Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein

In last week’s post we looked at aspects of puzzles of the form “What is the next number”. We are presented with a short list of numbers, for example ${1, 3, 5, 7, 9}$ and asked for the next number in the sequence. Arguments were given indicating why any number might be regarded as the next number.

Borwein integrals evaluated by Mathematica. The first seven integrals are all equal to ${\pi}$ . The eighth is a tiny bit less than this.

In this article we consider a sequence of seven ones: ${1, 1, 1, 1, 1, 1 ,1}$ . Most people would agree that the next number in the sequence is ${1}$ . We will show how the number ${1 - 1.47\times10^{-11} \approx 0.999\,999\,999\,985}$ could be the “correct” answer.

The Sequence of Integrals

The sinc function is defined thus:

$\displaystyle \mathrm{sinc\ } t = \begin{cases} \displaystyle{\frac{\sin t}{t}}, & t \ne 0 \\ \ \ 1, &t = 0 \end{cases}$

We note that ${\mathrm{sinc\ } t}$ oscillates with decaying amplitude as ${|t|}$ increases, and has zeros at ${\{\pm \pi, \pm 2\pi, \dots \}}$ . In 2001, father-and-son team David Borwein and Jonathan Borwein published a sequence of integrals with a very surprising property. We define the integrals

$\displaystyle J_n := \int_{-\infty}^{\infty} \prod_{k=1}^{n}\ \mathrm{sinc\ }\frac{t}{2k-1}\,\mathrm{d}t \,.$

Borwein and Borwein showed that, for ${n=1, 2, 3, \dots ,7}$ we have ${J_n =\pi}$ , but for ${n=8}$ the integral is a tiny amount less than this. Thus,

$\displaystyle \begin{array}{rcl} J_1 := \int_{-\infty}^{\infty} \mathrm{sinc\, } t\,\mathrm{d}t &=& \pi \\ J_2 := \int_{-\infty}^{\infty} \mathrm{sinc\, } t\ \mathrm{sinc\, }\frac{t}{3}\,\mathrm{d}t &=& \pi \\ J_3 := \int_{-\infty}^{\infty} \mathrm{sinc\, } t\ \mathrm{sinc\, }\frac{t}{3}\ \mathrm{sinc\, } \frac{t}{5}\,\mathrm{d}t &=& \pi \\ &\vdots& \\ J_7 := \int_{-\infty}^{\infty} \mathrm{sinc\, } t\ \mathrm{sinc\, }\frac{t}{3}\ \dots\ \mathrm{sinc\, }\frac{t}{13}\,\mathrm{d}t &=& \pi \end{array}$

However, for ${n=8}$ numerical experiments indicated that

$\displaystyle J_8 =\!\! \int_{-\infty}^{\infty}\!\!\!\! \mathrm{sinc\, } t\ \mathrm{sinc\, }\frac{t}{3} \dots \mathrm{sinc\, } \frac{t}{15}\,\mathrm{d}t = \frac{467\,807\,924\,713\,440\,738\,696\,537\,864\,469}{467\,807\,924\,720\,320\,453\,655\,260\,875\,000}\, \pi \,.$

That is, ${J_8 = (1-\epsilon)\pi}$ where ${\epsilon \approx 1.47\times 10^{-11}}$ . The Borweins provided an explanation of this most curious behaviour.

Analysis in Terms of Low-pass Filters

Frequency response function of low-pass filter ${H(\omega)}$ .

