中文名: 三角級數
原名: Trigonometric Series
作者: (美)Antoni Zygmund
資源格式: DJVU
版本: 掃描版
出版社: Cambridge University Press
書號: 0521890535
發行時間: 1959年
地區: 美國
語言: 英文
簡介:

內容簡介:
Zygmund教授的這部著作1935年於波蘭華沙首次出版時，便在學術界確立了其典范地位。第1版雖然對細節問題沒有展開詳盡討論，但對當時的主要研究成果都給予了簡要說明。1959年，劍橋大學出版社分兩卷出版了該書第2版，書中加進了自第1版以來在三角級數。傅裡葉級數以及純數學各相關分支中的研究成果，對原書做了重大擴充。而第3版是將第2版的兩卷合在一起，芝加哥大學數學系主任Robert Fefferman還特意為其作序，介紹作者的生平轶事、對數學分析的貢獻以及本書的學術價值。
盡管本書的重點是使讀者對主要概念有個全面的理解，但它同時還向讀者介紹了實際數據分析的方方面面。本書適合作為高等院校數學及相關專業高年級本科生或研究生概率統計課程的教材，同時也可作為相關領域科技人員的參考資料。
內容截圖:

目錄:
Preface
List of Symbols
CHAPTER I
TRIGONOMETRIC SERIES AND FOURIER SERIES.
AUXILIARY RESULTS
1. Trigonometric series
2. Summation by parts
3. Orthogonal series
4. The trigonometric system
5. Fourier-Stieltjes series
6. Completeness of the trigonometric system
7. Bessel's inequality and Parseval's formula
8. Remarks on series and integrals
9. Inequalities
10. Convex functions
11. Convergence in Lr
12. Sets of the first and second categories
13. Rearrangements of functions. Maximal theorems of Hardy and
Littlewood
Miscellaneous theorems and examples
CHAPTER II
FOURIER COEFFICIENTS. ELEMENTARY THEOREMS ON
THE CONVERGENCE OF S[f] AND S[f]
1. Formal operations on S[f]
2. Differentiation and integration of sir]
3. Modulus of continuity. Smooth functions
4. Order of magnitude of Fourier coefficients
5. Formulae for partial sums of S[f] and S[f]
6. The Dini test and the principle of localization
7. Some more formulae for partial sums
8. The Dirichlet-Jordan test
9. Gibbs's phenomenon p
10. The Dini-Lipschitz test
11. Lebesgue's test
12. Lebesgue constants
13. Poisson's summation formula
Miscellaneous theorems and examples
CHAPTER III
SUMMABILITY OF FOURIER SERIES
1. Summability of numerical series
2. General remarks about the summability of S[f] and S[f]
3. Summability of S[f] and S[f] by the method of the first arithmetic
mean
4. Convergence factors
5. Summability (C, a)
6. Abel summability
7. Abel summability (cont.)
8. Summability of S[dF] and S[dF]
9. Fourier series at simple discontinuities
10. Fourier sine series
11. Gibbs's phenomenon for the method (C, a)
12. Theorems of Rogosinski
13. Approximation to functions by trigonometric polynomials
Miscellaneous theorems and examples
CHAPTER IV
CLASSES OF FUNCTIONS AND FOURIER SERIES
1. The class L2
2. A theorem of Marcinkiewicz
3. Existence of the conjugate function
4. Classes of functions and (C, 1) means of Fourier series
5. Classes of functions and (C, 1) means of Fourier series (cont.)
6. Classes of functions and Abel means of Fourier series
7. Majorants for the Abel and Cesaro means of S[f]
8. Parseval's formula
9. Linear operations
10. Classes L
11. Conversion factors for classes of Fourier series
Miscellaneous theorems and examples
CHAPTER V
SPECIAL TRIGONOMETRIC SERIES
1. Series with coefficients tending monotonically to zero
2. The order of magnitude of functions represented by series with
monotone coefficients
3. A class of Fourier-Stieltjes series
4. The series
5. The series
6. Lacunary series
7. Riesz products
8. Rademacher series and their applications
9. Series with ' small' gaps
10. A power series of Salem
Miscellaneous theorems and examples
CHAPTER VI
THE ABSOLUTE CONVERGENCE OF TRIGONOMETRIC SERIES
1. General series
2. Sets N
3. The absolute convergence of Fourier series
4. Inequalities for polynomials
5. Theorems of Wiener and Levy
6. The absolute convergence of lacunary series
Miscellaneous theorems and examples
CHAPTER VII
COMPLEX METHODS IN FOURIER SERIES
1. Existence of conjugate functions
2. The Fourier character of conjugate series
3. Applications of Green's formula
4. Integrability B
5. Lipschitz conditions
6. Mean convergence of S[f] and S[f]
7. Classes Hp and N
8. Power series of bounded variation
9. Cauchy's integral
10. Conformal mapping
Miscellaneous theorems and examples
CHAPTER VIII
DIVERGENCE OF FOURIER SERIES
1. Divergence of Fourier series of continuous functions
2. Further examples of divergent Fourier series
3. Examples of Fourier series divergent almost everywhere
4. An everywhere divergent Fourier series
Miscellaneous theorems and examples
CHAPTER IX
RIEMANN'S THEORY OF TRIGONOMETRIC SERIES
1. General remarks. The Cantor-Lebesgue theorem
2. Formal integration of series
3. Uniqueness of the representation by trigonometric series
4. The principle of localization. Formal multiplication of trigonometric
series
5. Formal multiplication of trigonometric series (cont.)
6. Sets of uniqueness and sets of multiplicity
7. Uniqueness of summable trigonometric series
8. Uniqueness of summable trigonometric series (cont.)
9. Localization for series with coefficients not tending to zero
Miscellaneous theorems and examples
Notes