Hobson E.W.
Table of contents :
Title page ……Page 1
Date-line ……Page 2
Preface ……Page 3
CAMBRIDGE UNIVERSITY PRESS ……Page 5
Title ……Page 6
CONTENTS ……Page 7
1 Introduction ……Page 11
2-4 Non-convergent arithmetic series ……Page 12
5 The O—o notation ……Page 16
6 A general property of sequences ……Page 17
7-12 Convergence and divergence of series with positive terms ……Page 19
13-23 Criteria of convergence and divergence of series with positive terms ……Page 25
24-26 The convergence of series in general ……Page 44
27-28 Cesaro’s summation by arithmetic means ……Page 50
29 Series of transfinite type ……Page 53
30-36 Double sequences and double series ……Page 55
37-38 The convergence of the Cauchy-product of two series ……Page 66
39-43 The convergence of infinite products ……Page 68
44-46 The summability of series ……Page 75
47-54 Extension of Cesaro’s theory of summability ……Page 80
55-57 The equivalence of Cesaro’s and Holder’s methods of summation ……Page 95
58-60 The equivalence of Cesaro’s and Riesz’ methods of summation ……Page 100
61-63 Introduction ……Page 109
64-65 Functions related with a given function ……Page 112
66 Uniform convergence of sequences and series ……Page 114
67-68 Simply uniform convergence ……Page 115
69 Uniform divergence and uniform approach ……Page 118
70-76 Points of uniform and of non-uniform convergence ……Page 119
77-81 Tests of uniform convergence ……Page 125
82-85 The continuity of a sum-function at a point ……Page 133
86-89 The continuity of a sum-function in a domain ……Page 139
90-91 The measure of non-uniform convergence ……Page 143
92-96 The distribution of points of non-uniform convergence ……Page 145
97 Functions involving a parameter ……Page 151
98 The uniform convergence of infinite products ……Page 152
99-101 The convergence of a sequence in a measurable domain ……Page 154
102-107 Monotone sequences of functions ……Page 158
108-110 The extension of functions ……Page 164
111-113 Classes of monotone sequences ……Page 167
114-119 Uniform oscillation of a sequence of functions ……Page 170
120 Families of equi-continuous functions ……Page 177
121-123 Homogeneous oscillation ……Page 179
124-133 Introduction ……Page 182
134-135 Properties of power-series ……Page 202
136-138 The multiplication of power-series ……Page 204
139-140 Term by term differentiation and integration of power-series ……Page 206
141-150 Taylor’s series ……Page 208
151 Maxima and Minima of a function of one variable ……Page 222
152 Taylor’s theorem for functions of two variables ……Page 223
153-155 Maxima and Minima of functions of two variables ……Page 224
156-158 The limits of a series involving a parameter ……Page 231
159 Introduction ……Page 238
160-162 Weierstrass’ theorem for functions of two or more variables ……Page 240
163-165 Unbounded continuous functions ……Page 245
166-167 Standard sets of continuous functions ……Page 248
168-172 Convergence of sequences on the average ……Page 249
173-177 A classification of summable functions ……Page 259
178-180 Properties of a measurable function ……Page 264
181 Descriptive properties of sets of points ……Page 268
182-184 Sets of points of orders 1 and 2 ……Page 270
185-190 Functions representable by series or sequences of continuous functions ……Page 274
191-192 The convergence of monotone sequences of functions ……Page 284
193-196 Baire’s classification of functions ……Page 286
197 Property of a measurable function ……Page 292
198-200 The primitives of a function in a finite interval ……Page 294
201-213 The integration of series and sequences ……Page 299
214-218 Integration of series defined in an interval ……Page 313
219 Sequences of functions that are integrable ($R$) ……Page 322
220 Sequences of integrals of continuous functions ……Page 327
221-223 The oscillations of a sequence of integrals ……Page 328
224-230 The limit of an integral containing a parameter ……Page 332
231-235 The differentiation of series ……Page 342
236-238 Inversion of the order of repeated integrals ……Page 348
239-243 The inversion of repeated integrals over an infinite domain ……Page 354
244-251 Differentiation of an integral with respect to a parameter ……Page 363
252-255 Generalized Integrals ……Page 373
256-260 The method of monotone sequences ……Page 384
261 Tonelli’s theory of integration ……Page 390
262-263 Perron’s definition of an integral ……Page 392
264-266 The summability of integrals ……Page 394
267-269 The condensation of singularities ……Page 399
270 Cantor’s method of condensation of singularities ……Page 409
