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Continuity, Integration and Fourier Theory

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概要 This book is a textbook for graduate or advanced undergraduate students in mathematics and (or) mathematical physics. It is not primarily aimed, therefore, at specialists (or those who wish to become ...specialists) in integra tion theory, Fourier theory and harmonic analysis, although even for these there might be some points of interest in the book (such as for example the simple remarks in Section 15). At many universities the students do not yet get acquainted with Lebesgue integration in their first and second year (or sometimes only with the first principles of integration on the real line ). The Lebesgue integral, however, is indispensable for obtaining a familiarity with Fourier series and Fourier transforms on a higher level; more so than by us ing only the Riemann integral. Therefore, we have included a discussion of integration theory - brief but with complete proofs - for Lebesgue measure in Euclidean space as well as for abstract measures. We give some emphasis to subjects of which an understanding is necessary for the Fourier theory in the later chapters. In view of the emphasis in modern mathematics curric ula on abstract subjects (algebraic geometry, algebraic topology, algebraic number theory) on the one hand and computer science on the other, it may be useful to have a textbook available (not too elementary and not too spe cialized) on the subjects - classical but still important to-day - which are mentioned in the title of this book.続きを見る
目次 1. The Space of Continuous Functions
1. Subsets of ?k
2. The Space of Continuous Functions
3. Lattice Properties of the Space of Real Continuous Functions
2. Theorems of Korovkin and Stone-Weierstrass
4. Theorems of Korovkin and Weierstrass
5. The Stone-Weierstrass Theorem
6. The Complex Stone-Weierstrass Theorem
3. Fourier Series of Continuous Functions
7. Trigonometric Polynomials
8. Fourier Series
9. Fejér's Theorem on Uniform Convergence of Cesaro Means
10. Fourier Coefficients and Orders of Magnitude
4. Integration and Differentation
11. Measure
12. Integral
13. Product Integral, Fubini's Theorem and Convolution
14. Differentiation of the Integral
15. Measurability, Continuity and Differentiability
5. Spaces Lp and Convolutions
16. Hölder's Inequality
17. Spaces Lp
18. Convolution
19. Convolution and Approximate Identities
6. Fourier Series of Summable Functions
20. Fourier Coefficients and the Fourier Transform
21. Pointwise Convergence of Cesaro Means and Abel Means
22. Pointwise Convergence of Fourier Series
23. Hilbert Space and the Space L2
7. Fourier Integral
24. Some Useful Integrals
25. Inverse Fourier Transform
26. Convergence of the Inverse Fourier Transform
27. The Plancherel Theorem
8. Additional Results
28. The Wilbraham-Gibbs Phenomenon
29. Absolute Convergence
30. Positive Definite Functions
31. Equidistribution of Sequences
32. Functions of Analytic Type
33. The Hausdorff-Young Theorem
34. The Poisson Sum Formula
35. The Heat Equation
36. More on the Heat Equation
37. The Wave Equation
References.
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登録日 2020.06.27
更新日 2020.06.28