Sparse Averages of Partial Sums of Fourier Series

Ethan Goolish and Robert S. Strichartz

Project Summary: Slight modification to the Dirichlet Kernel yields an approximate identity of a function f through convolution with the partial Fourier expansion of f. We look at under which conditions does the sparse average of some subsequence continue to produce properties included boundedness on the period as well as limit behavior converging to zero when we take out some epsilon-neighborhood of the kernel.

First we look at the linear data: that is, subsequences where $$n_k = p*k \mid p \in \mathbb{N}$$ In the case where p = 1, we have exactly the standard case: that is, we have the exact average. In general, we find that this case is well behaved, providing some approximate identity for the function. Remember that our criteria includes a bounded integral overall and that the integral taken from some epsilon-neighborhood around zero must tend to zero. While the first case will be analyzed more below, we can see in the data below proof of the second criterion.

Here we provide log-log graphs of the integral from some epsilon neighborhood out the tail to pi for various epsilon values and for various values of p. The main thing of importance here is the fact that the integral tends to 0 in the long run. From the log-log values we then find the line of best fit for each epsilon-p value pair in the form $$ax + b$$ Here we note that the a-values are approximately one, which verifies the equation: ______. Similarly, the b-value tells us that ______. Finally, we provide various scales on the actual kernel.

Rather than deal solely with fixed linear sequences, we also examine the case of random linear sequences, where $$n_k = \{(p * k) - 1, p * k, (p * k) + 1\}$$ with equal probability. Here we compare the two sequences and the kernels they produce and find they appear quite similar. At the end, the specific random sequences used to generate the data are provided.

Now, we look at the first criterion, where we want the integral from 0 to pi of each sequence to stay bounded (i.e. we have epsilon = 0). Instead of looking solely at the linear case, we will also look at some other sequences as well.

The quadratic case, where $$n_k = k^2 \mid k \in [1, N]$$

The cubic case, where $$n_k = k^3 \mid k \in [1, N]$$

The exponential case, where $$n_k = 2^k \mid k \in [1, N]$$

The case where $$n_k = 2^{k^2} \mid k \in [1, N]$$

The case where $$n_k = 2^{1+k^3} \mid k \in [1, N]$$

From the data that follows, we can see that while the first three cases behave nicely and adhere to the necessary conditions, the last three begin to behave erratically, verifying the fact that not all subsequences can be used for sparse averaging. We provide first the ordinary boundedness graph followed by a log-log plot of the data that shows similar results.

Due to our interest in the linear case, we now look to see how varying the choice of p affects the properties of the subsequence. In the plot that follows, we look at p values between 1 and 5, and see that while for small choices of N, we have some variation, all cases converge to a tight band as N grows. Similarly, in the random linear case, we have the same result where small variation occurs near the beginning, but the cases converge in the long run. When we compare the fixed and the random cases, we see that by and large, the fixed case outperforms the random one. Finally, we provide the numerical integration data behind the graphs as well as the random subsequences that were used to generate the data.