The General Orthogonal Expansions

Abstract Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. We discuss applications to the Fourier extension problem, interpolation of functions with singularities or near singularities and the solution of Schrödinger’s equation with nondifferentiable or nearly nondifferentiable potentials.

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Orthogonal Expansions

Applied Differential Equations with Boundary Value Problems ◽

10.1201/9781315369785-10 ◽

2017 ◽

pp. 545-595

Author(s):

Vladimir A. Dobrushkin

Keyword(s):

Orthogonal Expansions

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Orthogonal Expansions in Curvilinear Coordinates

Engineering Applications of Noncommutative Harmonic Analysis ◽

10.1201/9781420041767.ch4 ◽

2000 ◽

Keyword(s):

Curvilinear Coordinates ◽

Orthogonal Expansions

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VI Vector Spaces, Orthogonal Expansions, and Fourier Transforms

Pure and Applied Mathematics ◽

10.1016/s0079-8169(08)61328-8 ◽

1983 ◽

pp. 438-534

Keyword(s):

Fourier Transforms ◽

Vector Spaces ◽

Orthogonal Expansions

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Physical nearest-neighbour models and non-linear time-series. II Further discussion of approximate solutions and exact equations

Journal of Applied Probability ◽

10.1017/s0021900200094705 ◽

1972 ◽

Vol 9 (01) ◽

pp. 76-86

Author(s):

M. S. Bartlett

Keyword(s):

Time Series ◽

Correlation Functions ◽

Linear Time ◽

Three Dimensional ◽

Approximate Solutions ◽

Nearest Neighbour ◽

Spatial Correlations ◽

Orthogonal Expansions ◽

Non Linear ◽

Time Series Approach

The approximate two- and three-dimensional solutions for spatial correlations, using the non-linear time-series approach for nearest-neighbour systems developed in my previous paper, are further discussed. Orthogonal expansions for the correlation functions are also developed which determine with this approach, though so far only in principle, the exact solutions.

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