Trigonometric polynomial

Key takeaways

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

Reference excerpt

In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients, there is no difference between such a function and a finite Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform. The term trigonometric polynomial for the real-valued case can be seen as using the analogy: the functions sin(nx) and cos(nx) are similar to the monomial basis for polynomials. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of e i x {\displaystyle e^{ix}} , i.e., Laurent polynomials in z {\displaystyle z} under the change of variables x ↦ z := e i x {\displaystyle x\mapsto z:=e^{ix}} .

Definition Any function T of the form

T ( x ) = a 0 + ∑ n = 1 N a n cos ⁡ n x + ∑ n = 1 N b n sin ⁡ n x ( x ∈ R ) {\displaystyle T(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos nx+\sum _{n=1}^{N}b_{n}\sin nx\qquad (x\in \mathbb {R} )}

with complex-valued coefficients a n {\displaystyle a_{n}} and b n {\displaystyle b_{n}} and at least one of the highest-degree coefficients a N {\displaystyle a_{N}} and b N {\displaystyle b_{N}} non-zero, is called a complex trigonometric polynomial of degree N. The cosine and sine are the even and odd parts of the exponential of an imaginary variable,

cos ⁡ n x = 1 2 ( e i n x + e − i n x ) , sin ⁡ n x = − 1 2 i ( e i n x − e − i n x ) , {\displaystyle \cos nx={\tfrac {1}{2}}{\bigl (}e^{inx}+e^{-inx}{\bigr )},\quad \sin nx=-{\tfrac {1}{2}}i{\bigl (}e^{inx}-e^{-inx}{\bigr )},}

so the trigonometric polynomial can alternately be written as

T ( x ) = ∑ n = − N N c n e i n x ( x ∈ R ) , {\displaystyle T(x)=\sum _{n=-N}^{N}c_{n}e^{inx}\qquad (x\in \mathbb {R} ),}

with complex coefficients ⁠ c 0 = a 0 {\displaystyle c_{0}=a_{0}} ⁠ and

c k = 1 2 ( a k − b k i ) , c − k = 1 2 ( a k + b k i ) , {\displaystyle \quad c_{k}={\tfrac {1}{2}}(a_{k}-b_{k}i),\quad c_{-k}={\tfrac {1}{2}}(a_{k}+b_{k}i),}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Trigonometric polynomial

Start with the simplest possible case. Write down what Trigonometric polynomial 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 Trigonometric polynomial 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 Trigonometric polynomial 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 Trigonometric polynomial

In research: Trigonometric polynomial 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 Trigonometric polynomial 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: Trigonometric polynomial is common in secondary-school and first-year university syllabi. It links to neighbouring topics Approximation theory, Fourier analysis, Polynomials, so understanding it makes those chapters shorter.
In everyday life: Look for Trigonometric polynomial 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 Trigonometric polynomial in 20 minutes

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

Frequently asked questions

What is Trigonometric polynomial in simple terms?

In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functi…

Why does Trigonometric polynomial 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 Trigonometric polynomial?

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 Trigonometric polynomial.