mathematics

Trigonometric Series

Trigonometric Series is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Trigonometric Series rather than just read about it. In short: Antoni Zygmund wrote a classic two-volume set of books entitled Trigonometric Series, which discusses many different aspects of trigonometric series. The first edition was a single volume, published in 1935 (under the slightly different title Trigonometrical Series).

Key takeaways

Trigonometric Series belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Trigonometric Series to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Trigonometric Series from memory before moving on to harder problems.

Reference excerpt

Antoni Zygmund wrote a classic two-volume set of books entitled Trigonometric Series, which discusses many different aspects of trigonometric 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.

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

Reviews 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

Worked examples

Example 1 — a first encounter with Trigonometric Series

Start with the simplest possible case. Write down what Trigonometric Series claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Trigonometric Series before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Trigonometric Series ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Trigonometric Series

In research: Trigonometric Series appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Trigonometric Series in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Trigonometric Series is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1935 non-fiction books, 1959 non-fiction books, 2002 non-fiction books, so understanding it makes those chapters shorter.
In everyday life: Look for Trigonometric Series outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Trigonometric Series in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Trigonometric Series means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Trigonometric Series out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Trigonometric Series in simple terms?

Why does Trigonometric Series matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Trigonometric Series?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Trigonometric Series.