# Trigonometric series

It is called a Fourier series if the terms and have the form:

A **trigonometric series** is a series of 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 19th century Europe. 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.[1]

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 *S*_{0} = *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): 281–334, doi:10.1007/BF01886630.

### Reviews of *Trigonometric Series*

*Trigonometric Series*

- Kahane, Jean-Pierre (2004), "Book review: Trigonometric series, Vols. I, II",
*Bulletin of the American Mathematical Society*,**41**(3): 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*

*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