Trigonometric series

A trigonometric series is a series of the form:

{\frac {A_{0}}{2}}+\displaystyle \sum _{n=1}^{\infty }(A_{n}\cos {nx}+B_{n}\sin {nx}).

It is called a Fourier series if the terms $A_{n}$ and $B_{{n}}$ have the form:

A_{{n}}={\frac {1}{\pi }}\displaystyle \int _{0}^{{2\pi }}\!f(x)\cos {nx}\,dx\qquad (n=0,1,2,3\dots )

B_{{n}}={\frac {1}{\pi }}\displaystyle \int _{0}^{{2\pi }}\!f(x)\sin {nx}\,dx\qquad (n=1,2,3,\dots )

where $f$ is an integrable function.

The zeros of a trigonometric series

The uniqueness and the zeros of trigonometric series was an active area of research in 19th century Europe. First, Georg Cantor proved that if a trigonometric series is convergent to a function $f(x)$ on the interval $[0,2\pi ]$ , which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.^[1]

Later Cantor proved that even if the set S on which $f$ is nonzero is infinite, but the derived set S' of S is finite, then the coefficients are all zero. In fact, he proved a more general result. Let S₀ = S and let S_k+1 be the derived set of S_k. If there is a finite number n for which S_n is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α such that S_α is finite, then the coefficients of the series are all zero. Cantor's work on the uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts α in S_α .^[2]

Zygmund's book

Antoni Zygmund wrote a classic two-volume set of books entitled Trigonometric Series, which discusses many different aspects of these series.The first edition was a single volume, published in 1935 (under the slightly different title "trigonometrical series"). The second edition of 1959 was greatly expanded, taking up two volumes, though it was later reprinted as a single volume paperback. The third edition of 2002 is similar to the second edition, with the addition of a preface by Robert A. Fefferman on more recent developments, in particular Carleson's theorem about almost everywhere pointwise convergence for square integrable functions.

References

↑
↑ Cooke, Roger (1993), "Uniqueness of trigonometric series and descriptive set theory, 1870–1985", Archive for History of Exact Sciences, 45 (4): 281, doi:10.1007/BF01886630.

Reviews of Trigonometric Series

Kahane, Jean-Pierre (2004), "Book review: Trigonometric series, Vols. I, II", Bulletin of the American Mathematical Society, 41: 377–390, doi:10.1090/s0273-0979-04-01013-4, ISSN 0002-9904
Salem, Raphael (1960), "Book Review: Trigonometric series", Bulletin of the American Mathematical Society, 66 (1): 6–12, doi:10.1090/S0002-9904-1960-10362-X, ISSN 0002-9904, MR 1566029
Tamarkin, J. D. (1936), "Zygmund on Trigonometric Series", Bull. Amer. Math. Soc., 42 (1): 11–13, doi:10.1090/s0002-9904-1936-06235-x

Publication history of Trigonometric Series

Zygmund, Antoni (1935), Trigonometrical series., Monogr. Mat., 5, Warszawa, Lwow: Subwencji Fundusz Kultury Narodowej., Zbl 0011.01703
Zygmund, Antoni (1952), Trigonometrical series, New York: Chelsea Publishing Co., MR 0076084
Zygmund, Antoni (1955), Trigonometrical series, Dover Publications, New York, MR 0072976
Zygmund, Antoni (1959), Trigonometric series Vols. I, II (2nd ed.), Cambridge University Press, MR 0107776
Zygmund, Antoni (1968), Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions (2nd ed.), Cambridge University Press, MR 0236587
Zygmund, Antoni (1977), Trigonometric series. Vol. I, II, Cambridge University Press, ISBN 978-0-521-07477-3, MR 0617944
Zygmund, Antoni (1988), Trigonometric series. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-35885-9, MR 0933759
Zygmund, Antoni (2002), Fefferman, Robert A., ed., Trigonometric series. Vol. I, II, Cambridge Mathematical Library (3rd ed.), Cambridge University Press, ISBN 978-0-521-89053-3, MR 1963498

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[1] ↑

[2] Cooke, Roger (1993), "Uniqueness of trigonometric series and descriptive set theory, 1870–1985", Archive for History of Exact Sciences, 45 (4): 281, doi:10.1007/BF01886630.