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High School Algebra II: Homework Help Resource26 chapters | 281 lessons | 2 flashcard sets

Instructor:
*Gerald Lemay*

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

This lesson looks at the remarkable Chebyshev polynomials and defines them using properties of recursion and orthogonality. Chebyshev polynomials have applications in science and engineering.

An extraordinary 19th Russian mathematician named Pafnuty Chebyshev had an uncommon way of thinking about the relationship between math theory and science applications. As students of math, we often ask ''What is this good for?'' or ''How can this math be used?'' Rather than viewing applications as the beneficiary of elegant mathematics, Chebyshev had a broader idea. He would see a wonderful application and ask if the math adequately described the science. In fact, he found many of his greatest theoretical mathematical discoveries by observing mechanical systems (like steam engines). When the mathematical theory at the time fell short of adequately explaining how an application worked, this inspired Chebyshev to create better math theory. It is out of this mindset he extended the idea of orthogonal polynomials to a set of polynomials that now bear his name. Fast forward to our present time and we find the Chebyshev polynomial intimately linked with digital filters, matching networks and other modern communication systems. There may be a Chebyshev filter in your smart phone or tablet!

Deciphering a word helps with understanding. What about the word **polynomial**? If you accept the word **nomial** as being a single term like 1, *x* or *x*2 then a polynomial is ''many'' of these terms together like 1 + *x*. We will get back to the word ''orthogonal'' later in this lesson. For now lets look at the Chebyshev polynomials.

The degree of a polynomial is the highest exponent of the variable in any of the terms.

The zeroth degree Chebyshev polynomial, To is:

When *x* is raised to the 0 power, the degree is 0. And *x*0 = 1.

By the way, Chebyshev's name is translated with a T instead of a C in some languages. That's why the letter ''T'' is used for his polynomials.

It is easy to plot To:

This is the zeroth-degree Chebyshev polynomial. How about a first-degree polynomial? Right, the highest power of *x* in any of the terms will be 1. The next example is Chebyshev's first-degree polynomial:

Let's plot T1 and To on the same graph.

A nice property of the Chebyshev polynomials is we can generate the rest of the Chebyshev polynomials using just these first two. The equation that tells us how to do this, is called a **recursion equation**. It's a way to get the next Chebyshev polynomial if we know the two previous ones. The Chebyshev recursion equation is, in words:

- Multiply the current polynomial by 2
*x* - From this result, subtract the previous polynomial

For example, to get T2

- Multiply T1 (which is
*x*) by 2*x*giving 2*x*2 - From 2
*x*2 subtract To

Thus,

Please take a moment and try calculating T3 on your own.

How did you do? To find T3, take 2*x* times T2 and then subtract T1.

This gives us 2*x*(2*x*2 - 1) - *x* which simplifies to:

Plotting these first four Chebyshev polynomials.

Instead of generating and plotting more and more of the Chebyshev polynomials, we will use what we have so far to delve into some fascinating properties. But first, let's look at the general way to write the recursion:

T*n*+1 is the next Chebyshev polynomial to find. The recursion steps are to multiply the current one, T*n*, by 2*x* and subtract the previous one, T*n* - 1.

As promised, lets delve into the word **orthogonal**. The word ''ortho'' means straight or right. When taken together with ''gonal'' we are describing two perpendicular lines that form a right angle.

A right angle is an angle of 90o. The cosine of 90o is zero. This is our test for orthogonality. Measure the angle between two lines, compute the cosine of this angle, and if we get 0, then the lines are orthogonal.

Amazingly, by the 19th century, it was already known how to do a similar orthogonality test with polynomials. The test:

- Multiply two polynomials together with a weighting function
- Integrate over some pre-defined interval
- If you get 0, the polynomials are orthogonal

The weighting function for the Chebyshev polynomials is 1/√(1 - *x*2).

If we take any two Chebyshev polynomials, multiply them together with the weighting function and then integrate over the values *x* = -1 to *x* = 1, we get zero. The demonstration of this integration is best shown using a plot.

Now, we integrate:

The area under the curve is the integral. Above the *x*-axis, the area is positive. Below the *x*-axis, the area is negative.

Do you see how adding equal amounts of positive and negative area gives zero? This happens when we integrate any two different Chebyshev polynomials from -1 to 1.

This property of the Chebyshev polynomials allows them to be used as a ''basis''. Briefly, if we have a set of polynomials that are a basis, we can approximate other functions as a weighted sum of these basis polynomials.

These ideas from the 19th century were later applied to approximating ideal filters with digital filters. Today, we have a nice sharing between applications and theory. Chebyshev would indeed approve.

A **nomial** is a term like 1, *x* and *x*2. Thus, a **polynomial** has many terms. In this lesson we explored two properties of the Chebyshev polynomials: recursion and orthogonality. All of the Chebyshev polynomials follow from the first two Chebyshev polynomials and a **recursion equation**. Analogous to perpendicular lines, the Chebyshev polynomials are **orthogonal**.

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14 in chapter 12 of the course:

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High School Algebra II: Homework Help Resource26 chapters | 281 lessons | 2 flashcard sets

- How to Evaluate a Polynomial in Function Notation 8:22
- Understanding Basic Polynomial Graphs 9:15
- Basic Transformations of Polynomial Graphs 7:37
- How to Graph Cubics, Quartics, Quintics and Beyond 11:14
- How to Add, Subtract and Multiply Polynomials 6:53
- Pascal's Triangle: Definition and Use with Polynomials 7:26
- The Binomial Theorem: Defining Expressions 13:35
- How to Divide Polynomials with Long Division 8:05
- How to Use Synthetic Division to Divide Polynomials 6:51
- Dividing Polynomials with Long and Synthetic Division: Practice Problems 10:11
- Operations with Polynomials in Several Variables 6:09
- How to Solve x^2 - 6x = 16
- Chebyshev Polynomials: Applications, Formula & Examples
- Chebyshev Polynomials: Definition, History & Properties
- Go to Algebra II - Polynomials: Homework Help

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