science

Sobolev orthogonal polynomials

Sobolev orthogonal polynomials is a science 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 Sobolev orthogonal polynomials rather than just read about it. In short: In mathematics, Sobolev orthogonal polynomials are orthogonal polynomials with respect to a Sobolev inner product, i.e. an inner product with derivatives. By having conditions on the derivatives, the Sobolev orthogonal polynomials in general no longer share some of the nice features that classical orthogonal polynomials have.

Key takeaways

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

Reference excerpt

In mathematics, Sobolev orthogonal polynomials are orthogonal polynomials with respect to a Sobolev inner product, i.e. an inner product with derivatives. By having conditions on the derivatives, the Sobolev orthogonal polynomials in general no longer share some of the nice features that classical orthogonal polynomials have. Sobolev orthogonal polynomials are named after Sergei Lvovich Sobolev.

Definition Let μ 0 , μ 1 , … , μ n {\displaystyle \mu _{0},\mu _{1},\dots ,\mu _{n}} be positive Borel measures on R {\displaystyle \mathbb {R} } with finite moments. Consider the inner product

⟨ p r , p s ⟩ W n , 2 = ∫ R p r ( x ) p s ( x ) d μ 0 + ∑ k = 1 n ∫ R p r ( k ) ( x ) p s ( k ) ( x ) d μ k {\displaystyle \langle p_{r},p_{s}\rangle _{W^{n,2}}=\int _{\mathbb {R} }p_{r}(x)p_{s}(x)\;\mathrm {d} \mu _{0}+\sum \limits _{k=1}^{n}\int _{\mathbb {R} }p_{r}^{(k)}(x)p_{s}^{(k)}(x)\;\mathrm {d} \mu _{k}}

and let W n , 2 {\displaystyle W^{n,2}} be the corresponding Sobolev space. The Sobolev orthogonal polynomials { p n } n ≥ 0 {\displaystyle \{p_{n}\}_{n\geq 0}} are defined as

⟨ p n , p s ⟩ W n , 2 = c n δ n , s {\displaystyle \langle p_{n},p_{s}\rangle _{W^{n,2}}=c_{n}\delta _{n,s}}

where δ n , s {\displaystyle \delta _{n,s}} denotes the Kronecker delta. One says that these polynomials are sobolev orthogonal.

Explanation Classical orthogonal polynomials are Sobolev orthogonal polynomials, since their derivatives are also orthogonal polynomials. Sobolev orthogonal polynomials in general are no longer commutative in the multiplication operator with respect to the inner product, i.e.

⟨ x p n , p s ⟩ W n , 2 ≠ ⟨ p n , x p s ⟩ W n , 2 {\displaystyle \langle xp_{n},p_{s}\rangle _{W^{n,2}}\neq \langle p_{n},xp_{s}\rangle _{W^{n,2}}}

Consequently neither Favard's theorem, the three term recurrence or the Christoffel-Darboux formula hold. There exist however other recursion formulas for certain types of measures. There exist a lot of literature for the case n = 1 {\displaystyle n=1} .

Literature Marcellán, Francisco; Xu, Yuan (2015). "On Sobolev orthogonal polynomials". Expositiones Mathematicae. 33 (3): 308–352. arXiv:1403.6249. Marcellán, Francisco; Moreno-Balcázar, Juan (2017). "WHAT IS... a Sobolev Orthogonal Polynomial?". Notices of the American Mathematical Society. 64: 873–875. doi:10.1090/noti1562.

References

Worked examples

Example 1 — a first encounter with Sobolev orthogonal polynomials

Start with the simplest possible case. Write down what Sobolev orthogonal polynomials claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Sobolev orthogonal polynomials 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 Sobolev orthogonal polynomials 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 Sobolev orthogonal polynomials

In research: Sobolev orthogonal polynomials appears in science 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 Sobolev orthogonal polynomials 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: Sobolev orthogonal polynomials is common in secondary-school and first-year university syllabi. It links to neighbouring topics Orthogonal polynomials, Sobolev spaces, so understanding it makes those chapters shorter.
In everyday life: Look for Sobolev orthogonal polynomials 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 Sobolev orthogonal polynomials in 20 minutes

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

Frequently asked questions

What is Sobolev orthogonal polynomials in simple terms?

Why does Sobolev orthogonal polynomials matter?

Because it connects several science 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 Sobolev orthogonal polynomials?

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 Sobolev orthogonal polynomials.