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.