近代解析應(yīng)用數(shù)學(xué)基礎(chǔ)

第一章 度量空間 1                  
 1.1 集合與映射 1                  
 1.1.1 集合的概念 1                  
 1.1.2 集合的表示和舉例 2                  
 1.1.3 集合的簡(jiǎn)單運(yùn)算 3                  
 1.1.4 映射 4                  
 1.1.5 可數(shù)集. 不可數(shù)集和集合的基數(shù) 6                  
 1.2 線性空間 7                  
 1.2.1 線性空間 7                  
 1.2.2 常見的線性空間 8                  
 1.2.3 線性子空間 10                  
 1.2.4 線性空間的維數(shù) 10                  
 1.3 度量空間 12                  
 1.3.1 度量空間 12                  
 1.3.2 常見的幾種度量線性空間 13                  
 1.3.3 幾個(gè)重要的不等式 15                  
 1.4 勒貝格(Lebesgue)積分和Lp空間 19                  
 1.4.1 測(cè)度, 可測(cè)集 19                  
 1.4.2 可測(cè)函數(shù) 22                  
 1.4.3 勒貝格積分 24                  
 1.4.4 Lp(E)函數(shù)空間 26                  
 1.5 度量空間的拓?fù)湫再|(zhì) 28                  
 1.5.1 點(diǎn)集的鄰域 29                  
 1.5.2 開集和閉集 30                  
 1.5.3 連續(xù)映射 32                  
 1.6 度量空間的可分性. 完備性和緊性 33                  
 1.6.1 度量空間的可分性 33                  
 1.6.2 序列的收斂和極限 34                  
 1.6.3 度量空間的完備性 37                  
 1.6.4 常見的完備度量空間 38                  
 1.6.5 度量空間的完備化 41                  
 1.6.6 度量空間的緊性和列緊性 42                  
 習(xí)題 43                  
 參考書目 46                  
                   
 第二章 賦范空間和內(nèi)積空間48                  
 2.1 賦范線性空間 49                  
 2.1.1 賦范線性空間. 巴拿赫空間 49                  
 2.1.2 常見的賦范線性空間 50                  
 2.1.3 賦范空間中的序列和級(jí)數(shù)的收斂 51                  
 2.1.4 賦范空間中的無(wú)窮級(jí)數(shù) 53                  
 2.1.5 有限維賦范空間和子空間 54                  
 2.1.6 賦范空間的同構(gòu)性 57                  
 2.2 內(nèi)積空間和希爾伯特空間 58                  
 2.2.1 內(nèi)積空間和希爾伯特空間 58                  
 2.2.2 常見的內(nèi)積空間 62                  
 2.3 內(nèi)積空間中的正交和投影 64                  
 2.3.1 正交性 65                  
 2.3.2 正交投影 66                  
 2.4 內(nèi)積空間的標(biāo)準(zhǔn)正交基 70                  
 2.4.1 標(biāo)準(zhǔn)正交集 70                  
 2.4.2 內(nèi)積空間的標(biāo)準(zhǔn)正交系 74                  
 2.4.3 內(nèi)積空間的標(biāo)準(zhǔn)正交基 77                  
 2.4.4 常用標(biāo)準(zhǔn)正交基舉例 81                  
 2.5 在逼近論中的應(yīng)用 85                  
 2.5.1 賦范空間中的逼近 85                  
 2.5.2 希爾伯特空間中的逼近 90                  
 習(xí)題 93                  
 參考書目 96                  
                   
 第三章 線性算子和線性泛函 97                  
 3.1 線性算子 97                  
 3.1.1 線性算子 97                  
 3.1.2 有限維線性空間的線性算子與矩陣表示 100                  
 3.1.3 逆算子 102                  
 3.2 有界線性算子 104                  
 3.2.1 有界線性算子 104                  
 3.2.2 線性算子的連續(xù)性 106                  
 3.2.3 線性算子空間 108                  
 3.2.4 下有界算子與逆算子 112                  
 3.3 有界線性泛函和對(duì)偶空間 113                  
 3.3.1 有界線性泛函 113                  
 3.3.2 對(duì)偶空間 115                  
 3.3.3 希爾伯特空間上泛函的一般形式 119                  
 3.3.4 雙線性泛函和二次泛函 123                  
 3.4 希爾伯特伴隨算子 126                  
 3.4.1 一個(gè)伴隨算子的特例 126                  
 3.4.2 希爾伯特伴隨算子 127                  
 3.4.3 希爾伯特伴隨算子的重要性質(zhì) 130                  
 3.5 希爾伯特空間的自伴算子. 酋算子和正規(guī)算子 131                  
 3.5.1 自伴算子 131                  
 3.5.2 酋算子和正規(guī)算子 135                  
 3.5.3 正算子 136                  
 3.6 投影算子 138                  
 3.6.1 投影算子 138                  
 3.6.2 投影算子的代數(shù)運(yùn)算 140                  
 3.7 希爾伯特空間中的無(wú)界線性算子 144                  
 3.7.1 無(wú)界線性算子的概念 145                  
 3.7.2 無(wú)界線性算子的伴隨算子 146                  
 3.7.3 對(duì)稱算子和自伴算子 148                  
 習(xí)題 150                  
 參考書目 154                  
                   
