正交多项式小波与de la Vallée Poussin型插值相关的[-1,1]区间
Orthonormal polynomial wavelets associated with de la Vallée Poussin-type interpolation on $[-1,1]$
AI总结：
本文提出了一种基于de la Vallée Poussin插值的正交多项式小波基，为非标准多分辨率分析提供良好的局部化正交基，并通过理论和实验验证其有效性。
摘要：
从de la Vallée Poussin型插值（VP）插值出发，作者最近引入了一族插值多项式尺度和小波基，生成非标准多分辨率分析的逼近和细节空间。受许多应用中正交基比插值基更受欢迎的启发，本研究开发了一族新的尺度和小波多项式，为相同的逼近和细节空间提供良好的局部化和正交基。我们证明所提出的基具有与已引入的插值基非常相似的行为，尽管它们不是插值的，但却是正交的。特别是，我们研究了所提出正交尺度基对应的傅里叶投影，并通过近似傅里叶-样式的系数引入其离散版本。对于连续和离散正交投影，我们证明了勒贝格常数的均匀有界性和具有与最佳均匀多项式逼近相同比例的均匀收敛性。数值实验验证了理论结果，并将新的正交VP尺度和小波基与作者先前处理的插值情况进行了比较。
英文摘要：
Starting from de la Vallée Poussin type (VP) interpolation, the authors have recently introduced a family of interpolating polynomial scaling and wavelet bases generating the approximation and detail spaces of a non-standard multiresolution analysis. Motivated by the fact that, in many applications, orthonormal rather than interpolating bases are preferable, the present study develops a new family of scaling and wavelet polynomials that provide well-localized and orthonormal bases for the same approximation and detail spaces. We show that the proposed new bases have a behavior very similar to the interpolating bases already introduced, presenting similar features although they are not interpolating but orthonormal. In particular, we study the Fourier projection corresponding to the proposed orthonormal scaling basis, and introduce a discrete version of it by approximating the Fourier--like coefficients. For both continuous and discrete orthogonal projections, we prove the uniform boundedness of the Lebesgue constants and the uniform convergence with an asymptotic rate comparable with the best uniform polynomial approximation. Numerical experiments confirm the theoretical results and compare the new orthonormal VP scaling and wavelet bases with the interpolating case previously treated by the authors.
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