Show that T3(x) = 4x3 − 3x. Also state THREE of its orthogonality conditions.

To show that \( T_3(x) = 4x^3 - 3x \), we can use the recurrence relation for Chebyshev polynomials, which is given by:

\[
T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)
\]

We start with the known base cases:
- \( T_0(x) = 1 \)
- \( T_1(x) = x \)

Now, we use the recurrence relation to find \( T_2(x) \) and \( T_3(x) \):

1. **Finding \( T_2(x) \)**:
\[
T_2(x) = 2xT_1(x) - T_0(x) = 2x(x) - 1 = 2x^2 - 1
\]

2. **Finding \( T_3(x) \)**:
\[
T_3(x) = 2xT_2(x) - T_1(x) = 2x(2x^2 - 1) - x = 4x^3 - 2x - x = 4x^3 - 3x
\]

Thus, we have shown that:
\[
T_3(x) = 4x^3 - 3x
\]

### Orthogonality Conditions

The Chebyshev polynomials \( T_n(x) \) are orthogonal with respect to the weight function \( \frac{1}{\sqrt{1 - x^2}} \) on the interval \( [-1, 1] \). The orthogonality condition is given by:

\[
\int_{-1}^{1} \frac{T_m(x) T_n(x)}{\sqrt{1 - x^2}} dx = 0 \quad \text{for} \quad m \neq n
\]

Three specific orthogonality conditions for Chebyshev polynomials are:

1. **For \( m \neq n \)**:
   \[
   \int_{-1}^{1} \frac{T_m(x) T_n(x)}{\sqrt{1 - x^2}} dx = 0
   \]

2. **For \( m = n = 0 \)**:
   \[
   \int_{-1}^{1} \frac{T_0(x) T_0(x)}{\sqrt{1 - x^2}} dx = \pi
   \]

3. **For \( m = n \neq 0 \)**:
   \[
   \int_{-1}^{1} \frac{T_n(x) T_n(x)}{\sqrt{1 - x^2}} dx = \frac{\pi}{2}
   \]

These conditions highlight the orthogonality of Chebyshev polynomials over the interval \( [-1, 1] \).

Would you like further details or have any questions?

Here are five related questions:
1. How can the recurrence relation for Chebyshev polynomials be derived?
2. What are Chebyshev polynomials used for in numerical analysis?
3. Can Chebyshev polynomials be generalized to other intervals besides \( [-1, 1] \)?
4. How do Chebyshev polynomials relate to minimizing polynomial approximation errors?
5. What is the weight function in the orthogonality conditions of Chebyshev polynomials, and why is it necessary?

**Tip:** Chebyshev polynomials are useful in approximating functions because they minimize the error compared to other polynomials in terms of the maximum norm.

Show that T3(x) = 4x^3 − 3x. Also state THREE of its orthogonality conditions.

Learn how to derive the Chebyshev polynomial T3(x) = 4x^3 − 3x using recurrence relations, and explore the orthogonality conditions of Chebyshev polynomials. This solution is suitable for undergraduate mathematics students.

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