Polynomials orthonormal

The Laguerre polynomials Ln(r) become orthonormal polynomials only after they are multiplied by a weighting function ( ) 1 exp( r) [11] ... 

Theorem. For any three consecutive orthonormal polynomials defined by Eq. (7.4), we have the recurrence formula... 

The orthonormalized polynomials P ( ) contain the powers of " with coefficient It may be convenient to consider the set of orthogo-... 

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 ... 

When this method is applied to the polynomial approximation of mechanisms, the function G is the response of the kinetic model calculated using the original detailed reaction mechanism, and (/> is a series of orthonormal polynomials constructed by a Gram-Schmidt orthonormalization process using the data set. The function F, defining the final algebraic model, is constructed in such a way that only the significant members of the summation are considered. 

T. Turdnyi, Parameterization of Reaction Mechanisms Using Orthonormal Polynomials, Comput. Chem. 18 (1994) 45-54. 

Table 1.1 The three sets of mutually-orthonormal polynomial functions required to provide basis functions for the three components of Tju symmetry in the Ij, regular representation.

In general, if an /-level factorial design is of interest, then the orthogonal basis for this factorial design will be formed by the set of polynomials from 0 to / - 1, evaluated at each of the treatment levels. The zeroth-order polynomial, Lq, will be used for the zero-order interaction (/ o)- The first-order polynomial, Lj, will be used for the factors whose powers are 1. Similarly, the second-order polynomials, L2, will be used for the factors whose powers are 2. The results can be obtained by solving Eqs. (4.35) and (4.36). In the following sections, the results for 1 = 2, 3, and 4 will be provided, as well as generalised orthonormal polynomials for 1 = 2 and 3. Examples will be provided as appropriate. 

To proceed consider some properties of orthonormal polynomials. A polynomial of order n, F (x), defined as... 

The weight function, w(x), and the corresponding orthonormal polynomials F of the two-electron integrals will be described in Section 5.3. 

Based on the HDMR expansion using orthonormal polynomial basis sets, the partial variances D, and D,y can be calculated (Li et al. 2002b Leng et al. 2004) ... 

The final step is the craiversion of the appropriate orthonormed polynomial to a usual polynomial ... 

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