Orthogonal functions, expansions

The time-independent case corresponds to fixed time t=0. The only constraints on this expansion are that the Hilbert space norm of the orthogonal functions v should be less than or equal to one, if the functions are to be interpreted as... 

A computationally efficient method of function fitting using an orthogonal polynomial expansion is presented for approximating continuous wall temperature profiles. 

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. 

Closely related to the strictly variational VB method described so far, there have been a number of recent approaches which use perturbation theory. All of these are characterized by the use of non-orthogonal functions and fully antisymmetrized wavefunctions. In addition the full Hamiltonian is used without a multipole expansion. 

If population coefficients are assigned to each multipole term, rather than orbital products,23 the expansion is now in terms of orthogonal functions. [TTie functions centered on different centers are not orthogonal to each other.] For this case correlation coefficients should be very small. Except for (2px )2, (2py )2, (2pz )2, and (2s)2, the orbital product population coefficient is proportional to the one assigned to a multipole. If we assign Py to FjCcos 0), Qx to P cos 20 and Qs to P%, then... 

The familiar Fourier series is only one special form of an expansion in terms of orthogonal functions. Figure 22-1, which gives a plot of the function... 

It may be mentioned that in some cases it is convenient to make use of complete sets of functions which are not mutually orthogonal. An arbitrary function can be expanded in terms of the functions of such a set the determination of the values of the coefficients is, however, not so simple as for orthogonal functions. An example of an expansion of this type occurs in Section 24. 

In certain applications of expansions in terms of orthogonal functions, we shall obtain expressions of the form... 

So far our treatment differs from the previous discussion of non-degenerate levels only in the use of x i instead of Vhhe., in the introduction of a general expression for unperturbed functions instead of the arbitrary set ik. In the next step we likewise follow the previous treatment, in which the quantities xp k and Hn4/ k were expanded in terms of the complete set of orthogonal functions . Here, however, we must in addition express x i in terms of the set by means of Equation 24-4, in which the coefficients < are so far arbitrary. Therefore we introduce the expansions... 

The orthogonal collocation technique is a simple numerical method which is easy to program for a computer and which converges rapidly. Therefore it is useful for the solution of many types of second order boundary value problems. This method in its simplest form as presented in this section was developed by Villadsen and Stewart (1967) as a modification of the collocation methods. In collocation methods, trial function expansion coefficients are typically determined by variational principles or by weighted residual methods (Finlayson, 1972). The orthogonal collocation method has the advantage of ease of computation. This method is based on the choice of suitable trial series to represent the solution. The coefficients of trial series are determined by making the residual equation vanish at a set of points called collocation points , in the solution domain. 

A consequence of the orthogonality relations is that the collocation functional expansion scheme becomes a discrete vector space with a unitary transformation between the discrete sampling points qt and the discrete functional base an. The matrix G is then unitary. 

Based on the Christoffel-Darboux formula (20) it can be shown that this procedure leads to a functional expansion which becomes an interpolation formula on the integration points, vK<7 ) = (<7 )- As an example consider an expansion by the Chebychev orthogonal polynomials g,(q) = T (q) with the constant weights W(qi) = 2/tt. The quadrature points q, are the zeros of the Chebychev polynomial of degree N + 1. On inserting Eq. (16) into the functional expansion Eq. (2) becomes... 

Thus, if we represent a function f(,x) in this interval by an expansion of such orthogonal functions... 

In recent years, the first order finite difference methods have been superseded by nodal methods. In these the flux in each mesh element, or node, is represented by a set of orthogonal functions, such as Legendre polynomials for each direction, or other types of expansion. Using such flux representations, a more accurate solution can be obtained using a coarser mesh. The matrix equations relating the various components of the flux become more complicated, involving a relationship between components of the flux inside the node and on the surfaces. 

Korenberg, M.J. 1988. Identifying nonlinear difference equation and functional expansion representations The fast orthogonal algorithm. Ann. Biomed. Eng. 16 123. 

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