Introduction to real functions and orthogonal expansions / by Béla Sz.-Nagy.
Idioma: Inglés Lenguaje original: Húngaro Series University texts in the mathematical sciencesEditor: New York : Oxford University Press, 1965Descripción: xi, 447 p. : il. ; 22 cmTítulos uniformes: Valós függvények és függvénysorok. Inglés Tema(s): Functional analysis | Orthogonal polynomialsOtra clasificación: 26-01 (42Cxx)Introduction [3] 1. SETS 1.1. Some fundamental notions [11] 1.1.1. The algebra of sets. — 1.1.2. Characteristic functions of sets. Envelopes of functions. — 1.1.3. Countable sets. — 1.1.4. Sets of higher power. — 1.1.5. Rings and cr-rings of sets. 1.2. Point sets [25] 1.2.1. Accumulation points. — 1.2.2. Closed sets. 1.2.3. Open sets. — 1.2.4. Borel’s covering theorem. — 1.2.5. The distance between point sets. — 1.2.6. The Cantor—Bendixson theorem. Cantor’s ternary set. 2. CONTINUOUS FUNCTIONS 2.1. Continuity [52] 2.1.1. Continuity and semi-continuity at a point. Limits of a function. — 2.1.2. Properties of functions which are continuous or semi-continuous on a compact point set. 2.2. Sequences of continuous functions [63] 2.2.1. Uniform and quasi-uniform convergence. —2.2.2. Necessary conditions for a sequence of continuous functions to have a continuous limit function. — 2.2.3. Monotone sequences of continuous functions. — 2.2.4. A classification of functions. 2.3. Approximation of continuous functions by polynomials [76] 2.3.1. The Weierstrass approximation theorem. — 2.3.2. The Weierstrass — Stone approximation theorem. — 2.3.3. Continuous extension of a continuous function. 2.4. Monotonic functions and functions of bounded variation [88] 2.4.1. One-sided limits. Discontinuities of the first kind. — 2.4.2. The continuous and the saltus part of a monotonic function. — 2.4.3. Functions of bounded variation. — 2.4.4. Majorization in variation. 3. DIFFERENTIATION 3.1. Differentiation of monotonic functions [101] 3.1.1. Example of a nondifferentiable continuous function. — 3.1.2. Lebesgue’s theorem on the differentiation of a monotonic function. Sets of measure zero. — 3.1.3. Proof of Lebesgue’s theorem (according to F. Riesz). — 3.1.4. Fubini’s theorem on the differentiation of series with monotonic terms. — 3.1.5. Density points of linear sets. 3.2. Dini derivates of arbitrary functions [119] 3.2.1. The Denjoy—Young—Saks theorem. — 3.2.2. The proof of the theorem. 4. INTERVAL FUNCTIONS. RIEMANN INTEGRAL 4.1. General theorems and their applications [127] 4.1.1. Integration and differentiation of interval functions. — 4.1.2. Darboux’s theorem. — 4.1.3. Differentiation of interval functions. — 4.1.4. The Riemann integral. — 4.1.5. Applications to functions of bounded variation and to the rectification of curves. 4.2. The Riemann integral [145] 4.2.1. Lebesgue’s criterion of Riemann integrability. — 4.2.2. Operations with Riemann integrable functions. — 4.2.3. Integral function, primitive function. — 4.2.4. Jordan measure. 4.3. Functions of several variables [159] 4.3.1. Functions of n-dimensional intervals. — 4.3.2. Riemann integral of a function of several variables. — 4.3.3. Iterated integration. 5. LEBESGUE INTEGRAL 5.1. Definition and fundamental properties [169] 5.1.1. Introduction. — 5.1.2. The integral of step functions. Two lemmas. — 5.1.3. The extension of the integral. — 5.1.4. Term by term integration of a monotonic sequence of functions and of a series of functions with constant sign. (Beppo Levi’s theorem.) — 5.1.5. Term by term integration of a majorized sequence. (Lebesgue’s theorem.) — 5.1.6. Fatou’s lemma and other theorems on the integrability of a limit function. — 5.1.7. The Riemann integral within the framework of the new theory. 5.2. Properties of the integral functions [195] 5.2.1. The total variation and the derivative of an integral function. — 5.2.2. Example of a monotonic continuous function whose derivative is zero almost everywhere. 5.2.3. Characterization of integral functions by absolute continuity. —5.2.4. Canonical decomposition of monotonic functions. — 5.2.5. Integration by parts and change of variables. 