Question

Consider using a DFT to interpolate the function \(f(x)=\log (x+1)\) on the interval \([0,2 \pi]\) as in the examples of Section \(13.2\). (a) Construct and plot the interpolant on \([0,2 \pi]\) for \(l=16\) and \(l=32\). Explain why the results look unsatisfactory. (b) Consider an even extension of \(f(x)\), defining $$ g(t)=\left\\{\begin{array}{ll} f(t), & 0 \leq t<2 \pi \\ f(4 \pi-t), & 2 \pi \leq t<4 \pi \end{array}\right. $$ Apply DFT interpolation to \(g(t)\) and plot the results on \([0,2 \pi] .\) Find maximum errors for \(l=16\) and \(l=32\). Are they better than before? Why?

Short answer

Based on the solution, provide a brief explanation of why using the even extension of the function produces a better DFT interpolation. Using the even extension of the function produces a better DFT interpolation because it creates a smooth periodic function on the domain considered, which deals with the singularity present in the original function at the boundary of the interval. DFT assumes the function is periodic, so by working with an even extension, we can obtain more accurate results due to the better behavior of the extended function on the periodic domain.

Step-by-step solution

Construct and plot the interpolant for \(l=16\) and \(l=32\)

We are given the function \(f(x) = \log(x+1)\), which we want to interpolate using the Discrete Fourier Transform (DFT). First, we need to compute the DFT coefficients for \(l=16\) and \(l=32\). Then, we will use these coefficients to find the interpolating polynomial and plot it on the interval \([0, 2\pi]\).

Explain why the results look unsatisfactory

Upon plotting the interpolating polynomial, we would notice that the results look unsatisfactory. This is because the function \(f(x)\) has a singularity (a logarithmic singularity) at the boundary of the interpolation interval (\(x=0\)). Since DFT assumes the function is periodic, this singularity causes undesirable artifacts in the interpolated polynomial, which makes the results unsuitable for this particular function.

Define the even extension of the function and apply DFT interpolation to it

To improve the interpolation results, we will consider an even extension of \(f(x)\), which is given by: $$ g(t) = \left\{\begin{array}{ll} f(t), & 0 \leq t < 2\pi \\ f(4\pi - t), & 2\pi \leq t < 4\pi \end{array}\right. $$ The function \(g(t)\) is now periodic with a period of \(4\pi\), and the interpolation should be better when we apply DFT to it. Perform DFT interpolation on \(g(t)\) and plot the results for the interval \([0, 2\pi]\).

Find maximum errors for \(l=16\) and \(l=32\) and compare them with the previous results

Now that we have applied DFT interpolation to the even extension of the function \(g(t)\), we can evaluate the errors for \(l=16\) and \(l=32\). To do this, find the maximum difference between the interpolated polynomial and the actual function on the interval \([0, 2\pi]\). We should observe that the errors are significantly smaller compared to the errors obtained from interpolating \(f(x)\) directly. This is because the even extension \(g(t)\) behaves better on the periodic domain, and the interpolation has a higher accuracy as a consequence.

Key concepts

Interpolation

Interpolation in mathematics is about finding a "middle path" or a smooth curve that best fits a set of data points. With Discrete Fourier Transform (DFT), interpolation involves using a set number of frequency components to create a continuous signal approximation from discrete data samples.

When we talk about interpolating the function \(f(x) = \log(x+1)\) over the interval \([0, 2\pi]\), we essentially generate a function that smoothly connects computed DFT coefficients. These coefficients represent our data points in the frequency domain, and we use them to reconstruct the signal in the spatial domain.

In the given problem, using a specific number \(l\) of samples like 16 or 32 alters the resolution of our interpolated curve. A higher \(l\) provides more data points for interpolation, theoretically leading to better approximation. But issues like singularities can still disrupt the quality.

Singularity

A singularity in a mathematical function is a point where it behaves in an unexpected or undefined way. For the function \(f(x) = \log(x+1)\), a singularity is present at \(x = 0\). Here, the behavior spikes sharply, which poses challenges for techniques like those involving Fourier Transform.

In the context of DFT interpolation, encountering singularities can lead to highly oscillatory artifacts, known as the Gibbs phenomenon. This occurs because DFT assumes the function is smooth and periodic over the interval, hence a singular point disrupts this assumption.

To effectively handle functions with singularities during interpolation, adjustments such as function modifications or extensions are often necessary. This ensures better approximation and minimizes erratic changes in the interpolated curve.

Even Extension

An even extension is a useful strategy to improve interpolation by eliminating errors due to singularities. It involves creating a symmetric version of the function across a specified interval, essentially "mirroring" the function to make it account for potential periodicity better.

In the practical exercise, \(g(t)\) represents an even extension of \(f(x)\) defined for \(t\) ranging from \(0\) to \(4\pi\). This means for \(t\) lying beyond \([0, 2\pi]\), we define \(g(t)\) as \(f(4\pi - t)\). Such an extension ensures that \(g(t)\) becomes periodic, smooth and reduces boundary discrepancies.

Subsequently applying DFT interpolation to an even extension often results in a closer match to the actual function, since it effectively addresses issues caused by non-periodicity and singularities in the original function.

Maximum Error

Maximum error is an important metric in interpolation, representing the largest deviation between the interpolated result and the true function over a specific interval.

In the scenario of using DFT for interpolation after extending the function evenly, maximum error calculations highlight improvement in approximation quality. Using \(l = 16\) or \(l = 32\), the maximum error is computed to compare performance between direct interpolation of \(f(x)\) and its even extension \(g(t)\).

Errors are typically smaller with the even extension, confirming that the symmetry helps handle boundary conditions and singularities better. This metric is crucial because it quantifies the reliability and accuracy of the approximation in applications where precision is necessary.