Mathematical Sciences: Research in Fourier Analysis

这个项目继续在傅立叶的现代方面的工作 分析相关的小波理论，加权不等式， 傅立叶变换及其限制定理。 加权傅里叶变换不等式的动机是 信号处理的一些核心问题。 以线性 系统理论，权重对应于能量中的各种滤波器 浓度问题。 另一个应用领域，预测 理论，加权Lebesgue空间出现对应的权重 平稳随机过程的功率谱。 对象 研究的目的是得到q次幂范数的界， 加权p次加权傅里叶变换 信号的范数（原始函数）。 该计划寻求最好的 可能的估计。 这种搜索带来了不确定性 原则不平等;这些提供了自然的约束， 确定频谱估计技术的有效性 用于信号处理。 小波理论方法在实际应用中发挥了重要作用， 在这项活动中发挥重要作用。 初期工作将侧重于 二次（加权）范数的特殊情况。 当有关的信息时，可以获得不等式。 权重的递减重排（的积分）可以是 计算。 同样的条件得到没有重新安排 不等式成立时的权重。 需要做的是 out是在必要和充分之间发生的事情 条件 来自balayage的技术结合 将采用底层帐篷空间的原子分解 以澄清不同的情况。

This project continues work on modern aspects of Fourier analysis related to wavelet theory, weighted inequalities for the Fourier transform and restriction theorems for the transform. Weighted Fourier transform inequalities are motivated by some of the central issues of signal processing. In linear system theory, weights correspond to various filters in energy concentration problems. Another area of application, prediction theory, weighted Lebesgue spaces arise for weights corresponding to power spectra of stationary stochastic processes. The object of the research is to obtain bounds on the q-th power norm of the weighted Fourier transform in terms of the weighted p-th power norm of the signal (original function). The program seeks best possible estimates. This search gives rise to uncertainty principle inequalities; these provide natural constraints for determining the effectiveness of spectrum estimation techniques used in signal processing. The methods of wavelet theory play a basic role in this activity. Initial efforts will focus on the special case of quadratic (weighted) norms. Inequalities may be obtained when information concerning the (integrals of) decreasing rearrangements of the weights can be computed. The same condition obtains without rearranging the weights when the inequalities are valid. What needs to be worked out is what happens between the necessary and sufficient conditions. Techniques derived from balayage combined with atomic decompositions of underlying tent spaces will be employed in efforts to clarify the different conditions.