5.3. Measurable functions and sets [211] 5.3.1. Measurable functions. — 5.3.2. Measurable sets. I 5.3.3. Relations between measurable sets and measurable functions. — 5.3.4. Lebesgue’s original definition of measurability, measure, and integral, and the proof of equivalence. — 5.3.5. Example of a non-measurable set. -**’5.3.6. Borel measurable sets and Baire functions. -* 5.3.7. The theorems of Egoroff and Lusin. 5.4. Functions of several variables [232] 5.4.1. Definition of the integral. Sets of measure zero. — 5.4.2. Fubini’s theorem on iterated integration. 6. THE STIELTJES INTEGRAL AND ITS GENERALIZATIONS 6.1. The Stieltjes integral and linear functionals on continuous functions [238] 6.1.1. The Stieltjes integral. — 6.1.2. The second theorem of the mean. — 6.1.3. Criteria for Stieltjes integrability with respect to a monotonic function. — 6.1.4. Linear functionals on continuous functions. — 6.1.5. The positive and negative parts of a linear functional. — 6.1.6. Uniqueness of the integral representation of linear functionals. 6.2. Generalizations of the Stieltjes integral [262] 6.2.1. The Lebesgue — Stieltjes integral. — 6.2.2. Relations between two Lebesgue — Stieltjes integrals. — 6.2.3. Generalizations to functions of several variables. 6.3. The Lebesgue integral on abstract spaces [269] 6.3.1. The definition of the integral. 6.3.2. Cartesian product of measure spaces. — 6.3.3. The Cartesian product of infinitely many measure spaces. 7. SPACES OF INTEGRABLE FUNCTIONS 7.1. The space L2 [285] 7.1.1. Definitions. Fundamental inequalities. — 7.1.2. The Riesz — Fischer theorem. — 7.1.3. Orthogonal sequences. Minimum property. Bessel inequality. — 7.1.4. The Parseval formula and the Riesz—Fischer theorem for orthogonal sequences. — 7.1.5. Hilbert space and its linear functionals. — 7.1.6. Gram — Schmidt orthogonalization procedure. — 7.1.7. Existence of complete orthogonal sequences in L2. 7.2. Fourier series [310] 7.2.1. Completeness of the trigonometric sequence. — 7.2.2. Fourier series. — 7.2.3. The complex form of a Fourier series. — 7.2.4. Application of the Parseval formula to the isoperimetric problem. 7.3. Other orthogonal sequences of functions [327] 7.3.1. Legendre polynomials. — 7.3.2. Orthogonal polynomials attached to a mass distribution. — 7.3.3. Classical orthogonal polynomials. — 7.3.4. The Haar orthogonal sequence, — 7.3.5. The Rademacher orthogonal sequence. 7.4. Fourier integrals [344] 7.4.1. Fourier integrals as formal limits of Fourier series. — 7.4.2. Fourier transforms of intergrable functions. — 7.4.3. Fourier transforms of L2 functions. 7.5. The Lp spaces [357] 7.5.1. Definitions. The Holder and Minkowski inequalities. — 7.5.2. Linear functionals on the space Lp, — 7.5.3. The concept of Banach space. 8. CONVERGENCE AND SUMMABILITY OF FOURIER SERIES 8.1. Historical comments. Some physical problems [375] 8.1.1. The problem of the vibrating string. — 8.1.2. A problem on heat conduction. — 8.1.3. Dirichlet’s problem for the circle. 8.2. Convergence theorems for Fourier series [385] 8.2.1. Some simple theorems. — 8.2.2. The Riemann — Lebesgue lemma. — 8.2.3. Dirichlet’s formula. — 8.2.4. Riemann’s localization theorems. — 8.2.5. The Dini and Lipschitz convergence theorems. — 8.2.6. The Dirichlet — Jordan theorem. — 8.2.7. The series 27(sin kx)/k, — 8.2.8. Fej^r’s example of a continuous function with divergent Fourier series. — 8.2.9. Conjugate series. Pringsheim’s convergence criterion. — 8.2.10. Lukacs’s theorem. 8.3. Summation methods [414] 8.3.1. Introduction. — 8.3.2. Fundamental theorems on the summability of series. 8.4. Summation of Fourier series by the method of arithmetical means [425] 8.4.1. Fejér’s theorem. — 8.4.2. Some consequences of Fejer’s theorem. — 8.4.3. Lebesgue’s theorem. — 8.4.4. The Lebesgue points of an integrable function. 8.5. Summation of Fourier series by the Abel — Poisson method [436] 8.5.1. Deduction from the Fejer and Lebesgue theorems. — 8.5.2. Direct proof on the basis of the Poisson integral formula. Index [443]
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Traducción de: Valós függvények és függvénysorok. 1961.
"The book is remarkable for its elegance and clarity. The choice and the ordering of the subjects are very original."--Math. Rev.
