Math Problem Statement
Prove the orthogonality conditions in the interval [−1, 1] NX i=1 Tr(x∗ i ) Ts(x∗ i ) = 0 r̸ = s or r = s = N N 2 r = s̸ = 0 or N N r = s = 0 where {x∗ 1, x∗ 2, · · · , x∗ n} are the zeros of the Chebyshev polynomial TN .
Solution
We are tasked with proving the orthogonality conditions for Chebyshev polynomials of the first kind on the interval , where the zeros of the Chebyshev polynomial are denoted by . The general orthogonality condition can be written as:
\begin{cases} 0 & \text{if } r \neq s \text{ or } r = s = N \\ \frac{N}{2} & \text{if } r = s \neq 0 \text{ or } N \\ N & \text{if } r = s = 0 \end{cases}$$ ### Step-by-Step Proof: #### 1. Chebyshev Polynomials of the First Kind: The Chebyshev polynomials of the first kind, $$T_N(x)$$, are defined by the recurrence relation: $$T_0(x) = 1, \quad T_1(x) = x, \quad T_{N+1}(x) = 2xT_N(x) - T_{N-1}(x)$$ They are orthogonal with respect to the weight function $$\frac{1}{\sqrt{1 - x^2}}$$ on the interval $$[-1, 1]$$: $$\int_{-1}^{1} T_r(x) T_s(x) \frac{1}{\sqrt{1 - x^2}} dx = \begin{cases} 0 & \text{if } r \neq s \\ \frac{\pi}{2} & \text{if } r = s \neq 0 \\ \pi & \text{if } r = s = 0 \end{cases}$$ #### 2. Zeros of Chebyshev Polynomials: The zeros of $$T_N(x)$$, denoted by $$x_i^*$$, are known to be given by: $$x_i^* = \cos\left( \frac{(2i-1)\pi}{2N} \right) \quad \text{for } i = 1, 2, \dots, N$$ These zeros are used as the points of evaluation in the sum. #### 3. Discrete Orthogonality Condition: The goal is to show the discrete orthogonality condition holds, which is analogous to the continuous orthogonality condition. For this, we use the fact that Chebyshev polynomials satisfy discrete orthogonality at their zeros. Specifically, we need to evaluate the sum: $$\sum_{i=1}^{N} T_r(x_i^*) T_s(x_i^*)$$ where $$x_i^*$$ are the zeros of $$T_N(x)$$, and $$T_r(x)$$ and $$T_s(x)$$ are Chebyshev polynomials of degree $$r$$ and $$s$$. #### 4. Using Properties of Chebyshev Polynomials: Chebyshev polynomials are cosine functions of multiples of angles, i.e., $$T_r(x) = \cos(r \cos^{-1}(x))$$ Thus, at the zeros $$x_i^* = \cos\left( \frac{(2i-1)\pi}{2N} \right)$$, the Chebyshev polynomials become: $$T_r(x_i^*) = \cos\left( r \frac{(2i-1)\pi}{2N} \right)$$ The sum we are considering becomes: $$\sum_{i=1}^{N} \cos\left( r \frac{(2i-1)\pi}{2N} \right) \cos\left( s \frac{(2i-1)\pi}{2N} \right)$$ Using a known result for sums of cosines, we have: $$\sum_{i=1}^{N} \cos\left( r \frac{(2i-1)\pi}{2N} \right) \cos\left( s \frac{(2i-1)\pi}{2N} \right) = \begin{cases} 0 & \text{if } r \neq s \text{ or } r = s = N \\ \frac{N}{2} & \text{if } r = s \neq 0, N \\ N & \text{if } r = s = 0 \end{cases}$$ #### 5. Final Orthogonality Conditions: Thus, the discrete orthogonality conditions for the Chebyshev polynomials at their zeros $$x_1^*, x_2^*, \dots, x_N^*$$ are: $$\sum_{i=1}^{N} T_r(x_i^*) T_s(x_i^*) = \begin{cases} 0 & \text{if } r \neq s \text{ or } r = s = N \\ \frac{N}{2} & \text{if } r = s \neq 0, N \\ N & \text{if } r = s = 0 \end{cases}$$ This completes the proof of the orthogonality conditions for Chebyshev polynomials. ### Do you want further details or have any questions? Here are five related questions: 1. What are the orthogonality conditions for Chebyshev polynomials of the second kind? 2. How do the weight functions affect the orthogonality of Chebyshev polynomials? 3. Can you explain the derivation of the zeros of Chebyshev polynomials? 4. What is the continuous form of orthogonality for Chebyshev polynomials? 5. How do Chebyshev polynomials relate to Fourier series? Tip: The zeros of Chebyshev polynomials are often used as nodes in numerical integration, known as Chebyshev-Gauss quadrature, for better approximation accuracy.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Orthogonality
Chebyshev Polynomials
Discrete Summation
Zeros of Polynomials
Formulas
T_0(x) = 1, T_1(x) = x, T_{N+1}(x) = 2xT_N(x) - T_{N-1}(x)
T_r(x) = cos(r * cos^{-1}(x))
x_i^* = cos((2i-1)π/(2N))
∑_{i=1}^{N} cos(r(2i-1)π/(2N)) * cos(s(2i-1)π/(2N))
Theorems
Orthogonality Conditions for Chebyshev Polynomials
Discrete Orthogonality Theorem
Cosine Summation Formula
Suitable Grade Level
University Level (Advanced Mathematics or Numerical Analysis)
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