mathematics

Hardy–Littlewood Tauberian theorem

Hardy–Littlewood Tauberian 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 Hardy–Littlewood Tauberian theorem rather than just read about it. In short: In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if the sequence a n ≥ 0 {\displaystyle a_{n}\geq 0} is such that there is an asymptotic equivalence ∑ n = 0 ∞ a n e − n y ∼ 1 y as y ↓ 0 {\displaystyle \sum _{n=0}^{\infty }a_{n}e^{-ny}\sim {…

Key takeaways

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

Reference excerpt

In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if the sequence

a n ≥ 0 {\displaystyle a_{n}\geq 0} is such that there is an asymptotic equivalence

∑ n = 0 ∞ a n e − n y ∼ 1 y as y ↓ 0 {\displaystyle \sum _{n=0}^{\infty }a_{n}e^{-ny}\sim {\frac {1}{y}}\ {\text{as}}\ y\downarrow 0}

then there is also an asymptotic equivalence

∑ k = 0 n a k ∼ n {\displaystyle \sum _{k=0}^{n}a_{k}\sim n}

as n → ∞ {\displaystyle n\to \infty } . The integral formulation of the theorem relates in an analogous manner the asymptotics of the cumulative distribution function of a function with the asymptotics of its Laplace transform. The theorem was proved in 1914 by G. H. Hardy and J. E. Littlewood. In 1930, Jovan Karamata gave a new and much simpler proof.

Statement of the theorem

Series formulation This formulation is from Titchmarsh. Suppose a n ≥ 0 {\displaystyle a_{n}\geq 0} for all n ∈ N {\displaystyle n\in \mathbb {N} } , and we have

∑ n = 0 ∞ a n x n ∼ 1 1 − x as x ↑ 1. {\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}\sim {\frac {1}{1-x}}\ {\text{as}}\ x\uparrow 1.}

Then as n → ∞ {\displaystyle n\to \infty } we have

∑ k = 0 n a k ∼ n . {\displaystyle \sum _{k=0}^{n}a_{k}\sim n.}

The theorem is sometimes quoted in equivalent forms, where instead of requiring a n ≥ 0 {\displaystyle a_{n}\geq 0} , we require a n = O ( 1 ) {\displaystyle a_{n}=O(1)} , or we require

a n ≥ − K {\displaystyle a_{n}\geq -K} for some constant K {\displaystyle K} . The theorem is sometimes quoted in another equivalent formulation (through the change of variable x = 1 / e y {\displaystyle x=1/e^{y}} ). If,

∑ n = 0 ∞ a n e − n y ∼ 1 y as y ↓ 0 {\displaystyle \sum _{n=0}^{\infty }a_{n}e^{-ny}\sim {\frac {1}{y}}\ {\text{as}}\ y\downarrow 0}

then

∑ k = 0 n a k ∼ n . {\displaystyle \sum _{k=0}^{n}a_{k}\sim n.}

Integral formulation The following more general formulation is from Feller. Consider a real-valued function F : [ 0 , ∞ ) → R {\displaystyle F:[0,\infty )\to \mathbb {R} } of bounded variation. The Laplace–Stieltjes transform of F {\displaystyle F} is defined by the Stieltjes integral

ω ( s ) = ∫ 0 ∞ e − s t d F ( t ) . {\displaystyle \omega (s)=\int _{0}^{\infty }e^{-st}\,dF(t).}

The theorem relates the asymptotics of ω with those of F {\displaystyle F} in the following way. If ρ {\displaystyle \rho } is a non-negative real number, then the following statements are equivalent

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hardy–Littlewood Tauberian theorem

Start with the simplest possible case. Write down what Hardy–Littlewood Tauberian 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 Hardy–Littlewood Tauberian 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 Hardy–Littlewood Tauberian 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 Hardy–Littlewood Tauberian theorem

In research: Hardy–Littlewood Tauberian 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 Hardy–Littlewood Tauberian 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: Hardy–Littlewood Tauberian theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Tauberian theorems, so understanding it makes those chapters shorter.
In everyday life: Look for Hardy–Littlewood Tauberian 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 Hardy–Littlewood Tauberian theorem in 20 minutes

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

Frequently asked questions

What is Hardy–Littlewood Tauberian theorem in simple terms?

Why does Hardy–Littlewood Tauberian 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 Hardy–Littlewood Tauberian 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 Hardy–Littlewood Tauberian theorem.