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Rational spaces: Bernstein Approximation, Dimension Elevation, and P\'olya-type Theorems on Positive Polynomials

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Abstract

On a given closed bounded interval we consider dimension elevation processes generated by nested sequences of rational spaces defined by infinite sequences of real poles outside the interval. We give necessary and sufficient conditions for the poles to ensure convergence of such processes to the underlying curves. Surprisingly, this typical CAGD issue is equivalent to determining all Polya positive sequences, in the sense of all infinite sequences of positive numbers which guarantee Polya-type results for univariate polynomials which are positive on the non-negative axis.

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