U.S.-Croatia Mathematics Research on the Theory of Reproducing Function Systems

这个由华盛顿大学的Guido Weiss和Edward Wilson以及萨格勒布大学的Hrvoje Sikic合作的美国-克罗地亚研究项目的目标是加强再现平方可积函数的函数系统所需的理论基础。这将通过汇集具有互补专业知识的高级和初级美国和克罗地亚数学家团队来完成，首先从代数的角度重新检查平移不变空间方法，其次，结合过滤器的思想。第三，他们将研究最近开发的统一方法的各种应用，以描述有限基族的函数，其中一组适当的平移，扩张和调制产生了分析和/或综合系统。结果有望完善小波和再现函数系统领域的理论框架，从而允许同时研究具有极大普遍性的系统。成功将提高我们进行高度精确的信号分析和信息传递的能力，对统计分析和可能应用于通信网络、编码系统、信息存储和检索、卫星和医疗成像数据分析都有影响。该数学项目实现了促进科学知识的计划目标，使美国和中欧南部的专家能够在共同感兴趣和能力强的领域结合互补的人才和共享研究资源。

The goal of this U.S.-Croatian cooperative research project between Guido Weiss and Edward Wilson of Washington University and Hrvoje Sikic of the University of Zagreb is to strengthen the theoretical foundation required for reproducing function systems of square integrable functions. This will be accomplished by bringing together a team of senior and junior U.S. and Croatian mathematicians with complementary expertise to first reexamine the shift invariant spaces approach from an algebraic perspective and second, incorporate the idea of a filter. Thirdly, they will examine various applications of a recently developed unified approach for characterizing those functions of finite families of foundations for which an appropriate set of translations, dilations and modulations produce an analyzing and/or synthesizing system.Results are expected to refine the theoretical framework for the field of wavelets and reproducing function systems thereby allowing systems of great generality to be studied simultaneously. Success would enhance our ability to conduct highly accurate signal analyses and information transfer, with implications for statistical analysis and likely applications in communication networks, coding systems, information storage and retrieval, and analysis of satellite and medical imaging data. . This mathematics project fulfills the program objective of advancing scientific knowledge by enabling experts in the United States and South Central Europe to combine complementary talents and share research resources in areas of strong mutual interest and competence.