271-275 The construction of non-differentiable functions ……Page 411
276-278 The construction of a differentiable everywhere-oscillating function ……Page 422
279-285 The general convergence theorem ……Page 432
286-287 The general convergence theorem in the case of non-summable functions ……Page 445
288-289 Necessity of the conditions of the general convergence theorem ……Page 448
290-291 Singular Integrals ……Page 453
292-297 The convergence of singular integrals ……Page 456
299 The failure of convergence or of uniform convergence of the singular integral ……Page 466
300-301 Applications of the theory ……Page 469
302-311 The convergence of the integrals of products of functions ……Page 474
313-314 The problem of vibrating strings ……Page 486
315 Special cases of trigonometrical scries ……Page 489
316-317 Later history of the theory ……Page 490
318-321 The formal expression of Fourier’s series ……Page 492
322 The general definition of a Fourier’s series ……Page 497
323 The partial sums of a Fourier’s series ……Page 499
324 The convergence of Fourier’s series ……Page 501
325-327 Particular cases of Fourier’s series ……Page 503
328-331 Dirichlet’s investigation of Fourier’s series ……Page 512
332-333 Application of the second mean value theorem ……Page 519
334-339 The limiting values of Fourier’s coefficients ……Page 524
340-347 Conditions of convergence at a point or in an interval ……Page 531
348-350 Sufficient conditions of uniform convergence of Fourier’s series ……Page 545
351-357 Points of non-convergence of Fourier’s series for a continuous function ……Page 549
358-359 The absolute convergence of trigonometrical series ……Page 558
360-364 The integration of Fourier’s series ……Page 561
365-370 The series of arithmetic means related to Fourier’s series ……Page 567
371 The properties of a certain class of functions ……Page 574
372-375 The summability $(C, k)$ of Fourier’s series ……Page 577
376 The Cesaro summation of a Fourier-Denjoy series ……Page 581
377-381 Properties of the Fourier’s constants ……Page 583
382-384 The substitution of a Fourier’s series in an integral ……Page 591
385-386 The formal multiplication of trigonometrical series ……Page 595
387 An extension of the theorem of arithmetic means ……Page 597
388-396 Extension and generalization of Parseval’s theorem ……Page 601
397-399 M. Riesz’ extension of Parseval’s theorem ……Page 620
400-405 Systems of Fourier’s constants ……Page 624
406-409 Convergence factors for Fourier’s series ……Page 633
410-414 Poisson’s method of summation ……Page 639
415-416 Approximate representation of functions by finite trigonometrical series ……Page 646
417-419 The differentiation of Fourier’s series ……Page 649
420-426 Riemann’s theory of trigonometrical series ……Page 655
427 Investigations subsequent to those of Riemann ……Page 666
428-432 The limits of the coefficients in a trigonometrical series ……Page 669
433-439 Properties of the generalized second derivative of a function ……Page 674
440 The convergence of a trigonometrical series at a point ……Page 682
441-450 The uniqueness of a trigonometrical series which represents a function ……Page 683
451-454 Restricted Fourier’s series ……Page 696
455-458 Convergence and summability of the series allied with a Fourier’s series ……Page 702
459-461 Double Fourier’s series ……Page 708
462-463 Functions of bounded variation ……Page 712
464-466 The convergence of the double scries ……Page 715
467 The integrated series ……Page 722
468 The Cesaro summation of a double Fourier’s series ……Page 725
469 The Poisson sum of the double series ……Page 727
470 Parseval’s theorem for the double series ……Page 728
471-472 Fourier’s single integral ……Page 730
473-474 Fourier’s repeated integral ……Page 735
475-477 The summability $(phi)$ of a Fourier’s repeated integral ……Page 738
478-480 The summability $(C, r)$ of Fourier’s repeated integral ……Page 747
481-488 Fourier transforms ……Page 752
489 Introduction ……Page 763
490 The convergence of the series of orthogonal functions ……Page 765
491 The failure of convergence at a particular point ……Page 767
492-493 Extension of the theorems of Parseval and Riesz-Fischer ……Page 769
494-495 The convergence of series of orthogonal functions ……Page 772
496 Series of Sturm-Liouville functions ……Page 781
CORRECTIONS AND ADDITIONS TO VOLUME I ……Page 783
LIST OF AUTHORS QUOTED IN VOLUME II ……Page 786
GENERAL INDEX TO VOLUME II ……Page 789
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