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Natanson I.P. — Theory of Functions of a Real Variable. Volume II
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Название: Theory of Functions of a Real Variable. Volume IIАвтор: Natanson I.P. Аннотация: This is a translation of Professor Natanson's Teoriya Funktsii Veshchestvennoi Peremennoi, Chapters X-XVI, XVIII of the second revised Russian edition, 1957; Chapter XVIII forms the content of Chapter XVII of the present volume. Chapter XVII of the Russian edition was omitted as such because the author follows quite closely (except for Appendix VI) the material in the Boron-Hewitt American edition of Chapters I-IX listed as Editor's Appendices and written by Professor Dr. Edwin Hewitt, dealing with the definition and basic properties of arbitrary measurable sets on the real line, the basic properties of measurable functions defined on sets which are not necessarily bounded, the definition of the Lebesgue integral for functions defined on unbounded sets, square-summable functions defined on arbitrary measurable sets on the real line, functions of finite variation on the infinite line (— infty, infty) and the Stieltjes integral of such functions. Appendices I, II, III are the same as in the second Russian edition. Appendix IV, Change of Variable in the Lebesgue Integral, is §5, Chapter IX, 2nd ed. Appendix V, Hausdorff's Theorem, is §5, Chapter XI, 1st ed. Appendix VI, Indefinite Integrals and Absolutely Continuous Set Functions, is §6, Chapter XVII, 2nd ed. Appendix VII, The Role of Russian and Soviet Mathematicians in the Development of the Theory of Functions of a Real Variable, is Chapter XVII, 1st ed.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1960Количество страниц: 263Добавлена в каталог: 07.04.2013Операции: Положить на полку |
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Open, sphere 199 Operator 217 Order type of the set of natural numbers 110 Order type of the set of rational numbers 110 Order types 107 Ordered sets 107 Ordinal number of the first type 118 Ordinal number of the second type 118 Ordinal numbers 115 Parsevafs equality 55 Perfect set 61 Perron 157 Perron, integral 158 160 Petrovsky 209 251 Pinsker 257 Plessner 253 254 256 257 Plessner of approximate continuity 16 Plessner, fixed (of an operator) 217 Plessner, interior 62 Plessner, isolated 60 Plessner, Lebesgue 16 Plessner, limit 60 198 Plessner, sets in two-dimensional spade 60 Poisson, integral 29 Poisson, singular integral 58 Principle of choice, general 124 Principle, Banach’s fixed-point 216 Principle, Cantor — Baire stationary 145 Privalov 22 28 253 254 257 Projective sets 250 Rademacher 27 253 radius of a circle 60 Raikov 256 Rajchman 54 57 Rapprochement operator 217 Rectifiable curve 225 Regular contractibility 103 Regularly ordered subset 125 Representation of a function by a singular integral 15 Riemann function 50 Riemann — Lebesgue theorem 15 Riemann’s lemma 48 Riemann’s theorem 51 Riesz 58 Romanov 255 Romanovsky 251 255 Romanovsky’s theorem 18 29 Rutman 255 257 Saks 157 158 Schwarz derivative 37 Schwarz derivative, derived number 38 Schwarz, derivatives 36 Schwarz, derived numbers 38 Schwarz’s lemma 55 Schwarz’s theorem 37 Second class of functions 128 Second number class 119 Section of a set 76 79 Segment of a set 111 Selivanovsky 250 Seliverstov 253 254 Semi-continuous functions 149 Sequence, lacunary 26 Series, Fourier 21 29 Series, lacunary trigonometric 27 Series, trigonometric 11 29 Set of type M 54 252 Set of type U 54 252 Set open 62 Set ordered 107 Set perfect 61 Set projective 250 Set segment of a 111 Set well-ordered 112 Set, absolutely continuous 94 Set, additive 94 Set, analytic 249 Set, closed 60 Set, derived 60 Set, diameter of a 103 201 Set, first category 142 Set, function 94 Set, measurable hull of a 95 Sets, similar 108 Shilov 256 Shmulyan 257 Shneider 253 Shnirelman 256 Sierpiiiski 124 127 245 Similar sets 108 Similarity mapping 108 Singular integral 12 Singular integral, Bernstein 58 Singular integral, de la Vallee Poussin 58 Singular integral, Dirichlet 22 Singular integral, Fejer 23 Singular integral, KantoroviC 58 Singular integral, Landau 58 Singular integral, Poisson 58 Smirnov 253 257 Solution of a denumberable system of equations 222 Space, Banach 195 Space, compact 199 204 Space, strictly normed Banach 257 Steinhaus 228 Steinhaus’s example 228 Steklov 254 Steklov, function 214 Stepanov 251 257 Strictly normed Banach space 257 subfunction 158 Subgraph of a function 83 Sudakov 216 Sum in the Riemann sense 51 Summation method, Abel — Poisson 28 Summation method, Cesaro 22 Summation method, Cesaro — Fejer 28 Superfunction 158 Suslin 249 Symmetric, derivative 95 Symmetric, derived number 95 Term-by-term integration of a Fourier series 33 Theorem on transfinite induction 119 Theorem, Abel’s 30 Theorem, Aleksandrov — Looman 185 189 Theorem, Arzela — Ascoli 207 Theorem, Baire’s 143 146 Theorem, Banach’s 217 Theorem, Bari — Menshov 255 Theorem, Bari’s 253 Theorem, Birnbaum — Orlicz 232 Theorem, Bolzano — Weierstrass 60 Theorem, Borel’s 61 Theorem, Cantor — Lebesgue 34 Theorem, Cantor’s 51 52 55 Theorem, de la Vallee Poussin’s 40 Theorem, du Bois-Reymond-de la Vallee Poussin 52 Theorem, Faddeyev’s 20 Theorem, Fejer — Lebesgue 25 Theorem, Fubini’s 86 Theorem, Hake’s 178 185 Theorem, Hausdorff’s 238 Theorem, Jordan’s 226 Theorem, Kolmogorov’s 26 27 215 Theorem, Lebesgue’s 13 15 137 141 Theorem, Levi’s 88 Theorem, Luzin — Denjoy 35 Theorem, Riemann — Lebesgue 15 Theorem, Riemann’s 51 Theorem, Romanovsky’s 18 29 Theorem, Schwarz’s 37 Theorem, Vitali’s 136 Theorem, Young’s 252 Theorem, Zermelo’s 125 Tikhonov 256 Tolstov 251 256 Totalization 157 Transfinite, alephs 122 Transfinite, induction 118 Transfinite, numbers 107 115 Transfinite, theorem on 119 Transformation affine 72 Transformation determinant of a 72 Transformation non-degenerate 72 Triangle inequality 193 Trichotomy 116 Trigonometric series 11 29 Trigonometric series, lacunary 27 Truncated function 87 Tulaikov 212 Uniqueness of trigonometric series expansion 48 Unit base vectors 206 Universal function 139 Unterfunktion 158 Upper derivative 158 Upper semi-continuous function 150 152 Value of an operator 217 Veniaminov 251 Verchenko 256 Vilenkin 254 Vitali covering 70 Vitali’s theorem 136 Vulikh 257 Weak convergence 15 Weierstrass singular integral 20 Well-ordered sets 112 Young 54 57 Young’s inequality 231 Young’s theorem 252 Yudin 257 Zarecki 251 Zermelo’s axiom 124 245 Zermelo’s theorem 125 Zhegalkin 127 257 Zygmund 22 28 54
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