Article

# On Double Interpolation in Polar Coordinates

Journal of Computer Science and Control Systems 01/2009;

Source: DOAJ

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**ABSTRACT:**A trivariate Lagrange interpolation method based on C1 cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The in- terpolation method is local and stable, provides optimal order approximation, and has linear complexity. - [Show abstract] [Hide abstract]

**ABSTRACT:**We give a simple, geometric and explicit construction of bivariate interpolation at certain points in a square (called Padua points), giving compact formulas for their fundamental Lagrange polynomials. We show that the Lebesgue constant has minimal order of growth, i.e. log square of the degree. To the best of our knowledge this is the first complete, explicit example of near optimal bivariate interpolation points.Journal of Approximation Theory 11/2006; · 0.76 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Under the assumption that the function ƒ is bounded on [−1, 1] and analytic at x = 0 we prove the local convergence of Lagrange interpolating polynomials of ƒ associated with equidistant nodes on [- 1, 1]. The classical results concerning the convergence of such interpolants assume the stronger condition that ƒ is analytic on [−1, 1]. A de Montessus de Ballore type theorem for interpolating rationals associated with equidistant nodes is also established without assuming the global analyticity of ƒ on [−1, 1].Journal of Approximation Theory 01/1994; 78(2):213-225. · 0.76 Impact Factor

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