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。关于帧的工作将寻求找到最佳帧，以尽量减少部分数据丢失和量化误差的影响。此外，我们将看看这些新的和高度复杂的框架产生的投影是否会对铺路猜想产生任何影响。研究的第二条线是关于其他函数代数的Nevanlinna-Pick插值问题的推广。对于磁盘代数的每一个有限余维子代数，我们相信可以构造出一组再现核希尔伯特空间，其作用与模自同构函数空间对多重连通区域的作用相同。最后，我们将继续研究注入包络在算子代数中各种问题的应用。我对框架的研究实际上是由以下问题引起的。一个信号，如声波或图像，本质上是一个无限维的物体。要完全准确地表示它，就需要无限多的实数，而要在计算机上以无限精确地存储一个实数，就需要无限多的信息位。在实践中，这样的信号首先用有限个实数来近似，比如d。现在假设我们希望将这个“信号”存储在一台仅使用N=Md条信息的二进制机器上。什么是“最好”的方法来做到这一点，以便尽可能准确地恢复d实数？过去，每个实数被单独处理并分配M个空间。这保证了每个数字的近似值具有一定的精度，但如果d非常大，则所有误差的总和可能非常大。更新的想法是把d个实数的集合想象成向量，这样它们就有了大小和方向。然后我们不再单独处理每个实数，而是看向量在N个不同方向上指向的距离，这样我们就得到了N个实数，我们将近似并存储它们。问题是找到这样一组最优的方向，并证明这些新方案与旧方法相比效果如何的估计。我对插值理论的研究是关于构造最小范数或“能量”的函数，给定关于函数的某些信息，例如它在几个点上的值和一些附加的侧条件。