Hardy space

Key takeaways

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

Reference excerpt

In complex analysis, the Hardy spaces (or Hardy classes) H p {\displaystyle H^{p}} are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz (Riesz 1923), who named them after G. H. Hardy, because of the paper (Hardy 1915). In real analysis Hardy spaces are spaces of distributions on the real n-space R n {\displaystyle \mathbb {R} ^{n}} , defined (in the sense of distributions) as boundary values of the holomorphic functions. Hardy spaces are related to the Lp spaces. For 1 ≤ p < ∞ {\displaystyle 1\leq p<\infty } these Hardy spaces are subsets of L p {\displaystyle L^{p}} spaces, while for 0 < p < 1 {\displaystyle 0<p<1} the L p {\displaystyle L^{p}} spaces have some undesirable properties, and the Hardy spaces are much better behaved. Hence, H p {\displaystyle H^{p}} spaces can be considered extensions of L p {\displaystyle L^{p}} spaces. Hardy spaces have a number of applications, both in mathematical analysis itself as well as in interdisciplinary areas such as control theory (e.g. H ∞ {\displaystyle H^{\infty }} methods) and scattering theory.

Definition

On the unit disk The Hardy space H p {\displaystyle H^{p}} for 0 < p < ∞ {\displaystyle 0<p<\infty } is the class of holomorphic functions f {\displaystyle f} on the open unit disk D = { z ∈ C : | z | < 1 } {\displaystyle \mathbb {D} =\{z\in \mathbb {C} :|z|<1\}} satisfying

sup 0 ⩽ r < 1 ( 1 2 π ∫ 0 2 π | f ( r e i θ ) | p d θ ) 1 p < ∞ . {\displaystyle \sup _{0\,\leqslant \,r\,<\,1}\left({\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{p}\;\mathrm {d} \theta \right)^{\frac {1}{p}}<\infty .}

If p ≥ 1 {\displaystyle p\geq 1} , this coincides with the definition of the Hardy space p {\displaystyle p} -norm, denoted by ‖ f ‖ H p . {\displaystyle \|f\|_{H^{p}}.}

The space H ∞ {\displaystyle H^{\infty }} is defined as the vector space of bounded holomorphic functions on the unit disk, with norm

‖ f ‖ H ∞ = sup | z | < 1 | f ( z ) | . {\displaystyle {\|f\|}_{H^{\infty }}=\sup _{|z|<1}\left|f(z)\right|.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hardy space

Start with the simplest possible case. Write down what Hardy space 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 Hardy space 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 Hardy space 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 Hardy space

In research: Hardy space 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 Hardy space 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: Hardy space is common in secondary-school and first-year university syllabi. It links to neighbouring topics Complex analysis, Function spaces, Hardy spaces, so understanding it makes those chapters shorter.
In everyday life: Look for Hardy space 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 Hardy space in 20 minutes

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

Frequently asked questions

What is Hardy space in simple terms?

In complex analysis, the Hardy spaces (or Hardy classes) H p {\displaystyle H^{p}} are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz (Riesz 1923), who named them after G.

Why does Hardy space 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 Hardy space?

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 Hardy space.