Introduction [3] --
1. SETS --
1.1. Some fundamental notions [11] --
1.1.1. The algebra of sets. --
— 1.1.2. Characteristic functions of sets. Envelopes of functions. --
— 1.1.3. Countable sets. --
— 1.1.4. Sets of higher power. --
— 1.1.5. Rings and cr-rings of sets. --
1.2. Point sets [25] --
1.2.1. Accumulation points. --
— 1.2.2. Closed sets. --
1.2.3. Open sets. --
— 1.2.4. Borel’s covering theorem. --
— 1.2.5. The distance between point sets. --
— 1.2.6. The Cantor—Bendixson theorem. Cantor’s ternary set. --
2. CONTINUOUS FUNCTIONS --
2.1. Continuity [52] --
2.1.1. Continuity and semi-continuity at a point. Limits of a function. --
— 2.1.2. Properties of functions which are continuous or semi-continuous on a compact point set. --
2.2. Sequences of continuous functions [63] --
2.2.1. Uniform and quasi-uniform convergence. --
—2.2.2. Necessary conditions for a sequence of continuous functions to have a continuous limit function. --
— 2.2.3. Monotone sequences of continuous functions. --
— 2.2.4. A classification of functions. --
2.3. Approximation of continuous functions by polynomials [76] --
2.3.1. The Weierstrass approximation theorem. --
— 2.3.2. The Weierstrass — Stone approximation theorem. --
— 2.3.3. Continuous extension of a continuous function. --
2.4. Monotonic functions and functions of bounded variation [88] --
2.4.1. One-sided limits. Discontinuities of the first kind. --
— 2.4.2. The continuous and the saltus part of a monotonic function. --
— 2.4.3. Functions of bounded variation. --
— 2.4.4. Majorization in variation. --
3. DIFFERENTIATION --
3.1. Differentiation of monotonic functions [101] --
3.1.1. Example of a nondifferentiable continuous function. --
— 3.1.2. Lebesgue’s theorem on the differentiation of a monotonic function. Sets of measure zero. --
— 3.1.3. Proof of Lebesgue’s theorem (according to F. Riesz). --
— 3.1.4. Fubini’s theorem on the differentiation of series with monotonic terms. --
— 3.1.5. Density points of linear sets. --
3.2. Dini derivates of arbitrary functions [119] --
3.2.1. The Denjoy—Young—Saks theorem. --
— 3.2.2. The proof of the theorem. --
4. INTERVAL FUNCTIONS. RIEMANN INTEGRAL --
4.1. General theorems and their applications [127] --
4.1.1. Integration and differentiation of interval functions. --
— 4.1.2. Darboux’s theorem. --
— 4.1.3. Differentiation of interval functions. --
— 4.1.4. The Riemann integral. --
— 4.1.5. Applications to functions of bounded variation and to the rectification of curves. --
4.2. The Riemann integral [145] --
4.2.1. Lebesgue’s criterion of Riemann integrability. --
— 4.2.2. Operations with Riemann integrable functions. --
— 4.2.3. Integral function, primitive function. --
— 4.2.4. Jordan measure. --
4.3. Functions of several variables [159] --
4.3.1. Functions of n-dimensional intervals. --
— 4.3.2. Riemann integral of a function of several variables. --
— 4.3.3. Iterated integration. --
5. LEBESGUE INTEGRAL --
5.1. Definition and fundamental properties [169] --
5.1.1. Introduction. --
— 5.1.2. The integral of step functions. Two lemmas. --
— 5.1.3. The extension of the integral. --
— 5.1.4. Term by term integration of a monotonic sequence of functions and of a series of functions with constant sign. (Beppo Levi’s theorem.) --
— 5.1.5. Term by term integration of a majorized sequence. (Lebesgue’s theorem.) --
— 5.1.6. Fatou’s lemma and other theorems on the integrability of a limit function. --
— 5.1.7. The Riemann integral within the framework of the new theory. --
5.2. Properties of the integral functions [195] --
5.2.1. The total variation and the derivative of an integral function. --
— 5.2.2. Example of a monotonic continuous function whose derivative is zero almost everywhere. --
5.2.3. Characterization of integral functions by absolute continuity. --
—5.2.4. Canonical decomposition of monotonic functions. --
— 5.2.5. Integration by parts and change of variables. --
5.3. Measurable functions and sets [211] --
5.3.1. Measurable functions. --
— 5.3.2. Measurable sets. --
I 5.3.3. Relations between measurable sets and measurable functions. --
