Best Approximation and Cyclic Variation Diminishing Kernels

We study best uniform approximation of periodic functions from

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where the kernelK(x, y) is strictly cyclic variation diminishing, and related problems including periodic generalized perfect splines. For various approximation problems of this type, we show the uniqueness of the best approximation and characterize the best approximation by extremal properties of the error function. The results are proved by using a characterization of best approximants from quasi-Chebyshev spaces and certain perturbation results.