Complex approximation on Riemann surfaces and in infinite dimensions
黎曼曲面和无限维上的复近似
基本信息
- 批准号:RGPIN-2016-04107
- 负责人:
- 金额:$ 1.31万
- 依托单位:
- 依托单位国家:加拿大
- 项目类别:Discovery Grants Program - Individual
- 财政年份:2019
- 资助国家:加拿大
- 起止时间:2019-01-01 至 2020-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The topic of this research is approximation theory. It is theoretical in nature, which means that it is potentially applicable to a wide variety of situations, but in order to apply to a particular field, say energy, collaboration will be required between applied mathematicians, physcists and engineers in order to make explicit how such mathematical entities as a, b and c can be interpreted as say force, potential energy and kinetic energy. More specifically, my research involves trying to approximate given functions by nicer ones. The functions we encounter in the world are often too complicated to deal with, so we seek simpler mathematical functions which approximate the real functions of the world. If these simpler functions approximate the real functions well enough, we can make the desired predictions within an acceptable degree of accuracy. For example, if we can make a watch which loses only one second in one hundred years, most people would be satisfied. Such a watch only approximates real time, but extremely well. The nice functions I wish to approximate with are the so-called analytic functions. These are functions f(x), which have the virtue that they can be written in the form of infinite series* f(x) = a + bx + cx2 + . . . ***where x is the variable and the coefficients a, b, c, . . . are constants. Analytic functions have extremely nice properties. For example, their global behaviour is uniquely determined by their local behavior. This means, for example, that if we know that some universal phenomenon behaves analytically, then, in principle, its behavior behind some distant star is completely determined by its behaviour on earth. Thus, (again in principle) we can know the behavior behind the star without going there, provided we can know it sufficiently well right here on earth. *** ***Our approach to approximating a given function g by an analytic function f is a localization process. We first approximate the function g in various local regions by analytic functions in these local regions. Then, we try to piece together these local analytic functions in order to get a global analytic function f which approximates our initial function g. This is hard to do, because the aforementioned uniqueness property of analytic functions makes it very difficult to patch analytic functions together and still retain analyticity. **
The topic of this research is approximation theory. It is theoretical in nature, which means that it is potentially applicable to a wide variety of situations, but in order to apply to a particular field, say energy, collaboration will be required between applied mathematicians, physcists and engineers in order to make explicit how such mathematical entities as a, b and c can be interpreted as say force, potential energy and kinetic energy. More specifically, my research involves trying to approximate given functions by nicer ones. The functions we encounter in the world are often too complicated to deal with, so we seek simpler mathematical functions which approximate the real functions of the world. If these simpler functions approximate the real functions well enough, we can make the desired predictions within an acceptable degree of accuracy. For example, if we can make a watch which loses only one second in one hundred years, most people would be satisfied. Such a watch only approximates real time, but extremely well. The nice functions I wish to approximate with are the so-called analytic functions. These are functions f(x), which have the virtue that they can be written in the form of infinite series* f(x) = a + bx + cx2 + . . . ***where x is the variable and the coefficients a, b, c, . . . are constants. Analytic functions have extremely nice properties. For example, their global behaviour is uniquely determined by their local behavior. This means, for example, that if we know that some universal phenomenon behaves analytically, then, in principle, its behavior behind some distant star is completely determined by its behaviour on earth. Thus, (again in principle) we can know the behavior behind the star without going there, provided we can know it sufficiently well right here on earth. *** ***Our approach to approximating a given function g by an analytic function f is a localization process. We first approximate the function g in various local regions by analytic functions in these local regions. Then, we try to piece together these local analytic functions in order to get a global analytic function f which approximates our initial function g. This is hard to do, because the aforementioned uniqueness property of analytic functions makes it very difficult to patch analytic functions together and still retain analyticity. **
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Gauthier, Paul其他文献
Gauthier, Paul的其他文献
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{{ truncateString('Gauthier, Paul', 18)}}的其他基金
Complex approximation on Riemann surfaces and in infinite dimensions
黎曼曲面和无限维上的复近似
- 批准号:
RGPIN-2016-04107 - 财政年份:2021
- 资助金额:
$ 1.31万 - 项目类别:
Discovery Grants Program - Individual
Complex approximation on Riemann surfaces and in infinite dimensions
黎曼曲面和无限维上的复近似
- 批准号:
RGPIN-2016-04107 - 财政年份:2020
- 资助金额:
$ 1.31万 - 项目类别:
Discovery Grants Program - Individual
Complex approximation on Riemann surfaces and in infinite dimensions
黎曼曲面和无限维上的复近似
- 批准号:
RGPIN-2016-04107 - 财政年份:2018
- 资助金额:
$ 1.31万 - 项目类别:
Discovery Grants Program - Individual
Complex approximation on Riemann surfaces and in infinite dimensions
黎曼曲面和无限维上的复近似
- 批准号:
RGPIN-2016-04107 - 财政年份:2017
- 资助金额:
$ 1.31万 - 项目类别:
Discovery Grants Program - Individual
Complex approximation on Riemann surfaces and in infinite dimensions
黎曼曲面和无限维上的复近似
- 批准号:
RGPIN-2016-04107 - 财政年份:2016
- 资助金额:
$ 1.31万 - 项目类别:
Discovery Grants Program - Individual
Analytic approximation
解析近似
- 批准号:
5597-2007 - 财政年份:2013
- 资助金额:
$ 1.31万 - 项目类别:
Discovery Grants Program - Individual
Analytic approximation
解析近似
- 批准号:
5597-2007 - 财政年份:2010
- 资助金额:
$ 1.31万 - 项目类别:
Discovery Grants Program - Individual
Analytic approximation
解析近似
- 批准号:
5597-2007 - 财政年份:2009
- 资助金额:
$ 1.31万 - 项目类别:
Discovery Grants Program - Individual
Analytic approximation
解析近似
- 批准号:
5597-2007 - 财政年份:2008
- 资助金额:
$ 1.31万 - 项目类别:
Discovery Grants Program - Individual
Analytic approximation
解析近似
- 批准号:
5597-2007 - 财政年份:2007
- 资助金额:
$ 1.31万 - 项目类别:
Discovery Grants Program - Individual
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