— 5.3.4. Lebesgue’s original definition of measurability, measure, and integral, and the proof of equivalence. --
— 5.3.5. Example of a non-measurable set. --
-**’5.3.6. Borel measurable sets and Baire functions. --
-* 5.3.7. The theorems of Egoroff and Lusin. --
5.4. Functions of several variables [232] --
5.4.1. Definition of the integral. Sets of measure zero. --
— 5.4.2. Fubini’s theorem on iterated integration. --
6. THE STIELTJES INTEGRAL AND ITS GENERALIZATIONS --
6.1. The Stieltjes integral and linear functionals on continuous functions [238] --
6.1.1. The Stieltjes integral. --
— 6.1.2. The second theorem of the mean. --
— 6.1.3. Criteria for Stieltjes integrability with respect to a monotonic function. --
— 6.1.4. Linear functionals on continuous functions. --
— 6.1.5. The positive and negative parts of a linear functional. --
— 6.1.6. Uniqueness of the integral representation of linear functionals. --
6.2. Generalizations of the Stieltjes integral [262] --
6.2.1. The Lebesgue — Stieltjes integral. --
— 6.2.2. Relations between two Lebesgue — Stieltjes integrals. --
— 6.2.3. Generalizations to functions of several variables. --
6.3. The Lebesgue integral on abstract spaces [269] --
6.3.1. The definition of the integral. --
6.3.2. Cartesian product of measure spaces. --
— 6.3.3. The Cartesian product of infinitely many measure spaces. --
7. SPACES OF INTEGRABLE FUNCTIONS --
7.1. The space L2 [285] --
7.1.1. Definitions. Fundamental inequalities. --
— 7.1.2. The Riesz — Fischer theorem. --
— 7.1.3. Orthogonal sequences. Minimum property. Bessel inequality. --
— 7.1.4. The Parseval formula and the Riesz—Fischer theorem for orthogonal sequences. --
— 7.1.5. Hilbert space and its linear functionals. --
— 7.1.6. Gram — Schmidt orthogonalization procedure. --
— 7.1.7. Existence of complete orthogonal sequences in L2. --
7.2. Fourier series [310] --
7.2.1. Completeness of the trigonometric sequence. --
— 7.2.2. Fourier series. --
— 7.2.3. The complex form of a Fourier series. --
— 7.2.4. Application of the Parseval formula to the isoperimetric problem. --
7.3. Other orthogonal sequences of functions [327] --
7.3.1. Legendre polynomials. --
— 7.3.2. Orthogonal polynomials attached to a mass distribution. --
— 7.3.3. Classical orthogonal polynomials. --
— 7.3.4. The Haar orthogonal sequence, --
— 7.3.5. The Rademacher orthogonal sequence. --
7.4. Fourier integrals [344] --
7.4.1. Fourier integrals as formal limits of Fourier series. --
— 7.4.2. Fourier transforms of intergrable functions. --
— 7.4.3. Fourier transforms of L2 functions. --
7.5. The Lp spaces [357] --
7.5.1. Definitions. The Holder and Minkowski inequalities. --
— 7.5.2. Linear functionals on the space Lp, --
— 7.5.3. The concept of Banach space. --
8. CONVERGENCE AND SUMMABILITY OF FOURIER SERIES --
8.1. Historical comments. Some physical problems [375] --
8.1.1. The problem of the vibrating string. --
— 8.1.2. A problem on heat conduction. --
— 8.1.3. Dirichlet’s problem for the circle. --
8.2. Convergence theorems for Fourier series [385] --
8.2.1. Some simple theorems. --
— 8.2.2. The Riemann — Lebesgue lemma. --
— 8.2.3. Dirichlet’s formula. --
— 8.2.4. Riemann’s localization theorems. --
— 8.2.5. The Dini and Lipschitz convergence theorems. --
— 8.2.6. The Dirichlet — Jordan theorem. --
— 8.2.7. The series 27(sin kx)/k, --
— 8.2.8. Fej^r’s example of a continuous function with divergent Fourier series. --
— 8.2.9. Conjugate series. Pringsheim’s convergence criterion. --
— 8.2.10. Lukacs’s theorem. --
8.3. Summation methods [414] --
8.3.1. Introduction. --
— 8.3.2. Fundamental theorems on the summability of series. --
8.4. Summation of Fourier series by the method of arithmetical means [425] --
8.4.1. Fejér’s theorem. --
— 8.4.2. Some consequences of Fejer’s theorem. --
— 8.4.3. Lebesgue’s theorem. --
— 8.4.4. The Lebesgue points of an integrable function. --
8.5. Summation of Fourier series by the Abel — Poisson method [436] --
8.5.1. Deduction from the Fejer and Lebesgue theorems. --
— 8.5.2. Direct proof on the basis of the Poisson integral formula. --
Index [443] --
MR, 31 #5938
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