Fourier Analysis : An Introduction (Princeton Lectures in Analysis, No. 1) (Hardcover)

Elias M. Stein, Rami Shakarchi

  • 出版商: Princeton University
  • 出版日期: 2003-04-06
  • 售價: $1,580
  • 貴賓價: 9.8$1,548
  • 語言: 英文
  • 頁數: 328
  • 裝訂: Hardcover
  • ISBN: 069111384X
  • ISBN-13: 9780691113845
  • 相關分類: 物理學 Physics
  • 立即出貨(限量) (庫存=1)

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商品描述

This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions.

The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression.

In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest.

The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.

商品描述(中文翻譯)

這本書的第一卷是這個主題的三部分介紹,適合具備初步數學分析知識並有動機探索傅立葉分析的學生。它從一個簡單的信念開始,即傅立葉在19世紀初研究物理科學問題時得出的結論,即任意函數可以表示為最基本的三角函數的無窮和。

第一部分以傅立葉級數的收斂和可加性概念來實現這個想法,同時突出應用,如等周不等式和均勻分佈。第二部分涉及傅立葉變換及其在經典偏微分方程和Radon變換中的應用;對該主題的清晰介紹有助於避免技術困難。書末介紹了有限可達群的傅立葉理論,並應用於算術進行中的質數。

在組織他們的講解時,作者精心平衡了對關鍵概念洞察力的強調與提供嚴謹分析的技術基礎之間的需求。數學、物理、工程和其他科學的學生將發現本卷涵蓋的理論和應用非常有趣。

《普林斯頓分析講座》是一個持續努力,旨在介紹數學分析的核心領域,同時展示它們之間的有機統一性。計劃中的四卷書中的第一卷《傅立葉分析》以大量的例子和應用突顯了分析中某些思想對其他數學領域和各種科學的深遠影響。斯坦和沙卡爾奇從介紹傅立葉級數和積分到深入探討復分析;測度和積分理論,以及希爾伯特空間;最後,進一步討論了功能分析、分布和概率論的要素。