A trigonometric series is a series of the form:
It is called a Fourier series if the terms and have the form:
where 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 the 19th century. First, Georg Cantor proved that if a trigonometric series is convergent to a function on the interval , which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.
Later Cantor proved that even if the set S on which 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 Sk+1 be the derived set of Sk. If there is a finite number n for which Sn is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α such that Sα 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 Sα .
Antoni Zygmund wrote a classic two-volume set of books entitled Trigonometric Series, which discusses many different aspects of these series, which we will not attempt to discuss here.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.
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 933759
- 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