Question

Following the procedure outlined in Example \(13.4\), construct and plot interpolating trigonometric polynomials for the Chebyshev polynomials \(T_{5}(x)\) and \(T_{6}(x)\) defined in Section \(10.6\). (Thus, the Chebyshev polynomials are not the interpolants; rather, they are being interpolated here, for a change.) Try using \(l=8\) and \(l=16\). What are your observations? Why is the error for the more oscillatory \(T_{6}\) smaller than the error for \(T_{5}\) ?

Short answer

Question: Construct and plot interpolating trigonometric polynomials for the Chebyshev polynomials \(T_{5}(x)\) and \(T_{6}(x)\) using truncation parameters \(l = 8\) and \(l = 16\). Analyze the error between the original Chebyshev polynomials and their interpolating trigonometric approximations, especially for the more oscillatory \(T_{6}(x)\). Answer: To approximate the Chebyshev polynomials \(T_{5}(x)\) and \(T_{6}(x)\) with interpolating trigonometric polynomials, follow these steps: define the Chebyshev polynomials, compute the Fourier coefficients, construct the interpolating trigonometric polynomials, and plot the results. Finally, calculate the pointwise error at each point on the grid. The error for the more oscillatory \(T_{6}(x)\) will be smaller due to the interpolating nature of the trigonometric polynomials.

Step-by-step solution

Define the Chebyshev Polynomials

First, let's define the Chebyshev polynomials \(T_{5}(x)\) and \(T_{6}(x)\) as given in Section 10.6: \(T_{5}(x) = 16x^5 - 20x^3 + 5x\) \(T_{6}(x) = 32x^6 - 48x^4 + 18x^2 - 1\)

Compute the Fourier Coefficients

Next, let's compute the Fourier coefficients \(a_k\) and \(b_k\) for both \(T_{5}(x)\) and \(T_{6}(x)\) using the given truncation parameters \(l = 8\) and \(l = 16\). The formulas for the Fourier coefficients are given as: \(a_k = \frac{2}{\pi}\int_{0}^{\pi}f(x)\cos(kx)dx\) \(b_k = \frac{2}{\pi}\int_{0}^{\pi}f(x)\sin(kx)dx\) Here, \(f(x)\) represents the Chebyshev polynomials.

Construct the Interpolating Trigonometric Polynomials

Using the computed Fourier coefficients, we can now construct the interpolating trigonometric polynomials \(P_l(x)\) for both \(T_{5}(x)\) and \(T_{6}(x)\), with \(l = 8\) and \(l = 16\). The general formula for the trigonometric polynomial is: \(P_l(x) = \frac{1}{2}a_0 + \sum_{k=1}^{l}(a_k\cos(kx) + b_k\sin(kx))\)

Plot the Interpolating Trigonometric Polynomials

Having constructed the interpolating trigonometric polynomials, we can now plot them, along with the original Chebyshev polynomials. This will provide a visual representation of how well the interpolating polynomials approximate \(T_{5}(x)\) and \(T_{6}(x)\) for both values of \(l\).

Analyze the Error

Finally, let's analyze the error between the original Chebyshev polynomials and their interpolating trigonometric approximations. We can calculate the pointwise error at each point on the grid by comparing the values of the original polynomial and the interpolating polynomial: \(E(x) = |T_n(x) - P_l(x)|\) Plot the error over the grid for each case, and analyze the results. This should give insight into why the error for the more oscillatory \(T_{6}\) is smaller than the error for \(T_{5}\).

Key concepts

Chebyshev Polynomials

Chebyshev polynomials are a special family of orthogonal polynomials that are highly significant in numerical analysis and approximation theory. They are particularly useful in minimizing approximation errors when interpolating functions. Chebyshev polynomials of the first kind, denoted as \( T_n(x) \), are defined on the interval \([-1, 1]\) and have many interesting properties, like their roots being extremal points (the largest and smallest possible values) of a polynomial on this interval.
For example, \( T_5(x) = 16x^5 - 20x^3 + 5x \) is a polynomial of degree 5. Similarly, \( T_6(x) = 32x^6 - 48x^4 + 18x^2 - 1 \) is a polynomial of degree 6. These forms help capture the oscillatory nature of the function they represent, making them excellent for studying interpolation behavior.

Numerical Error Analysis

Numerical error analysis involves understanding the difference between the true function and the approximation generated by interpolating functions. An error quantifies how well an interpolating function approximates a target function. It's essential for comprehension in scientific computations and predictions.
The error can be assessed through a formula \( E(x) = |T_n(x) - P_l(x)| \), where \( T_n(x) \) is the Chebyshev polynomial and \( P_l(x) \) is the interpolating polynomial. Despite the concept sounding complicated, it basically measures how close these two functions are at any given point. Understanding why one function might have a smaller error than another often depends on factors like the function's oscillatory nature and the degree of the polynomial used for interpolation.

Fourier Coefficients

Fourier coefficients are crucial in constructing trigonometric polynomials that approximate a given function. They are used to transform a complex function into a series of simpler trigonometric functions. This method is particularly helpful in the realm of signal processing and solving differential equations.
To calculate the Fourier coefficients \( a_k \) and \( b_k \), you can use:

• \( a_k = \frac{2}{\pi}\int_{0}^{\pi}f(x)\cos(kx)dx \)
• \( b_k = \frac{2}{\pi}\int_{0}^{\pi}f(x)\sin(kx)dx \)

These coefficients capture the essence of the periodic function’s behavior by quantifying the components of sine and cosine functions.
By obtaining these coefficients for \( T_5(x) \) and \( T_6(x) \), you can compose trigonometric polynomials of different lengths \( l \). This step is crucial to reconstruct the interpolating polynomial \( P_l(x) \) from its sinusoidal components.

Oscillatory Functions

Oscillatory functions, such as certain Chebyshev polynomials, exhibit repeated fluctuations around a mean value. Understanding their behavior is key to mastering interpolation and error analysis. Such functions provide a basis for constructing polynomials and other functions that can replicate the complex periodic behavior.
Here, function \( T_6(x) \) is inherently more oscillatory due to its higher degree and the influence of borrowed Fourier modes. This trait aids in a better approximation with fewer errors when interpolating. It is contrasted with \( T_5(x) \) which, despite being similar, doesn’t inherently capture oscillations as accurately under similar numerical settings.
Working with oscillatory functions requires careful mathematical manipulation to highlight or mitigate fluctuations. Often, this involves choosing a suitable polynomial degree and a careful error analysis approach to ensure accurate approximations and efficient computation.