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佳帧，以使部分数据丢失和量化误差的影响最小化。此外，我们还将看到这些新的高度复杂的框架所产生的投影是否会对ε铺路猜想产生任何影响。第二行的研究是关于获得推广的Nevanlinna-Pick插值问题的其他功能代数。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，它们在多连通区域中起着与模自守函数空间相同的作用。最后，我们将继续研究内射包络在算子代数中的各种问题中的应用。信号，如声波或图像，本质上是一个无限维的对象。要完全精确地表示它，就需要无穷多个真实的数字，而要在计算机上以无穷大的精确度存储一个真实的数字，就需要无穷多比特的信息。在实践中，这样的信号首先由多个（比如说d个）真实的数来近似。现在假设我们希望将这个“信号”存储在一个二进制机器上，只使用N=Md条信息。什么是“最好的”方法来做到这一点，使d真实的数字可以恢复尽可能准确？在过去，每个真实的数被单独处理，并分配M个空间。这保证了每个数字都以一定的精度近似，但是如果d非常大，那么所有误差的总和可能非常大。较新的想法是把d个真实的数的集合想象成向量，这样它们既有大小又有方向。然后，我们不再单独处理每个真实的数，而是看看向量在N个不同方向上的指向距离，这就给了我们N个真实的数，我们将近似并存储它们。问题是要找到最佳的这样一组方向，并证明估计，将告诉如何以及这些新计划的工作相比，旧的方法。我的研究插值理论是有关构造函数的最小范数或“能量”给定某些片段的功能，如它的值在几个点和一些额外的侧条件。

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.