绝对连续和霍尔德连续函数的傅里叶衰减：无限或有限振荡
Fourier decay of absolutely and Hölder continuous functions with infinitely or finitely many oscillations
AI总结：
本文研究了绝对连续和均匀霍尔德连续函数的傅里叶系数衰减率，证明了在有限差函数在有界区间内不无限振荡时，衰减率可精确估计，且霍尔德连续性可预测，但其他条件非必要。
摘要：
本文的主要结果是，若函数绝对连续且均匀霍尔德连续，且其有限差函数在有界区间内不无限振荡，则其傅里叶系数的衰减率可精确估计。该衰减率预测了相同的均匀霍尔德连续性，但另外两个条件并非必要。文献中的多个例子以及作者的研究表明，无法放松任何假设而不削弱某些函数的衰减率。研究了霍尔德连续的啁啾函数及其傅里叶系数的衰减。然后将主要结果应用于通过离散傅里叶变换计算数值威利分数导数的误差估计，并扩展到傅里叶变换。
英文摘要：
The main result of this paper is, that if we suppose that a function is absolutely continuous and uniformly Hölder continuous and that its finite difference function does not oscillate infinitely often on a bounded interval, then the decay rate of its Fourier coefficients can be estimated exactly. This rate of decay predicts the same uniform Hölder continuity but the two other conditions are not necessary. Several examples from literature and by the author show that none of the assumptions can be relaxed without weakening the decay for some functions. The uniform Hölder continuity of chirps and the decay of their Fourier coefficients are studied. The main result is then applied in the estimation of the error of numerical Weyl fractional derivatives calculated using the discrete Fourier transform. The main result is also extended to Fourier transforms.
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