Frames, Interpolation and Injective Envelopes

框架、插值和内射包络

• 批准号: 0600191
• 负责人: Vern Paulsen
• 金额: $ 14.56万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600191&HistoricalAwards=false
• 关键词: Frames; Interpolation; Injective; Envelopes

The proposed research will follow three main directions. The work on frames will seek to find optimal frames for minimizing the effects of partial data loss and of quantization errors. In addition, we will see if the projections that arise from these new and highly complex frames can have any impact on the epsilon-paving conjecture. The second line of research is concerned with obtaining generalizations of the Nevanlinna-Pick interpolation problem for other function algebras. For each finite codimension subalgebra of the disk algebra, we believe that one can construct a family of reproducing kernel Hilbert spaces that play the same role as the spaces of modulus automorphic functions play for multiply-connected regions. Finally, we will continue our study of applications of injective envelopes to various problems in operator algebras.My research on frames is really motivated by the following problem. A signal, such as a sound wave or an image, is inherently an infinite dimensional object. To represent it with complete accuracy, one would need infinitely many real numbers and to store even a single real number on a computer with infinite accuracy would require infinitely many bits of information. In practice such a signal is first approximated by finitely many, say d, real numbers. Now suppose that we wish to store this "signal" on a binary machine using only N=Md pieces of information. What is the "best" way to do this so that the d real numbers can be recovered as accurately as possible? In the past, each real number was treated separately and alloted M spaces. This guarantees that each number is approximated with a certain accuracy, but if d is very large, then the sum of all the errors could be very large. The newer idea is to imagine sets of d real numbers as vectors, so that they have both a magnitude and a direction. Then instead of treating each real number separately, we will look at how far the vector points in N different directions, which now gives us N real numbers, that we will approximate and store. The problem is to find the optimal such set of directions and to prove estimates that will tell how well these new schemes work compared to the old methods.My research on interpolation theory is concerned with constructing functions of minimum norm or "energy" given certain pieces of information about the function, such as its values at just a few points and some additional side conditions.

拟议的研究将遵循三个主要方向。框架上的工作将寻求找到最佳的框架，以最大程度地减少部分数据丢失和量化错误的影响。此外，我们将看看这些新的且高度复杂的框架产生的预测是否会对Epsilon-Paiving猜想产生任何影响。第二道研究与获得其他功能代数的Nevanlinna-Pick插值问题的概括有关。对于磁盘代数的每个有限的编成子代数，我们认为可以构建一个繁殖核心希尔伯特空间的家族，这些家族与模量自动形态功能的空间相同的作用起着多种连接区域的作用。最后，我们将继续研究操作员代数中各种问题的注射信封的应用。我对框架的研究确实是由以下问题激发的。信号（例如声波或图像）本质上是无限的尺寸对象。为了完全准确地表示它，一个人将需要无限的实数，甚至以无限精度存储一个实数将需要无限的信息。实际上，这种信号首先被有限的许多（例如D，实数）近似。现在，假设我们希望仅使用n = md的信息将此“信号”存储在二进制计算机上。这样做的“最佳”方法是什么，以便D实数可以尽可能准确地恢复？过去，每个实际数字都进行了分别处理和分配的M空间。这确保每个数字都具有一定精度近似，但是如果D非常大，那么所有错误的总和可能很大。更新的想法是想象一组D实数是向量，以便它们既有大小又有方向。然后，我们将查看n个不同方向的向量点多远，而不是分别处理每个实数，而现在为我们提供了n个实数，我们将近似并存储。问题是要找到最佳的此类方向，并证明与旧方法相比，这些新方案的效果如何。我对插值理论的研究与构建有关该功能的某些信息的最低规范或“能量”的功能有关，例如其值，例如其值，例如在几点点和其他侧面条件。