Stable Multiscale Analytic Sampling Approximation To Periodic Function And Its Fast Algorithm

Compared with traditional linear Fourier atoms, nonlinear Fourier atoms can depict the time-vary ing characteristics of non-stationary signal. In ad-dition, mon-component signal has nonnegative instantaneous frequency. It makes an important physical sense to decompose signal into the sum of mon-component. M(?)bius transformation function is a mon-component. And it has nonlinear instantaneous frequency. By applying the multiscale method to M(?)bius transformation function, we will construct multiscale analytic sam-pling approximation to H2(Td). And its main contents are as follows:Firstly, Using M(?)bius transformation function to construct the multi-scale analytic sampling approximation, we will give the concrete expression of approximation formulas (?)kf, and estimate its approximation error.Secondly, when noise included in multisclae analytic samples, we will give the approximation error with noise. In particular, if the noise is stochas-tic, we give the estimation of expectation and variance of the approximation error with noise. It prove that this approximation method has stronger stabil-ity.Thirdly, under the multiscale analytic sampling approximation, we prove that the numerical calculation formula of the uniform points has multilevel Hankel matrices structure. And by applying this special structure, we develop the fast algorithm of numerical calculation.Fourthly, we prove that the multiscale analytic samples can be written as the form of d-level circulant matrices structure. Then by the property of special d-level circulant matrix, we will establish the fast extraction algorithm of multisclae analytic samples.In the last, numerical experiments in H2(T) and H2(T2) are carried out to demonstrate the efficiency of the multiscale analytic sampling approximation.