Orthonormal vector polynomials in a unit circle, Part II : completing the basis set
نویسندگان
چکیده
منابع مشابه
Orthonormal vector polynomials in a unit circle, Part II : Completing the basis set.
Zernike polynomials provide a well known, orthogonal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. A related set of orthogonal functions is given here which represent vector quantities, such as mapping distortion or wavefront gradient. Previously, we have developed a basis of functions generated from gradients of Zern...
متن کاملOrthonormal vector polynomials in a unit circle, Part I: Basis set derived from gradients of Zernike polynomials.
Zernike polynomials provide a well known, orthogonal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. A related set of orthogonal functions is given here which represent vector quantities, such as mapping distortion or wavefront gradient. These functions are generated from gradients of Zernike polynomials, made orthonorm...
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Stieltjes-type polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of co...
متن کاملStieltjes-type Polynomials on the Unit Circle
Stieltjes-type polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of co...
متن کاملBuilding an Orthonormal Basis from a 3D Unit Vector Without Normalization
I present two tools that save the computation of a dot product and a reciprocal square root in operations that are used frequently in the core of many rendering programs. The first tool is a formula for rotating a direction sampled around the z-axis to a direction sampled around an arbitrary unit vector. This is useful in Monte Carlo rendering techniques, such as path tracing, where directions ...
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ژورنال
عنوان ژورنال: Optics Express
سال: 2008
ISSN: 1094-4087
DOI: 10.1364/oe.16.006586