mathematics

Müntz–Szász theorem

Müntz–Szász 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 Müntz–Szász theorem rather than just read about it. In short: The Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász in 1916. Roughly speaking, the theorem shows to what extent the Weierstrass theorem on polynomial approximation can be extended by restricting certain coefficients in the polynomials to be zero.

Key takeaways

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

Reference excerpt

The Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász in 1916. Roughly speaking, the theorem shows to what extent the Weierstrass theorem on polynomial approximation can be extended by restricting certain coefficients in the polynomials to be zero. The form of the result had been conjectured by Sergei Bernstein before it was proved. The theorem, in a special case, states that a necessary and sufficient condition for the monomials

x n , n ∈ S ⊂ N {\displaystyle x^{n},\quad n\in S\subset \mathbb {N} }

to span a dense subset of the Banach space C[a,b] of all continuous functions with complex number values on the closed interval [a,b] with a > 0, with the uniform norm, is that the sum

∑ n ∈ S 1 n {\displaystyle \sum _{n\in S}{\frac {1}{n}}\ }

of the reciprocals, taken over S, should diverge, i.e. S is a large set. For an interval [0, b], the constant functions are necessary: assuming therefore that 0 is in S, the condition on the other exponents is as before. More generally, one can take exponents from any strictly increasing sequence of positive real numbers, and the same result holds. Szász showed that for complex number exponents, the same condition applied to the sequence of real parts. There are also versions for the Lp spaces.

See also Erdős conjecture on arithmetic progressions

References Müntz, Ch. H. (1914). "Über den Approximationssatz von Weierstrass". H. A. Schwarz's Festschrift. Berlin. pp. 303–312.{{cite book}}: CS1 maint: location missing publisher (link) Scanned at University of Michigan Szász, O. (1916). "Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen". Math. Ann. 77 (4): 482–496. doi:10.1007/BF01456964. S2CID 123893394. Scanned at digizeitschriften.de Shen, Jie; Wang, Yingwei (2016). "Müntz-Galerkin methods and applications to mixed Dirichlet-Neumann boundary value problems". SIAM Journal on Scientific Computing. 38 (4): A2357–A2381. Bibcode:2016SJSC...38A2357S. doi:10.1137/15M1052391.

Worked examples

Example 1 — a first encounter with Müntz–Szász theorem

Start with the simplest possible case. Write down what Müntz–Szász 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 Müntz–Szász 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 Müntz–Szász 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 Müntz–Szász theorem

In research: Müntz–Szász 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 Müntz–Szász 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: Müntz–Szász theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Functional analysis, Theorems in approximation theory, so understanding it makes those chapters shorter.
In everyday life: Look for Müntz–Szász 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 Müntz–Szász theorem in 20 minutes

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

Frequently asked questions

What is Müntz–Szász theorem in simple terms?

Why does Müntz–Szász 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 Müntz–Szász 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 Müntz–Szász theorem.