Finding a closed form for constant

Question

As part of my research I've evaluated an integral numerically and I'm trying to find a closed form for the output. The relevant answer is $\alpha \approx 0.6869812930331146009494130384156977196866429945014483$ (with accuracy in at least the first 20 decimal places). I've tried using ISC and RIES however both have failed to give a good answer to this.

Thanks!

Edit: I was trying to maximise $$\frac{\displaystyle\int_{-\infty}^{\infty}\left[\displaystyle\int_{-\infty}^{\infty} f(x)f(x-y) \ \mathrm{d}x \right]^2 \ \mathrm{d}y}{\left[\displaystyle \int_{-\infty}^{\infty} f(x)^2 \ \mathrm{d}x\right]^2}$$ where $f(x)$ is even, real, $f(0)=1$ and $f(x)$ is zero for $|x| > 1/2$. (Note that $f(x)$ does not have to equal $0$ at $x=\pm 1/2$).

Answer 1

Let's turn this into an answer, at least sort of. Taken generally, I understand the question to be

I have this number to many decimals. How do I find (a) a closed form expression [understood broadly: perhaps a rational approximation, a continued fraction, or something similar]; or (b) whatever else is known about it?

For (a), there are some tools like WolframAlpha and ries as noted by the OP and in comments. Sure, try them! This might succeed or not. Perhaps there is a closed form but too complicated for the tools. Perhaps your decimal approximation is too imprecise for meaningful results. Perhaps there is no closed form but still there is some existing knowledge. So what about (b)?

Some constants can be found in OEIS as sequences of their decimals: try 3.14159265. But it's only the few best-known constants you will find this way. No luck with our current number.

Some you can find in Steven Finch's book Mathematical Constants. It is a wonderful book with pages and pages of decimal expansions in increasing order (and then a pointer to more knowledge). To give you a flavor, here's what we could find around what we are now searching for:

0.6867778344... With Kalmár's constant [5.5]
0.6903471261... One of the iterated exponential constants [6.11]

No luck this time: our 0.68698... mismatches already in the fourth decimal. (But still I recommend trying this book.)

What else can one try? Surprisingly: Google. Perhaps your number has already appeared in mathematical literature. The catch is formatting and number of decimals: Google does not search for substrings but "words". If a 1969 article contains your constant, how do you know how many decimals they gave? You don't, so the best you can do is search for different truncations, like .68698129303 (no matches), .686981293033114600949 (no matches) ... are we desperate yet? Try once more: .686981293033114600949413 and yay! we find Garsia et al. (1969), On Some Extremal Positive Definite Functions. We have a match of 24 decimals: highly unlikely to be a coincidence.

So if all else fails: Try google. Although you may need to try several variations of your search term, it may be worth it. Be persistent in searching! Also, you get different results when you include the initial zero and when you don't, so try both.

Oh, I have to confess one thing. I only tried one arbitrary truncation 0.686981293033114600949413 (just selected "some reasonable number of decimals" with the mouse) and was lucky on my first attempt. (So much for motivational stories about persistence.)