Continuity, Integration and Fourier Theory

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.

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.