Recently, several authors have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either the unwinding series or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of the boundary value of an analytic function \(F: \partial {\mathbb {D}}\rightarrow {\mathbb {C}}\) and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be efficiently approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was initiated by Coifman and Steinerberger, under the assumption that the complex valued function F has an analytic extension to \({\mathbb {D}}_{1+\epsilon }\) for some \(\epsilon >0\). This provided bounds on weighted Hardy norms involving a single zero, \(\alpha \in {\mathbb {D}}\), of F and its Blaschke decomposition. That work also noted that in many specific examples, the unwinding series of F converges at an exponential rate to F, which when coupled with an efficient algorithm to compute a Blaschke decomposition, has led to the hope that this process will provide a new and efficient way to approximate signals. In this work, we continue the study of Blaschke decompositions on weighted Hardy Spaces for functions in the larger space \({\mathcal {H}}^2({\mathbb {D}})\) under the assumption that the function has finitely many roots in \({\mathbb {D}}\). This is meaningful, since there are many functions that meet this criterion but do not extend analytically to \({\mathbb {D}}_{1+\epsilon }\) for any \(\epsilon >0\), for example \(F(z)=\log (1-z)\). By studying the growth rate of the weights, we improve the bounds provided by Coifman and Steinerberger to obtain new estimates containing all roots of F in \({\mathbb {D}}\). This provides us with new insights into Blaschke decompositions on classical function spaces including the Hardy–Sobolev spaces and weighted Bergman spaces, which correspond to making specific choices for the aforementioned weights. Further, we state a sufficient condition on the weights for our improved bounds to hold for any function in the Hardy space, \({\mathcal {H}}^2({\mathbb {D}})\), in particular functions with an infinite number of roots in \({\mathbb {D}}\). These results may help to better explain why the exponential convergence of the unwinding series is seen in many numerical examples.

中文翻译：

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加权Hardy空间上的Blaschke分解

最近，有几位作者在信号分析中考虑了傅立叶级数的非线性模拟，称为展开级或自适应傅立叶分解。在这些过程中，信号通过解析方法\（F：\ partial {\ mathbb {D}} \ rightarrow {\ mathbb {C}} \}的边界值的实分量表示，并且通过执行迭代方法为了获得一系列的Blaschke分解，仅需使用几个项就可以有效地近似信号。为了更好地理解这些方法的收敛性，Coifman和Steinerberger在假设复数值函数F具有\（{\ mathbb {D}} _ {的解析扩展的前提下，开始了对加权Hardy空间的Blaschke分解的研究。1+ \ epsilon} \）对于一些\（\ epsilon> 0 \）。这为包含单个零的F （\（\ alpha \ in {\ mathbb {D}} \））的加权Hardy范数及其Blaschke分解提供了界限。该工作还指出，在许多特定的示例中，F的展开序列以指数速率收敛到F，当与有效的算法一起计算Blaschke分解时，F导致该过程将提供一种新的，有效的方法。近似信号的方法。在这项工作中，我们继续研究在较大空间\（{\ mathcal {H}} ^ 2（{\ mathbb {D}}）\）中加权Hardy空间上的Blaschke分解假设函数在\（{\ mathbb {D}} \）中具有有限的根。这是有意义的，因为对于任何\（\ epsilon> 0 \），例如，有许多函数都满足此条件，但没有解析地扩展到\（{\ mathbb {D}} _ {1+ \ epsilon} \）\（F（z）= \ log（1-z）\）。通过研究权重的增长速度，我们改善夸夫曼和Steinerberger提供的范围，以获得含有新的估计所有的根˚F在\（{\ mathbb {d}} \）。这为我们提供了有关经典函数空间上的Blaschke分解的新见解，包括Hardy-Sobolev空间和加权Bergman空间，它们对应于为上述权重做出特定选择。此外，我们在权重上说明了一个充分的条件，以使我们的改进边界能够适用于Hardy空间中的任何函数\（{\ mathcal {H}} ^ 2（{\ mathbb {D}}）\），特别是函数在\（{\ mathbb {D}} \）中具有无限数量的根。这些结果可能有助于更好地解释为什么在许多数值示例中都能看到展开系列的指数收敛。