Mathematical Sciences: Approximation Theory and Potentials

Totik将研究近似理论一般领域中的几个问题，对势和多项式的应用特别感兴趣。主要主题包括外场存在下的对数势、变权多项式逼近、速降多项式、正交多项式的渐近、多项式的零和正算子逼近。谐波分析结合了最能体现综合思想的数学元素。人们试图把复杂的问题分解成基本的组成部分。然后分析这些组件的基本特性。最后，通过组件的重组重构解。傅里叶级数和傅里叶变换是在这种情况下使用的工具的例子；一个是离散的，另一个是连续的。正交多项式在调和分析的某些部分起着重要的作用，因为(a)它们在微分方程的研究中自然出现(b)它们可以用来表示大的函数类(c)它们通常很容易计算。

DMS-9415657 Totik Totik will investigate several problems in the general areas of approximation theory with special interest in applications of potentials and in polynomials. The main topics include logarithmic potentials in the presence of an external field, polynomial approximation with varying weights, fast decreasing polynomials, asymptotic for orthogonal polynomials, zeros of polynomials, and approximation by positive operators. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one discrete , the other representing a continuous decomposition. Orthogonal polynomials play an important role in certain segments of harmonic analysis because (a) they arise naturally in the study of differential equations (b) they can be used to represent large classes of functions and (c) they are often easily computable.