13 research outputs found
Optimal sampling patterns for Zernike polynomials
A pattern of interpolation nodes on the disk is studied, for which the inter-
polation problem is theoretically unisolvent, and which renders a minimal
numerical condition for the collocation matrix when the standard basis of
Zernike polynomials is used. It is shown that these nodes have an excellent
performance also from several alternative points of view, providing a numer-
ically stable surface reconstruction, starting from both the elevation and the
slope data. Sampling at these nodes allows for a more precise recovery of the
coefficients in the Zernike expansion of a wavefront or of an optical surface.
Keywords:
Interpolation
Numerical condition
Zernike polynomials
Lebesgue constant
Approximation and Integration on Compact Subsets of Euclidean Space
This thesis deals with approximation of real valued functions. It considers interpolation and numerical integration of functions. It also looks at error and precision, the Weierstrass Theorem and Taylors Theorem. In addition, spherical harmonics, the Laplacian, Hilbert spaces and linear projections are considered with respect to the unit sphere. An example of distributing points equally on a sphere is illustrated and a covering theorem for a unit sphere is proved
Polynomial Interpolation On The Unit Sphere
The problem of interpolation at (n + 1) points on the unit sphere S by spherical polynomials of degree at most n is studied. Many sets of points that admit unique interpolation are given explicitly. The proof is based on a method of factorization of polynomials. A related problem of interpolation by trigonometric polynomials is also solved
Polynomial Interpolation on the Unit Sphere II.
The problem of interpolation at points on the unit sphere by spherical polynomials of degree at most is proved to have a unique solution for several sets of points. The points are located on a number of circles on the sphere with even number of points on each circle. The proof is based on a method of factorization of polynomials