 第四章 泛函的極值問(wèn)題 155                  
 4.1 泛函極值問(wèn)題的提法 155                  
 4.1.1 泛函極值問(wèn)題的由來(lái)和意義 155                  
 4.1.2 經(jīng)典變分法中兩個(gè)著名的極值問(wèn)題 156                  
 4.1.3 泛函極值問(wèn)題的提法 158                  
 4.2 泛函的微分(變分) 159                  
 4.2.1 函數(shù)的微分 159                  
 4.2.2 經(jīng)典變分法中變分的概念 161                  
 4.2.3 算子的加脫微分 163                  
 4.2.4 算子的弗雷謝微分 165                  
 4.3 泛函的無(wú)約束極值 168                  
 4.3.1 泛函極值及其必要條件 168                  
 4.3.2 歐拉-拉格朗日方程 169                  
 4.3.3 捷線問(wèn)題的解 172                  
 4.3.4 自由邊界和自然邊界條件 174                  
 4.4 泛函的約束極值問(wèn)題 175                  
 4.4.1 有限維約束問(wèn)題 175                  
 4.4.2 無(wú)窮維約束問(wèn)題 180                  
 4.4.3 約束極值問(wèn)題舉例 185                  
 4.5 求泛函極值的下降法 188                  
 4.5.1 下降法的一般原理 188                  
 4.5.2 最速下降法 189                  
 4.5.3 共軛方向法 193                  
 4.5.4 共軛梯度法 197                  
 習(xí)題 201                  
 參考書目 203                  
                   
 第五章 線性算子方程 204                  
 5.1 壓縮映射與不動(dòng)點(diǎn)原理 204                  
 5.1.1 線性算子方程 204                  
 5.1.2 不動(dòng)點(diǎn) 206                  
 5.1.3 壓縮映射原理 207                  
 5.2 線性算子的譜 212                  
 5.2.1 特征值概念的回顧 213                  
 5.2.2 線性算子譜的概念 216                  
 5.2.3 有界線性算子譜的某些性質(zhì) 217                  
 5.2.4 希爾伯特自伴算子譜的性質(zhì) 220                  
 5.2.5 無(wú)界自伴算子譜的性質(zhì) 224                  
 5.3 微分算子方程 227                  
 5.3.1 自伴二階線性常微分算子 227                  
 5.3.2 二階線性偏微分算子 230                  
 5.3.3 常見二階自伴線性微分算子方程 231                  
 5.4 積分算子方程 234                  
 5.4.1 積分方程的來(lái)源和分類 234                  
 5.4.2 積分方程的逐次逼近解法 239                  
 5.4.3 弗雷德霍姆定理 244                  
 5.4.4 全連續(xù)(緊)積分算子 249                  
 5.5 算子方程的變分原理 252                  
 5.5.1 自伴算子的確定性方程 252                  
 5.5.2 非自伴算子的確定性方程 256                  
 5.5.3 特征值問(wèn)題 258                  
 5.6 變分方程的瑞利-里茲(Rayleigh-Ritz)解法 259                  
 5.6.1 自伴問(wèn)題的瑞利-里茲法 260                  
 5.6.2 瑞利-里茲法求解二階自伴常微分方程邊值問(wèn)題 262                  
 5.6.3 非自伴問(wèn)題的瑞利-里茲法 264                  
 5.7 基于變分原理的有限元法 265                  
 5.7.1 問(wèn)題的提法和出發(fā)點(diǎn) 266                  
 5.7.2 區(qū)域的剖分 267                  
 5.7.3 插值函數(shù)的構(gòu)造 267                  
 5.7.4 單元?jiǎng)偠确治?nbsp;269                  
 5.7.5 有限元方程 271                  
 5.8 加權(quán)余量法 273                  
 5.8.1 加權(quán)余量法的基本公式 273                  
 5.8.2 矩量法--內(nèi)域積分形式的加權(quán)余量法 274                  
 5.8.3 基于加權(quán)余量法的有限元法和邊界元法 278                  
 習(xí)題 284                  
 參考書目 286                  
                   
