Titchmarsh convolution theorem

Key takeaways

Titchmarsh convolution 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 Titchmarsh convolution theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Titchmarsh convolution theorem from memory before moving on to harder problems.

Reference excerpt

The Titchmarsh convolution theorem describes the properties of the support of the convolution of two functions. It was proven by Edward Charles Titchmarsh in 1926.

Titchmarsh convolution theorem If φ ( t ) {\textstyle \varphi (t)\,} and ψ ( t ) {\textstyle \psi (t)} are integrable functions, such that

φ ∗ ψ = ∫ 0 x φ ( t ) ψ ( x − t ) d t = 0 {\displaystyle \varphi *\psi =\int _{0}^{x}\varphi (t)\psi (x-t)\,dt=0}

almost everywhere in the interval 0 < x < κ {\displaystyle 0<x<\kappa \,} , then there exist λ ≥ 0 {\displaystyle \lambda \geq 0} and μ ≥ 0 {\displaystyle \mu \geq 0} satisfying λ + μ ≥ κ {\displaystyle \lambda +\mu \geq \kappa } such that φ ( t ) = 0 {\displaystyle \varphi (t)=0\,} almost everywhere in 0 < t < λ {\displaystyle 0<t<\lambda } and ψ ( t ) = 0 {\displaystyle \psi (t)=0\,} almost everywhere in 0 < t < μ . {\displaystyle 0<t<\mu .}

As a corollary, if the integral above is 0 for all x > 0 , {\textstyle x>0,} then either φ {\textstyle \varphi \,} or ψ {\textstyle \psi } is almost everywhere 0 in the interval [ 0 , + ∞ ) . {\textstyle [0,+\infty ).} Thus the convolution of two functions on [ 0 , + ∞ ) {\textstyle [0,+\infty )} cannot be identically zero unless at least one of the two functions is identically zero. As another corollary, if φ ∗ ψ ( x ) = 0 {\displaystyle \varphi *\psi (x)=0} for all x ∈ [ 0 , κ ] {\displaystyle x\in [0,\kappa ]} and one of the function φ {\displaystyle \varphi } or ψ {\displaystyle \psi } is almost everywhere not null in this interval, then the other function must be null almost everywhere in [ 0 , κ ] {\displaystyle [0,\kappa ]} . The theorem can be restated in the following form:

Let φ , ψ ∈ L 1 ( R ) {\displaystyle \varphi ,\psi \in L^{1}(\mathbb {R} )} . Then inf supp ⁡ φ ∗ ψ = inf supp ⁡ φ + inf supp ⁡ ψ {\displaystyle \inf \operatorname {supp} \varphi \ast \psi =\inf \operatorname {supp} \varphi +\inf \operatorname {supp} \psi } if the left-hand side is finite. Similarly, sup supp ⁡ φ ∗ ψ = sup supp ⁡ φ + sup supp ⁡ ψ {\displaystyle \sup \operatorname {supp} \varphi \ast \psi =\sup \operatorname {supp} \varphi +\sup \operatorname {supp} \psi } if the right-hand side is finite. Above, supp {\displaystyle \operatorname {supp} } denotes the support of a function f (i.e., the closure of the complement of f−1(0)) and inf {\displaystyle \inf } and sup {\displaystyle \sup } denote the infimum and supremum. This theorem essentially states that the well-known inclusion supp ⁡ φ ∗ ψ ⊂ supp ⁡ φ + supp ⁡ ψ {\displaystyle \operatorname {supp} \varphi \ast \psi \subset \operatorname {supp} \varphi +\operatorname {supp} \psi } is sharp at the boundary. The higher-dimensional generalization in terms of the convex hull of the supports was proven by Jacques-Louis Lions in 1951:

If φ , ψ ∈ E ′ ( R n ) {\displaystyle \varphi ,\psi \in {\mathcal {E}}'(\mathbb {R} ^{n})} , then c . h . ⁡ supp ⁡ φ ∗ ψ = c . h . ⁡ supp ⁡ φ + c . h . ⁡ supp ⁡ ψ {\displaystyle \operatorname {c.h.} \operatorname {supp} \varphi \ast \psi =\operatorname {c.h.} \operatorname {supp} \varphi +\operatorname {c.h.} \operatorname {supp} \psi }

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Titchmarsh convolution theorem

Start with the simplest possible case. Write down what Titchmarsh convolution 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 Titchmarsh convolution 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 Titchmarsh convolution 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 Titchmarsh convolution theorem

In research: Titchmarsh convolution 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 Titchmarsh convolution 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: Titchmarsh convolution theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in complex analysis, Theorems in harmonic analysis, Theorems in real analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Titchmarsh convolution 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 Titchmarsh convolution theorem in 20 minutes

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

Frequently asked questions

What is Titchmarsh convolution theorem in simple terms?

The Titchmarsh convolution theorem describes the properties of the support of the convolution of two functions. It was proven by Edward Charles Titchmarsh in 1926.

Why does Titchmarsh convolution 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 Titchmarsh convolution 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 Titchmarsh convolution theorem.