Trigonometric series

A trigonometric series is a series of the form:

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

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 the 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. But almost half a millennium back the Indian Mathematicians, notably from Kerala school of astronomy and mathematics like Madhava of Sangamagrama and Neelakanta Somayaji had already created the whole basis of the same theory. Due to the imperialism that occurred in India most of the information was hidden from the outside world.

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 .^[1]

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

1. 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 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