one-dimensional case,... 

A related approach is the approximation of peak-shaped fimctions by means of orthogonal polynomials, described by Scheeren et al. A function f(t), in this case the chromatographic signal, can be expanded in a series ... 

The alternative recursion (57) involves the monic Lanczos states ] ) Here, the term monic" serves to indicate that for any given integer n, the highest state ) in the finite sum, which defines the vector jr ), always has an overall multiplying coefficient equal to unity similarly to a monic polynomial [2], The Lanczos states ) are orthogonal, but unnormalized, as opposed to orthonormalized Lanczos states ( linear combinations of powers of the operator U(r) acting on the initial state 0)- Therefore, due to the relation 4> ) = U (r) 0) from Eq. (36), the vectors tjrn) and [ ) are certain sums of Schrodinger states ). ... 

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 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... 

To prove the important general property that orthogonal polynomials are connected by a three-term recurrence relation, we consider the polynomial h +ii +i( )—jEP (jE), whose degree is n provided the are defined by (7.11). liius it can be expanded as a linear combination of polynomials up to P (E) ... 

B. Relation between Orthogonal Polynomials and Continued Fractions... 

The expansions in even powers of normal frequencies are of special interest, because they provide means for obtaining explicit relations between the equations of motion and the thermodynamic quantities, through the use of the method of moments The sum of over all the normal vibrations can be expressed as the trace, or the sum of all the diagonal elements, of a matrix H" obtained by multiplying the Hamiltonian matrix H of the system by itself (n — 1) times. Such expansions thus enable us to estimate the thermodynamic functions and their isotope effects from known force fields and structures without solving the secular equations, or alternatively, to estimate the force fields from experimental data on the thermodynamic quantities and their isotope effects. The expansions explicitly correlate the motions of particles with the thermodynamic quantities. They can also be used to evaluate analytically a characteristic temperature associated with the system, such as the cross-over temperature of an isotope exchange equilibrium. Such possible applications, however, are useful only if the expansion yields a sufficiently close approximation. The precision of results obtainable with orthogonal polynomial expansions will be explored later. 

With the known 2n moments, the singularities of Ig e), in accordance with Eqn. (2.34), may be examined with a respective function q Z). Standard relations from the classical problem of moments and formulas from the Kristoffel-Darbu theory of orthogonal polynomials [14] allow us to establish the functions... 

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