Radial orthogonality and Lebesgue constants on the disk

Optimal sampling patterns for Zernike polynomials

“…It is well known that in the multivariate case the unisolvence of the interpolation problem for arbitrary nodes is not guaranteed, so not every sampling pattern is acceptable (several configurations of interpolation points on the disk that guarantee unisolvence are well-known and can be found in the literature, see e.g. [20,21,22,23]). Moreover, the error in approximating a function by its interpolating polynomial depends on the interpolation nodes: standard upper bounds for the error are based on the so-called Lebesgue constants corresponding to these nodes, which give the norm of the interpolation as a projection operator onto the polynomial subspace (see e.g.…”

“…Moreover, the error in approximating a function by its interpolating polynomial depends on the interpolation nodes: standard upper bounds for the error are based on the so-called Lebesgue constants corresponding to these nodes, which give the norm of the interpolation as a projection operator onto the polynomial subspace (see e.g. [23,24]).…”

“…The constraint r 1 < 1 is for a practical reason: in actual measurements the data in the periphery is usually much less reliable. A similar problem, but minimizing the Lebesgue constants for the nodes {P i }, was considered in [23,32,36,37]. In [23], the radii were studied in connection with the notion of spherical orthogonality for multivariate polynomials, and values of r j 's, given by the zeros of certain Gegenbauer orthogonal polynomials, were analyzed experimentally.…”

“…A similar problem, but minimizing the Lebesgue constants for the nodes {P i }, was considered in [23,32,36,37]. In [23], the radii were studied in connection with the notion of spherical orthogonality for multivariate polynomials, and values of r j 's, given by the zeros of certain Gegenbauer orthogonal polynomials, were analyzed experimentally. In [36], along with the Lebesgue constants, the condition numbers κ ∞ (A n ) = A n ∞ A −1 n ∞ were used to optimize the radii.…”

Affine THDMR on the unit disk

“…We assume that M and N in (6) are taken as M + 1 = (m + 1)(m + 2)/2 and N +1 = (n+1)(n+2)/2, where the rational approximant is of total degree m in the numerator and n in the denominator. The following set of polynomials are mutually orthogonal on the unit disk and can be used as a basis for the space of bivariate polynomials of total degree at most max{m, n} (see [5]). …”