mathematics

Wiener's lemma

Wiener's lemma 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 Wiener's lemma rather than just read about it. In short: In mathematics, Wiener's lemma is a well-known identity which relates the asymptotic behaviour of the Fourier coefficients of a Borel measure on the circle to its discrete part. This result admits an analogous statement for measures on the real line.

Key takeaways

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

Reference excerpt

In mathematics, Wiener's lemma is a well-known identity which relates the asymptotic behaviour of the Fourier coefficients of a Borel measure on the circle to its discrete part. This result admits an analogous statement for measures on the real line. It was first discovered by Norbert Wiener.

Definition Consider the space M ( T ) {\displaystyle M(\mathbb {T} )} of all (finite) complex Borel measures on the unit circle T {\displaystyle \mathbb {T} } and the space C ( T ) {\displaystyle C(\mathbb {T} )} of continuous functions on T {\displaystyle \mathbb {T} } as its dual space. Then C ( T ) ⊂ L p ( T ) {\displaystyle C(\mathbb {T} )\subset L^{p}(\mathbb {T} )} for all 1 ≤ p < ∞ {\displaystyle 1\leq p<\infty } and L 1 ( T ) ⊂ M ( T ) {\displaystyle L^{1}(\mathbb {T} )\subset M(\mathbb {T} )} . Given μ ∈ M ( T ) {\displaystyle \mu \in M(\mathbb {T} )} , let

μ p p = ∑ j c j δ z j , {\displaystyle \mu _{pp}=\sum _{j}c_{j}\delta _{z_{j}},}

be its discrete part (meaning that μ ( { z j } ) = c j ≠ 0 {\displaystyle \mu (\{z_{j}\})=c_{j}\neq 0} and μ ( { z } ) = 0 {\displaystyle \mu (\{z\})=0} for z ∉ { z j } {\displaystyle z\not \in \{z_{j}\}} . Then

lim N → ∞ 1 2 N + 1 ∑ n = − N N | μ ^ ( n ) | 2 = ∑ j | c j | 2 , {\displaystyle \lim _{N\to \infty }{\frac {1}{2N+1}}\sum _{n=-N}^{N}|{\widehat {\mu }}(n)|^{2}=\sum _{j}|c_{j}|^{2},}

where μ ^ ( n ) = ∫ T z − n d μ ( z ) {\displaystyle {\widehat {\mu }}(n)=\int _{\mathbb {T} }z^{-n}\,d\mu (z)} is the n {\displaystyle n} -th Fourier-Stieltjes coefficient of μ {\displaystyle \mu } . Similarly, on the real line R {\displaystyle \mathbb {R} } , the space M ( R ) {\displaystyle M(\mathbb {R} )} is the dual space of C 0 ( R ) {\displaystyle C_{0}(\mathbb {R} )} , continuous functions which vanish at infinity, and C 0 ( R ) ⊂ L p ( R ) {\displaystyle C_{0}(\mathbb {R} )\subset L^{p}(\mathbb {R} )} for all 1 ≤ p ≤ ∞ {\displaystyle 1\leq p\leq \infty } . Given μ ∈ M ( R ) {\displaystyle \mu \in M(\mathbb {R} )} , let

μ p p = ∑ j c j δ x j , {\displaystyle \mu _{pp}=\sum _{j}c_{j}\delta _{x_{j}},}

its discrete part. Then

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Wiener's lemma

Start with the simplest possible case. Write down what Wiener's lemma 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 Wiener's lemma 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 Wiener's lemma 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 Wiener's lemma

In research: Wiener's lemma 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 Wiener's lemma 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: Wiener's lemma is common in secondary-school and first-year university syllabi. It links to neighbouring topics Lemmas in mathematical analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Wiener's lemma 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 Wiener's lemma in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Wiener's lemma 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 Wiener's lemma out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Wiener's lemma in simple terms?

Why does Wiener's lemma 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 Wiener's lemma?

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 Wiener's lemma.