mathematics

Hartogs–Rosenthal theorem

Hartogs–Rosenthal theorem 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 Hartogs–Rosenthal theorem rather than just read about it. In short: In mathematics, the Hartogs–Rosenthal theorem is a classical result in complex analysis on the uniform approximation of continuous functions on compact subsets of the complex plane by rational functions. The theorem was proved in 1931 by the German mathematicians Friedrich Hartogs and Arthur Rosenthal and has been widely applied, particularly in operator theory.

Key takeaways

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

Reference excerpt

In mathematics, the Hartogs–Rosenthal theorem is a classical result in complex analysis on the uniform approximation of continuous functions on compact subsets of the complex plane by rational functions. The theorem was proved in 1931 by the German mathematicians Friedrich Hartogs and Arthur Rosenthal and has been widely applied, particularly in operator theory.

Statement The Hartogs–Rosenthal theorem states that if K is a compact subset of the complex plane with Lebesgue measure zero, then any continuous complex-valued function on K can be uniformly approximated by rational functions.

Proof By the Stone–Weierstrass theorem any complex-valued continuous function on K can be uniformly approximated by a polynomial in z {\displaystyle z} and z ¯ {\displaystyle {\overline {z}}} . So it suffices to show that z ¯ {\displaystyle {\overline {z}}} can be uniformly approximated by a rational function on K. Let g(z) be a smooth function of compact support on C equal to 1 on K and set

f ( z ) = g ( z ) ⋅ z ¯ . {\displaystyle f(z)=g(z)\cdot {\overline {z}}.}

By the generalized Cauchy integral formula

f ( z ) = 1 2 π i ∬ C ∖ K ∂ f ∂ w ¯ d w ∧ d w ¯ w − z , {\displaystyle f(z)={\frac {1}{2\pi i}}\iint _{C\backslash K}{\frac {\partial f}{\partial {\bar {w}}}}{\frac {dw\wedge d{\bar {w}}}{w-z}},}

since K has measure zero. Restricting z to K and taking Riemann approximating sums for the integral on the right hand side yields the required uniform approximation of z ¯ {\displaystyle {\bar {z}}} by a rational function.

See also Runge's theorem Mergelyan's theorem

Notes

References Conway, John B. (1995), Functions of one complex variable II, Graduate Texts in Mathematics, vol. 159, Springer, p. 197, ISBN 0387944605 Conway, John B. (2000), A course in operator theory, Graduate Studies in Mathematics, vol. 21, American Mathematical Society, pp. 175–176, ISBN 0821820656 Gamelin, Theodore W. (2005), Uniform algebras (2nd ed.), American Mathematical Society, pp. 46–47, ISBN 0821840495 Hartogs, Friedrichs; Rosenthal, Arthur (1931), "Über Folgen analytischer Funktionen", Mathematische Annalen, 104: 606–610, doi:10.1007/bf01457959, S2CID 179177370

Worked examples

Example 1 — a first encounter with Hartogs–Rosenthal theorem

Start with the simplest possible case. Write down what Hartogs–Rosenthal 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 Hartogs–Rosenthal 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 Hartogs–Rosenthal 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 Hartogs–Rosenthal theorem

In research: Hartogs–Rosenthal 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 Hartogs–Rosenthal 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: Hartogs–Rosenthal theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Rational functions, Theorems in approximation theory, Theorems in complex analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Hartogs–Rosenthal 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.

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How to study Hartogs–Rosenthal theorem in 20 minutes

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

Frequently asked questions

What is Hartogs–Rosenthal theorem in simple terms?

Why does Hartogs–Rosenthal 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 Hartogs–Rosenthal 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 Hartogs–Rosenthal theorem.