Hermite-Shannon-Cosine Interval Wavelet and Its Application in Adaptive Distribute Interpolation on Curves

Abstract

<p indent="0mm">The dynamic curve is a common tool to describe the transformation law of various phenomena. To capture the characteristic points dynamically and approximate the dynamic curve accurately, a Shannon-Cosine wavelet function with interpolation, smoothness, tight support, and symmetry is constructed. Firstly, by means of the wavelike property and continuity of the Shannon wavelet function, a parameterized window function is designed based on the integral median theorem, in which the support and smoothness of Shannon-Cosine wavelet can be chosen to meet the requirement of the approximated curves. Secondly, based on the theory analysis and numerical experiments, the discontinuity near the boundary of the curve is identified as the major factors which result in the boundary effect appeared on the wavelet transform on the curve, and then the interval wavelet is constructed by using the two-point cubic Hermite interpolation function. Finally, the multi-scale Shannon-Cosine wavelet is used to perform multi-scale adaptive subdivision and approximation of the propagation of shock wave curve and the reflected Burger curve. It is able to capture the characteristic points of the curve adaptively, which can be used to reconstruct the curve. Burger curve is taken as the example to test the proposed method, the numerical experiments show that, compared with other methods, the method of Hermite Shannon-Cosine interval wavelet approximation curve has higher numerical accuracy and lower algorithm complexity.