Mathematical Sciences: Applications of Fourier Analysis to Banach Space Theory

经典巴拿赫空间的子空间研究仍然是巴拿赫空间理论最重要的方向之一。基于Banach空间嵌入Lp、正定函数和稳定测度之间联系的方法是由P. Levy在20世纪30年代发现的，此后得到了广泛的发展。作者在这方面的贡献包括：解决了1938年关于正定函数的Schoenberg问题，构造了Lp空间的公共子空间，提出了Lp中等距嵌入的傅里叶变换判据，解决了1937年的p . Levy问题，解决了1969年关于p < 1的Lp的Banach子空间的Kwapien问题的等距版本。这些结果都是用傅立叶分析方法得到的。该项目解决了这个方向上的开放性问题。这些问题包括正定范数相关函数的描述、非稳定测度混合的各向同性测度的存在性问题、Lp和类群子空间的表征、Kwapien问题的同构版本和公共子空间上的问题、傅里叶变换在计算Banach- mazur距离中的应用、范数的傅里叶变换与Banach空间对偶性之间的联系、以及傅里叶变换和单位球的截面体积之间的关系。将傅里叶变换技术推广到某些无限维情况的问题在随机过程和电气工程中都有应用。傅里叶变换作为一种将信号分解成简单频率组合的方法，已被证明是数学、物理、工程和统计学不同领域的重要工具。作者十多年来一直致力于傅里叶变换在巴拿赫空间理论中的应用，他的贡献包括解决了1938年以来一直开放的关于正定函数的勋伯格问题，将傅里叶变换技术推广到无限维情况，构造了巴拿赫空间的公子空间等。在这个方向上的工作提供了泛函分析、调和分析和概率论之间的重要联系。例如，计算范数的傅里叶变换可以在任意维空间中找到凸体截面的体积，或者构造在求和下稳定的随机变量。需要计算巴拿赫空间范数（距离）的傅里叶变换的情况出现在不同的应用领域。例如，如果不可能直接测量一个信号，并且人们只知道信号到某个巴拿赫空间范数中固定函数的距离信息，那么识别信号的方法之一就是求解包含范数的某个卷积方程，通常可以使用傅里叶变换技术来求解。当由于噪声，人们只能看到信号的最大值（最高norm）时，就会发生这种情况，这种情况经常发生在电气工程中。在纤维强度的研究中，重要的是找到主要缺陷的频率，而人们无法测量每个点的强度，只能知道某些材料的最薄弱点。寻找强度的谱函数是范数傅里叶变换在材料研究中如何工作的一个例子。

DMS-9531594 Alex Koldobsky University of Texas @ San Antonio The study of subspaces of classical Banach spaces remains one of the most important directions of the Banach space theory. An approach based on the connection between embedding of Banach spaces into Lp, positive definite functions and stable measures was discovered by P. Levy in the 30's, and since then has been under intensive development. The author's contributions to this direction include a solution to the 1938 Schoenberg's problem on positive definite functions, construction of common subspaces of Lp-spaces, a Fourier transform criterion of isometric embedding into Lp contributing to the 1937 P. Levy's problem, a solution to the isometric version of the 1969 Kwapien's problem on Banach subspaces of Lp with p less than 1. All these results were obtained by using methods of Fourier analysis. The project addresses open problems in this direction. The problems include the description of positive definite norm dependent function, the related question on the existence of isotropic measures which are not mixtures of stable measures, characterization of subspaces of Lp and zonoids, isomorphic versions of Kwapien's problem and the problem on common subspaces, application of the Fourier transform to computing the Banach-Mazur distance, connections between the Fourier transform of norms and duality of Banach spaces, and between the Fourier transform and the volume of sections of the unit ball. The problem of extending the Fourier transform technique to certain infinite dimensional situations has applications to stochastic processes and electrical engineering. The Fourier transform, as a method of decomposing a signal into a combination of simple frequencies, has proved to be an important tool in different areas of mathematics, physics, engineering and statistics. The author has been working on applications of the Fourier transform to the Banach space theory for more than ten years, and his contributions include a solution to the Schoenberg problem on positive definite functions which had been open since 1938, extension of the Fourier transform technique to infinite dimensional situations, construction of common subspaces of Banach spaces and other results. Work in this direction provides important connections between functional analysis, harmonic analysis and probability theory. For example, calculating the Fourier transform of a norm one can find the volume of sections of convex bodies in spaces of arbitrary dimension, or construct random variables which are stable under summation. Situations where one has to calculate the Fourier transform of the norm (distance) of a Banach space occur in different applied areas. For example, if it is impossible to measure a signal directly, and one only has the information about the distance from the signal to fixed functions in the norm of a certain Banach space, then one of the ways to identify the signal is to solve a certain convolution equation involving the norm, which can usually be solved using the Fourier transform technique. This happens when, because of noise, one can only see the maximum of a signal (the supremum-norm), which often happens in electrical engineering. In the study of the strength of fibers, it is important to find the frequencies of the main defects while one can not measure the strength at every point and only knows the weakest points of certain pieces of the material. Finding the spectral function of the strength is an example of how the Fourier transform of norms works for the study of materials.