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Smoothing periodic curves by a method of regularization. (English) Zbl 0699.65095
The theory of polynomial smoothing splines is developed for the statistical estimation of a periodic curve based on discrete and noisy observations. Here, the splines are called as the periodic \(\alpha\)- splines, The estimates are obtained by a method of regularization type with a penalty based on the Fourier coefficients. This penalty is motivated by the smoothness of a periodic function. Asymptotic bias and variance of the estimates are discussed for equispaced design points. It is also shown that the impulse response functions for \(\alpha\)-splines are well approximated by a kernel-type impulse function, when the “folded-version” of the kernel is used. The extension of \(\alpha\)- splines to convolution type inverse problems and multidimensional problems are briefly mentioned. Some numerical examples concerning the choice of penalty are presented.
Reviewer: K.Uosaki

MSC:
65C99 Probabilistic methods, stochastic differential equations
65D07 Numerical computation using splines
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
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