Oscillatory Sums and Radon - Fourier Analysis

Oskolkov将继续他的调查振荡总和和积分涉及虚指数的真实的代数多项式。 多项式的系数被认为是独立的真实的变量。这些求和和积分建立了这些表面上 遥远的领域，如调和分析，解析数论和偏微分方程。他还将阐述 多元函数的Radon-Fourier分析。特别强调的是脊近似的相关问题，包括最佳拟合线性的选择 平面波型函数的组合。这些问题是高度非线性的性质，特别是相对于波矢量的最佳选择。 这些研究的直接后果包括神经 网络模型在工业、工程、计算机科学、金融和医学中得到了广泛的应用。Radon变换是一种公认的 强大的工具，在几个应用领域， 经典量子力学和光学到X射线断层扫描。 在计算X射线断层摄影中，脊近似可以被解释为非均匀性（例如肿瘤）图像的重建。另一个截然不同的领域是识别大型系统（如细菌种群或股票市场）进化的主要趋势，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.