q_{0} = 1 , q_{1} = x , q_{2} = x ^ 2 - 1/3 where \{q_{i}\} is the set of orthogonal monomials in the interval [-1,1] with respect to the standard square norm of the differentiable func- tions. Compute q_{3} and then express x ^ 3 as a q_{0} to q_{3}

To solve this problem, we need to compute \( q_3 \), the next orthogonal polynomial in the sequence, and then express \( x^3 \) as a linear combination of the polynomials \( q_0, q_1, q_2, q_3 \).

### Step 1: Orthogonality Condition
The polynomials \( \{q_i\} \) are orthogonal with respect to the standard square norm on the interval \([-1, 1]\), which means that for any two different polynomials \( q_i \) and \( q_j \):

\[
\int_{-1}^{1} q_i(x) q_j(x) \, dx = 0 \quad \text{for } i \neq j
\]

### Step 2: Determine \( q_3(x) \)
Assume that \( q_3(x) \) has the general form of a cubic polynomial:

\[
q_3(x) = x^3 + a_2 x^2 + a_1 x + a_0
\]

The orthogonality condition requires \( q_3(x) \) to be orthogonal to \( q_0(x) = 1 \), \( q_1(x) = x \), and \( q_2(x) = x^2 - \frac{1}{3} \).

Let's calculate the coefficients \( a_2 \), \( a_1 \), and \( a_0 \) by ensuring orthogonality with respect to \( q_0(x) \), \( q_1(x) \), and \( q_2(x) \).

#### Orthogonality with \( q_0(x) = 1 \):

\[
\int_{-1}^{1} q_3(x) \cdot 1 \, dx = \int_{-1}^{1} \left( x^3 + a_2 x^2 + a_1 x + a_0 \right) dx = 0
\]

Evaluating the integrals:

\[
\int_{-1}^{1} x^3 \, dx = 0, \quad \int_{-1}^{1} x^2 \, dx = \frac{2}{3}, \quad \int_{-1}^{1} x \, dx = 0, \quad \int_{-1}^{1} 1 \, dx = 2
\]

This gives:

\[
a_2 \cdot \frac{2}{3} + 2a_0 = 0 \quad \Rightarrow \quad a_0 = -\frac{a_2}{3}
\]

#### Orthogonality with \( q_1(x) = x \):

\[
\int_{-1}^{1} q_3(x) \cdot x \, dx = \int_{-1}^{1} \left( x^4 + a_2 x^3 + a_1 x^2 + a_0 x \right) dx = 0
\]

Evaluating the integrals:

\[
\int_{-1}^{1} x^4 \, dx = \frac{2}{5}, \quad \int_{-1}^{1} x^3 \, dx = 0, \quad \int_{-1}^{1} x^2 \, dx = \frac{2}{3}, \quad \int_{-1}^{1} x \, dx = 0
\]

This simplifies to:

\[
a_1 \cdot \frac{2}{3} = 0 \quad \Rightarrow \quad a_1 = 0
\]

#### Orthogonality with \( q_2(x) = x^2 - \frac{1}{3} \):

\[
\int_{-1}^{1} q_3(x) \cdot \left(x^2 - \frac{1}{3}\right) dx = \int_{-1

Learn how to compute q3, the next orthogonal polynomial in the sequence, and express x^3 as a linear combination of orthogonal polynomials q0, q1, q2, and q3. This problem involves orthogonal polynomials, integral calculus, and polynomial orthogonality concepts.

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