Series of orthogonal functions

In both calculations, the boundary conditions are linear with respect to 0 and its first-order derivatives. The solution of the Fourier equation, with respect to the space variables, may be developed in a series of orthogonal functions, winch are exponential with respect to the time variable [for the solution of similar problems, see (45)]- The time-dependance of the temperature distribution along a single space variable r, resulting from a unit pulse, is therefore given by... 

The most commonly used approach to the problem is to expand the correlation functions and their Fourier transforms in a series of orthogonal functions, usually the spherical harmonics. This approach was pioneered by Chen and Steele in the case of the Percus-Yevick approximation for hard diatomic fluids. More recently, the approach has been generalized to arbitrary... 

Kuo (K2) has recently obtained a solution to the nonlinear equations of cellular convection by expanding the dependent variables in series of orthogonal functions and by expanding the coefficients of these functions in a power series of an amplitude parameter. His solution also predicts a heat transfer rate proportional to the 5/4 power of the temperature difference. 

The source term is then expanded in a series of orthogonal functions ... 

The next step in our process is to expand b / /bz and u each into a series of suitable functions. For this purpose it seems natural to use the set of solutions of the equation of the unperturbed atom, i.e., the set w, tt), as was done by Schrodinger. Unfortunately, this set is not a complete orthogonal set unless a continuous range of complicated functions corresponding to imaginary values of / are included. To avoid this complication we follow a procedme analogous to that used by Epstein for a similar purpose, i.e., we use for our expansion another set of functions, T V, w, )> defined as follows... 

The orientation is not strictly identical for all structural units and is rather spread over a certain statistical distribution. The distribution of orientation can be fully described by a mathematical function, N(6, q>, >//), the so-called ODF. Based on the theory of orthogonal polynomials, Roe and Krigbaum [1,2] have shown that N(6, generalized spherical harmonics that form a complete set of orthogonal functions, so that... 

A remarkable series of CH functionalizations has been described whereby the regiochemical outcome of the reaction is determined by the catalyst employed. Directed and nondirected C-H functionalizations on 2-phenylimi-dazole were observed. This orthogonal approach is excellent for introducing diversity and may have applications in library generation in areas including medicinal chemistry (Scheme 18). 

The addition of more terms does not influence the values of the already calculated terms. In this aspect, orthogonal polynomials are superior to other polynomials calculation of the coefficients is simple and fast. Moreover, according to the Gram-Schmidt theory every function can be expressed as a series of orthogonal polynomials, using the weighting function w(t). 

The orthogonal collocation method has several important differences from other reduction procedures. Jn collocation, it is only necessary to evaluate the residual at the collocation points. The orthogonal collocation scheme developed by Villadsen and Stewart (1967) for boundary value problems has the further advantage that the collocation points are picked optimally and automatically so that the error decreases quickly as the number of terms increases. The trial functions are taken as a series of orthogonal polynomials which satisfy the boundary conditions and the roots of the polynomials are taken as the collocation points. A major simplification that arises with this method is that the solution can be derived in terms of its value at the collocation points instead of in terms of the coefficients in the trial functions and that at these points the solution is exact. 

Da] Davis, H.F., Fourier Series and Orthogonal Functions, Dover, New York, 1989. (Unabridged republication of tbe edition pubUsbed by Allyn and Bacon, Boston, 1963.)... 

FOURIER SERIES AND ORTHOGONAL FUNCTIONS, Harry F. Davis. An incisive text combining theory and practical example to introduce Fourier series, orthogonal functions and applications of the Fourier method to boundary-value problems. 570 exercises. Answers and notes. 4I6pp. 5H x 8b. 65973-9 Pa. 8.95... 

An extensive study of analytical techniques used in conduction heat transfer requires a background in the theory of orthogonal functions. Fourier series are one example of orthogonal functions, as are Bessel functions and other special functions applicable to different geometries and boundary conditions. The interested reader may consult one or more of the conduction heat-transfer texts listed in the references for further information on the subject. 

The final step is to choose the A so that w(r, 7) satisfies the initial condition 777(7, 0) = -(1 - r2)/4. The general Sturm-Louiville theory16 guarantees that the eigenfunctions (3-106) form a complete set of orthogonal functions. Thus it is possible to express the smooth initial condition (1 -r2) by means of the Fourier-Bessel series (3-109) with 7 = 0, that is,... 

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... 

To do this, the PCLD is expanded in an orthogonal series of special functions, in this case Laguerre polynomials. The coefficients of the series are given in terms of the moments of the original CLD, as illustrated below. 

According to the Karhunen-Loeve (K-L) theorem, a stochastic process on a bounded interval can be represented as an infinite linear combination of orthogonal functions, the coefficients of which constitute uncorrelated random variables. The basis functions in K-L expansions are obtained by eigendecomposition of the autocovariance function of the stochastic process and are shown to be its most optimal series representation. The deterministic basis functions, which are orthonormal, are the eigenfunctions of the autocovariance function and their magnitudes are the eigenvalues. The Karhunen-Loeve expansion converges in the mean-square sense for any distribution of the stochastic process (Papoulis and Pillai 2002). A K-L representation of a zero-mean stochastic process f(t, 6) can be represented in the form... 

The great importance of orthogonal functions lies in the possibility of expanding arbitrary functions in a series of these orthogonal functions. Suppose that/(a ) is any function and that it is possible to expand f(x) in the interval (a, b) in a series... 

According to Muller (43), the function N(0) may be expanded in terms of a series of orthogonal polynomials, notably Legendre polynomials ... 

Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods finite difference, orthogonal collocation, and Galerkin finite element methods. 

Just as a vector is projected as components on orthogonal axes, a given function defined on a given domain can be projected onto an orthogonal set of functions. The Fourier series decomposition of a function /(x) defined over the interval [ —X, X] is a convenient example... 

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 ... 

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