相關主題
商品描述
Now considered a classic text on the topic, Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis by first developing the theory of measure and integration in the simple setting of Euclidean space, and then presenting a more general treatment based on abstract notions characterized by axioms and with less geometric content.
Published nearly forty years after the first edition, this long-awaited Second Edition also:
- Studies the Fourier transform of functions in the spaces L1, L2, and Lp, 1 p
- Shows the Hilbert transform to be a bounded operator on L2, as an application of the L2 theory of the Fourier transform in the one-dimensional case
- Covers fractional integration and some topics related to mean oscillation properties of functions, such as the classes of Hölder continuous functions and the space of functions of bounded mean oscillation
- Derives a subrepresentation formula, which in higher dimensions plays a role roughly similar to the one played by the fundamental theorem of calculus in one dimension
- Extends the subrepresentation formula derived for smooth functions to functions with a weak gradient
- Applies the norm estimates derived for fractional integral operators to obtain local and global first-order Poincaré-Sobolev inequalities, including endpoint cases
- Proves the existence of a tangent plane to the graph of a Lipschitz function of several variables
- Includes many new exercises not present in the first edition
This widely used and highly respected text for upper-division undergraduate and first-year graduate students of mathematics, statistics, probability, or engineering is revised for a new generation of students and instructors. The book also serves as a handy reference for professional mathematicians.
商品描述(中文翻譯)
現在被視為該主題的經典著作，測度與積分：實分析導論通過首先在歐幾里得空間的簡單環境中發展測度和積分的理論，然後基於以公理為特徵的抽象概念，並且幾何內容較少，來介紹實分析。
這本期待已久的第二版於首次出版近四十年後發行，並且：
- 研究了空間L1、L2和Lp中函數的傅立葉變換，1
- 顯示希爾伯特變換在L2上是一個有界算子，作為一維情況下L2傅立葉變換理論的應用
- 涵蓋分數積分及一些與函數的均值振盪性質相關的主題，例如霍爾德連續函數的類別和有界均值振盪函數的空間
- 推導出一個子表示公式，在更高維度中，其作用大致類似於一維中微積分基本定理的作用
- 將為光滑函數推導的子表示公式擴展到具有弱梯度的函數
- 應用為分數積分算子推導的範數估計，以獲得局部和全局的一階Poincaré-Sobolev不等式，包括端點情況
- 證明了多變數Lipschitz函數圖形的切平面存在性
- 包括許多在第一版中未出現的新習題
這本廣泛使用且備受尊敬的教材適用於數學、統計學、概率論或工程學的高年級本科生和一年級研究生，經過修訂以適應新一代的學生和教師。該書也作為專業數學家的便捷參考。
作者簡介
Richard L. Wheeden is Distinguished Professor of Mathematics at Rutgers University, New Brunswick, New Jersey, USA. His primary research interests lie in the fields of classical harmonic analysis and partial differential equations, and he is the author or coauthor of more than 100 research articles. After earning his Ph.D. from the University of Chicago, Illinois, USA (1965), he held an instructorship there (1965-1966) and a National Science Foundation (NSF) Postdoctoral Fellowship at the Institute for Advanced Study, Princeton, New Jersey, USA (1966-1967).
Antoni Zygmund was Professor of Mathematics at the University of Chicago, Illinois, USA. He was earlier a professor at Mount Holyoke College, South Hadley, Massachusetts, USA, and the University of Pennsylvania, Philadelphia, USA. His years at the University of Chicago began in 1947, and in 1964, he was appointed Gustavus F. and Ann M. Swift Distinguished Service Professor there. He published extensively in many branches of analysis, including Fourier series, singular integrals, and differential equations. He is the author of the classical treatise Trigonometric Series and a coauthor (with S. Saks) of Analytic Functions. He was elected to the National Academy of Sciences in Washington, District of Columbia, USA (1961), as well as to a number of foreign academies.
作者簡介(中文翻譯)
Richard L. Wheeden 是美國新澤西州羅格斯大學（Rutgers University, New Brunswick, New Jersey）的數學特聘教授。他的主要研究興趣集中在古典調和分析和偏微分方程領域，並且是超過100篇研究文章的作者或合著者。在1965年從美國伊利諾伊州芝加哥大學（University of Chicago, Illinois）獲得博士學位後，他在該校擔任講師（1965-1966），並在美國新澤西州普林斯頓的高等研究所（Institute for Advanced Study, Princeton, New Jersey）獲得國家科學基金會（NSF）博士後獎學金（1966-1967）。
Antoni Zygmund 曾是美國伊利諾伊州芝加哥大學的數學教授。他之前曾在美國麻薩諸塞州南哈德利的霍利奧克學院（Mount Holyoke College, South Hadley, Massachusetts）和美國賓夕法尼亞州費城的賓夕法尼亞大學（University of Pennsylvania, Philadelphia）任教。他在芝加哥大學的任教生涯始於1947年，並於1964年被任命為古斯塔夫斯·F·和安·M·斯威夫特特聘服務教授（Gustavus F. and Ann M. Swift Distinguished Service Professor）。他在許多分析分支領域發表了大量著作，包括傅立葉級數、奇異積分和微分方程。他是經典著作《三角級數》（Trigonometric Series）的作者，並與S. Saks合著了《解析函數》（Analytic Functions）。他於1961年當選為美國華盛頓國家科學院（National Academy of Sciences in Washington, District of Columbia）成員，以及多個外國學院的成員。