Let us define the integrand of ${J_\ell}$ to be

$\displaystyle S_\ell(t) = \prod_{k=1}^{\ell}\mathrm{sinc\ }\frac{t}{2k-1} = \mathrm{sinc\ }t\times \mathrm{sinc\, }\frac{t}{3}\times \cdots \ \times\mathrm{sinc\, } \frac{t}{(2\ell-1)} \,.$

We note that the Fourier transform of ${S_1( t )}$ is ${\textstyle{ \frac{1}{2}\mathrm{H}(\frac{\omega}{2}) }}$ , where

$\displaystyle \mathrm{H}(\omega) = \begin{cases} 1, & \omega \le \textstyle{\frac{1}{2}} \\ 0, & \omega > \textstyle{\frac{1}{2}} \end{cases}$

We define the low-pass response functions ${H_\ell(\omega) = H((2\ell-1)\omega)}$ . By the convolution theorem, the Fourier transform of ${S_\ell(t)}$ is the convolution of the transforms of the factors:

$\displaystyle \mathcal{F}\{ S_\ell \}(\omega) = \mathcal{H}_\ell(\omega) = [ H_1 \star H_2 \star \cdots \star H_\ell ](\omega) \,.$

This is the frequency response function for successive application of low-pass filters ${H_k(\omega)}$ with half-width ${1/[2(2k-1)]}$ . We note the special case of the Fourier transform when ${\omega = 0}$ implies

$\displaystyle J_\ell = \mathcal{F}\{ S_\ell \}(0) = \mathcal{H}_\ell(0)$

We can consider the convolution as the action of the filters ${\{H_2, H_3, \dots , H_\ell\}}$ upon the first factor, ${\mathcal{H}_1(\omega) = H_1(\omega)}$ . This has jump discontinuities at ${\omega = \pm\frac{1}{2}}$ (see Figure above). It is near these points that subsequent filtering has smoothing effects. Results for successive application of further factors are shown in Figure below. We see that the sudden jumps become smoother transitions. As consecutive filters have decreasing band-width, the range over which each one acts decreases, but the overall range of impact increases: the effect of filtering moves closer to ${\omega=0}$ as ${\ell}$ increases. The range for ${H_\ell(\omega)}$ is ${\sum_{k=1}^{\ell} 1/(2(2\ell-1))}$ . The numerical values of this sum for ${\ell= 2, 3, \dots 8}$ are

$\displaystyle \{ 0.167,0.267,0.338,0.394,0.439,0.478,0.511 \}$

The sum exceeds ${0.5}$ when ${\ell=8}$ . This implies that the value of ${\mathcal{H}_\ell(0)}$ remains equal to ${1}$ until ${\ell=8}$ . But this value is equal to the integral of ${S_\ell (t)}$ over the full range of ${t}$ , and that is just the definition of ${J_\ell }$ . However, when ${\ell=8}$ , filtering reduces ${\mathcal{H}_\ell(0)}$ to a value slightly below ${1}$ , which explains why ${J_8}$ is marginally less than ${\pi}$ .

Response functions of ${H_\ell(\omega)}$ for ${\ell=1,2,3}$ and ${4}$ .

Conclusion

When computer scientists first obtained the (apparently anomalous) value of ${J_8}$ , they suspected a software bug. It seemed extraordinary that the first seven terms of the sequence ${J_\ell}$ should have the value ${\pi}$ , but that the eighth should deviate ever-so-slightly from this value. However, the analysis of Borwein and Borwein (2001) made it clear that the numerical results were correct. The first figure above shows some output from a Mathematica notebook, confirming the results. Schmid (2014) provides another analysis of the phenomenon, and an excellent video in the 3Blue1Brown series of Grant Sanderson gives further insight into the strange behaviour of the Borwein integrals.

Sources

${\bullet}$ Baez, John C., 2018: Patterns That Eventually Fail. Link.

${\bullet}$ Borwein, David and Borwein, Jonathan M., 2001: Some remarkable properties of sinc and related integrals. The Ramanujan Journal, 5(1), 73–89, doi.

${\bullet}$ Sanderson, Grant, 2022: Researchers thought this was a bug (Borwein integrals). YouTube Video.

${\bullet}$ Schmid, Hanspeter, 2014: Two curious integrals and a graphic proof. Elemente der Mathematik, 69, 11–17. PDF.

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