Oscillatory Sums and Radon - Fourier Analysis

Oskolkov将继续研究涉及实代数多项式的虚指数的振荡和和积分。将多项式的系数视为独立的实变量。这些和和积分在调和分析、解析数理论和偏微分方程等看似遥远的领域之间建立了相互联系。他还将详细阐述多变量函数的Radon-Fourier分析。文中将特别强调脊形近似的相关问题，包括平面波型函数的最佳拟合线性组合的选择。这些问题是高度非线性的，特别是关于波矢的最优选择。这类研究的直接后果包括神经网络模型，这些模型在工业、工程、计算机科学、金融和医学中得到了广泛应用。Radon变换在从经典量子力学和光学到X射线层析成像的几个应用领域中都是公认的强大工具。在计算X射线层析成像中，脊线近似可以解释为对肿瘤等非均质性图像的重建。另一个非常不同的领域是认识大系统进化的主要趋势，如细菌种群或股票市场，其中必须对大量变量进行建模。

Oskolkov will continue his investigations of oscillatory sums and integrals involving imaginary exponentials of real algebraic polynomials. The coefficients of the polynomials are considered as independent real variables. These sums and integrals establish interconnections between such apparently distant fields as Harmonic Analysis, Analytic Number Theory and Partial Differential Equations. He will also elaboratee on Radon-Fourier analysis for functions of several variables. Special emphasis will be made on associated problems of ridge approximation, which consist in selection of best fit linear combinations of planar wave type functions. These problems are of highly non-linear nature, especially with respect to optimal selection of wave vectors. Immediate ramifications of such research include neural network models that have found wide applications in industry, engineering, computer science, finance and medicine. The Radon transform is an acknowledged powerful tool in several applied areas ranging from classical quantum mechanics and optics to X-ray tomography. In computational X-ray tomography, ridge approximation can be interpreted as reconstruction of images of non-homogeneities such as tumors. Another extremely different field is recognition of main trends in evolution of large systems such as bacteria populations or the stock market, where an enormous number of variables must be modeled.