S. Chandrasekaran

S. Chandrasekaran, H. N. Mhaskar

Let $q\ge 1$ be an integer. Given $M$ samples of a smooth function of $q$ variables, $2π$--periodic in each variable, we consider the problem of constructing a $q$--variate trigonometric polynomial of spherical degree $Ø(M^{1/q})$ which interpolates the given data, remains bounded (independent of $M$) on $[-π,π]^q$, and converges to the function at an optimal rate on the set where the data becomes dense. We prove that the solution of an appropriate optimization problem leads to such an interpolant. Numerical examples are given to demonstrate that this procedure overcomes the Runge phenomenon when interpolation at equidistant nodes on $[-1,1]$ is constructed, and also provides a respectable approximation for bivariate grid data, which does not become dense on the whole domain.

S. Chandrasekaran, C. H. Gorman, H. N. Mhaskar

We study the problem of reconstructing a function on a manifold satisfying some mild conditions, given data on the values and some derivatives of the function at arbitrary points on the manifold. While the problem of finding a polynomial of two variables with total degree $\le n$ given the values of the polynomial and some of its derivatives at exactly the same number of points as the dimension of the polynomial space is sometimes impossible, we show that such a problem always has a solution in a very general situation if the degree of the polynomials is sufficiently large. We give estimates on how large the degree should be, and give explicit constructions for such a polynomial even in a far more general case. As the number of sampling points at which the data is available increases, our polynomials converge to the target function on the set where the sampling points are dense. Numerical examples in single and double precision show that this method is stable and of high-order.