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Periodic orthonormal quasi-wavelet bases

机译：周期正交准小波基

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摘要

It is well known that many objects and problems in mathematics or in mathematic physics have property of periodicity. For example, in solving some differential equations in two-dimensional domain, we can reduce them to periodic problems of various orthonormal wavelet bases in various spaces. In recent years, the study of periodic wavelets presents active tendency. For the purpose of varieties of applications, the author constructed the periodic orthonormal quasi-wavelet bases in different spline spaces, for instance, the periodic polynomial spline functions, the spline functions on the circle (of complex variable z=e~(ix)), and the periodic trigonometric splines. Besides, we have established the refinable equations. The decomposition and reconstruction formulas for the corresponding filter coefficients were also found. The error bounds were estimated. The anti-periodic wavelets were also studied. We are surprised that in the periodic case, the refinable equation involves only two terms; on the other hand, in the antiperiodic case, the refinable equation has three terms.
机译：众所周知，数学或数学物理学中的许多对象和问题都具有周期性。例如，在求解二维域中的一些微分方程时，我们可以将它们简化为各种空间中各种正交小波基的周期问题。近年来，对周期性小波的研究呈现出活跃的趋势。为了适应多种应用，作者在不同样条空间中构造了周期正交准小波基，例如，周期多项式样条函数，圆上的样条函数（复变量z = e〜（ix）） ，以及周期性三角样条曲线。此外，我们建立了可精炼的方程。还找到了相应滤波器系数的分解和重建公式。估计误差范围。还研究了反周期小波。令我们惊讶的是，在周期性情况下，可精简方程只涉及两个项。另一方面，在反周期的情况下，可精化方程具有三个项。

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