 第六章 廣義函數(shù) 288                  
 6.1 引入廣義函數(shù)的必要性 288                  
 6.1.1 古典函數(shù)的局限性 288                  
 6.1.2 微分方程古典解的局限性 290                  
 6.1.3 廣義函數(shù)概念大意 293                  
 6.2 基本空間和廣義函數(shù) 295                  
 6.2.1 C(Ω)和Cm(Ω)上的極限及其完備性 295                  
 6.2.2 基本函數(shù)空間C∞(Ω)和C∞0(Ω) 298                  
 6.2.3 廣義函數(shù)和廣義函數(shù)空間 300                  
 6.2.4 廣義函數(shù)的支集 303                  
 6.3 廣義函數(shù)的基本運(yùn)算 304                  
 6.3.1 廣義函數(shù)的導(dǎo)數(shù) 305                  
 6.3.2 廣義函數(shù)的極限 312                  
 6.3.3 廣義函數(shù)的乘法運(yùn)算 315                  
 6.3.4 廣義函數(shù)的卷積 318                  
 6.4 廣義函數(shù)的傅里葉(Fourier)變換 321                  
 6.4.1 普通函數(shù)傅里葉變換回顧 321                  
 6.4.2 速降函數(shù)和緩增廣義函數(shù) 325                  
 6.4.3 緩增廣義函數(shù)的傅里葉變換 331                  
 6.5 偏微分方程的廣義解 335                  
 6.5.1 偏微分方程的廣義解 336                  
 6.5.2 線性偏微分方程弱解存在的條件 338                  
 6.5.3 偏微分方程的基本解 340                  
 6.6 索伯列夫(Sobolev)空間 347                  
 6.6.1 空間Wm,p(Ω) 347                  
 6.6.2 Hm(Ω)空間 349                  
 6.6.3 嵌入定理 351                  
 6.6.4 二階橢圓型方程狄里赫雷問(wèn)題在H10(Ω)中的可解性 352                  
 習(xí)題 356                  
 參考書目 357                  
                   
 第七章 小波分析 358                  
 7.1 窗口傅里葉變換 358                  
 7.1.1 傅里葉變換的局限性 358                  
 7.1.2 窗口傅里葉變換 360                  
 7.1.3 窗口傅里葉變換的時(shí)域-頻域局域性 362                  
 7.2 連續(xù)小波變換 365                  
 7.2.1 連續(xù)小波變換 365                  
 7.2.2 小波變換的時(shí)-頻局域性 370                  
 7.2.3 連續(xù)小波變換的基本性質(zhì) 372                  
 7.3 離散小波變換 375                  
 7.3.1 離散小波變換 375                  
 7.3.2 二進(jìn)制離散小波變換 376                  
 7.3.3 Haar小波基和Shannon小波基 378                  
 7.4 多分辨分析和小波正交基 382                  
 7.4.1 正交多分辨分析 382                  
 7.4.2 小波子空間 383                  
 7.4.3 L2(R)的正交小波基的構(gòu)造 385                  
 7.4.4 利用MRA構(gòu)造Haar和Shannon小波基 391                  
 7.5 緊支集正交小波基 394                  
 7.5.1 有限長(zhǎng)雙尺度方程及其求解 394                  
 7.5.2 有限長(zhǎng)雙尺度方程存在解的條件 397                  
 7.5.3 緊支集正交小波基的構(gòu)造 403                  
 7.6 小波框架 405                  
 7.6.1 框架概念 406                  
 7.6.2 希爾伯特空間的框架理論 408                  
 7.6.3 小波框架 412                  
 7.7 小波分解與重構(gòu)算法 415                  
 7.7.1 函數(shù)的多尺度分解 415                  
 7.7.2 分解算法與重構(gòu)算法 418                  
 7.8 小波與取樣定理 483                  
 7.8.1 Shannon取樣定理 423                  
 7.8.2 小波取樣定理 425                  
 7.9 二維正交小波基 429                  
 7.10 小波與算子方程計(jì)算 433                  
 7.10.1 函數(shù)按Haar基展開的算法 434                  
 7.10.2 二維小波基展開和積分算子的近似對(duì)角化 438                  
 參考書目 441