Frames, Interpolation and Injective Envelopes

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优的帧，以最小化部分数据丢失和量化误差的影响。此外，我们将看看从这些新的和高度复杂的框架中产生的投影是否会对epsilon铺路猜想产生任何影响。第二个研究方向是得到其他函数代数的Nevanlinna-Pick插值问题的推广。对于圆盘代数的每个有限余维子代数，我们相信可以构造一族再生核Hilbert空间，其作用与多连通区域的模自同构函数空间所起的作用相同。最后，我们将继续研究内射包络在算子代数中各种问题中的应用。我对框架的研究真正受到以下问题的启发。信号，如声波或图像，本质上是无限维的物体。要完全准确地表示它，一个人需要无限多个实数，而在计算机上以无限精度存储即使是一个实数也需要无限多比特的信息。在实践中，这样的信号首先被有限多个例如d个实数近似。现在，假设我们希望仅使用N=md条信息在二进制机器上存储该“信号”。为了尽可能准确地恢复d个实数，“最佳”方法是什么？在过去，每个实数被分开处理，并被分配M空间。这保证了每个数字都以一定的精度进行近似，但是如果d非常大，那么所有误差的总和可能非常大。较新的想法是将d个实数集想象为向量，这样它们既有大小又有方向。然后，我们将不再单独处理每个实数，而是查看向量在N个不同方向上指向的距离，这现在给出了N个实数，我们将对其进行近似和存储。问题是找到最优的方向集，并证明与旧方法相比，这些新方案的工作效果如何的估计。我对内插理论的研究涉及构造给定函数的某些信息，如它在几个点的值和一些附加的边条件下的最小范数或“能量”函数。

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.