Abstract
Sections 10.1 and 10.2 are devoted to the general theory of orthogonal systems. Besides basic results (Bessel’s inequality, Riesz–Fischer theorem, etc.) we consider various examples of orthogonal systems (trigonometric system, Rademacher functions, Legendre polynomials, etc.). At the end of Sect. 10.2, we consider orthogonal series of independent functions, which play an important role in probability theory.
In Sects. 10.3 and 10.4 we discuss facts related to trigonometric Fourier series. For such series, we establish convergence conditions, the possibility of termwise integration, and the uniqueness theorem (for both summable functions and measures). We consider summation methods for Fourier series (including the classical methods of Fejér and Abel–Poisson) and the corresponding approximate identities. At the end of Sect. 10.4, we discuss comparatively recent results showing that the results concerning trigonometric series for functions of one variable cannot be carried over to Fourier series of functions of several variables. The rest of Sect. 10.4 is devoted to multiple Fourier series. Along with the counterparts of certain statements in the one-dimensional case, we discuss some facts (failure of the localization principle, etc.) showing that certain classical results cannot be carried over to the multi-dimensional case.
Section 10.5 is devoted to the Fourier transform, which is one of the most important concepts in harmonic analysis. We consider both \(\mathcal{L}^{1}\) and \(\mathcal{L}^{2}\)- theory of the Fourier transform, and, in particular, prove the inversion formula, Plancherel’s theorem, and the uncertainty principle. We also study the Fourier transforms of finite Borel measures. Using the Fourier transform, we prove that the \(\mathcal{L}^{1}\)-norms of the Dirichlet kernels for balls (such kernels arise when studying the multiple Fourier series) have power rate of growth.
In Sect. 10.6, we discuss the Poisson summation formula and its applications. In particular, we show how this formula can be used to estimate the number of integer points in a ball (Gauss’s problem).
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Notes
- 1.
Pythagoras (Πυϑαγóρας) (circa 570–500 BC)—Greek philosopher and mathematician.
- 2.
Friedrich Wilhelm Bessel (1784–1846)—German mathematician.
- 3.
Jean Baptiste Joseph Fourier (1768–1830)—French mathematician.
- 4.
Ernest Sigismund Fisher (1875–1954)—German mathematician.
- 5.
Marc-Antoine Parseval (1755–1836)—French mathematician.
- 6.
Fedor L’vovich Nazarov (born 1967)—Russian mathematician.
- 7.
Adolf Hurwitz (1859–1919)—German mathematician.
- 8.
Charles Hermite (1822–1901)—French mathematician.
- 9.
Joseph Leonard Walsh (1895–1973)—American mathematician.
- 10.
Andrei Nikolaevich Kolmogorov (1903–1987)—Russian mathematician.
- 11.
Edmond Nicolas Laguerre (1834–1886)—French mathematician.
- 12.
Ulisse Dini (1845–1918)—Italian mathematician.
- 13.
Marie Ennemond Camille Jordan (1838–1922)—French mathematician.
- 14.
Lennart Axel Edvard Carleson (born 1928)—Swedish mathematician.
- 15.
Arnaud Denjoy (1884–1974)—French mathematician.
- 16.
Ernesto Cesaro (1859–1906)—Italian mathematician.
- 17.
Lipót Fejér (1880–1959)—Hungarian mathematician.
- 18.
Charles Louis Fefferman (born 1949)—American mathematician.
- 19.
Sergei Natanovich Bernstein (1880–1968)—Russian mathematician.
- 20.
Michel Plancherel (1885–1967)—Swiss mathematician.
- 21.
Edmund Georg Hermann Landau (1877–1938)—German mathematician.
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Makarov, B., Podkorytov, A. (2013). Fourier Series and the Fourier Transform. In: Real Analysis: Measures, Integrals and Applications. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5122-7_10
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