Paley–Wiener theorem

Key takeaways

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

Reference excerpt

In mathematics, a Paley–Wiener theorem is a theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. It is named after Raymond Paley (1907–1933) and Norbert Wiener (1894–1964) who, in 1934, introduced various versions of the theorem. The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz. These theorems heavily rely on the triangle inequality (to interchange the absolute value and integration). The original work by Paley and Wiener is also used as a namesake in the fields of control theory and harmonic analysis; introducing the Paley–Wiener condition for spectral factorization and the Paley–Wiener criterion for non-harmonic Fourier series respectively. These are related mathematical concepts that place the decay properties of a function in context of stability problems.

Holomorphic Fourier transforms The classical Paley–Wiener theorems make use of the holomorphic Fourier transform on classes of square-integrable functions supported on the real line. Formally, the idea is to take the integral defining the (inverse) Fourier transform

f ( ζ ) = ∫ − ∞ ∞ F ( x ) e i x ζ d x {\displaystyle f(\zeta )=\int _{-\infty }^{\infty }F(x)e^{ix\zeta }\,dx}

and allow ζ {\displaystyle \zeta } to be a complex number in the upper half-plane. One may then expect to differentiate under the integral in order to verify that the Cauchy–Riemann equations hold, and thus that f {\displaystyle f} defines an analytic function. However, this integral may not be well-defined, even for F {\displaystyle F} in L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} ; indeed, since ζ {\displaystyle \zeta } is in the upper half plane, the modulus of e i x ζ {\displaystyle e^{ix\zeta }} grows exponentially as x → − ∞ {\displaystyle x\to -\infty } ; so differentiation under the integral sign is out of the question. One must impose further restrictions on F {\displaystyle F} in order to ensure that this integral is well-defined. The first such restriction is that F {\displaystyle F} be supported on R + {\displaystyle \mathbb {R} _{+}} : that is, F ∈ L 2 ( R + ) {\displaystyle F\in L^{2}(\mathbb {R} _{+})} . The Paley–Wiener theorem now asserts the following: The holomorphic Fourier transform of F {\displaystyle F} , defined by

f ( ζ ) = ∫ 0 ∞ F ( x ) e i x ζ d x {\displaystyle f(\zeta )=\int _{0}^{\infty }F(x)e^{ix\zeta }\,dx}

for ζ {\displaystyle \zeta } in the upper half-plane is a holomorphic function. Moreover, by Plancherel's theorem, one has

∫ − ∞ ∞ | f ( ξ + i η ) | 2 d ξ ≤ ∫ 0 ∞ | F ( x ) | 2 d x {\displaystyle \int _{-\infty }^{\infty }\left|f(\xi +i\eta )\right|^{2}\,d\xi \leq \int _{0}^{\infty }|F(x)|^{2}\,dx}

and by dominated convergence,

lim η → 0 + ∫ − ∞ ∞ | f ( ξ + i η ) − f ( ξ ) | 2 d ξ = 0. {\displaystyle \lim _{\eta \to 0^{+}}\int _{-\infty }^{\infty }\left|f(\xi +i\eta )-f(\xi )\right|^{2}\,d\xi =0.}

Conversely, if f {\displaystyle f} is a holomorphic function in the upper half-plane satisfying

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Paley–Wiener theorem

Start with the simplest possible case. Write down what Paley–Wiener theorem 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 Paley–Wiener theorem 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 Paley–Wiener theorem 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 Paley–Wiener theorem

In research: Paley–Wiener theorem 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 Paley–Wiener theorem 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: Paley–Wiener theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Generalized functions, Hardy spaces, Theorems in Fourier analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Paley–Wiener theorem 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.

How to study Paley–Wiener theorem in 20 minutes

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

Frequently asked questions

What is Paley–Wiener theorem in simple terms?

In mathematics, a Paley–Wiener theorem is a theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. It is named after Raymond Paley (1907–1933) and Norbert Wiener (1894–1964) who, in 1934, introduced various versions of the theorem.

Why does Paley–Wiener theorem 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 Paley–Wiener theorem?

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 Paley–Wiener theorem.