# Series acceleration for more complicated types of oscillating series

Question: It is well known that the greatest integer function has a Fourier series representation. Since the greatest integer function itself is not periodic, the representation is derived from the sawtooth wave function, $p(x)=x-[x]-1/2$, which has a pure sine series that converges to $p(x)$ for all $x\notin Z$. What other infinite oscillating series representations for the greatest integer function exist and can we express more complicated types of infinite oscillating series in terms of the greatest integer function? More specifically, can we express the greatest integer function as a Fourier-Bessel series i.e. as an infinite sum of Bessel functions.

Background: If the greatest integer function can be expressed in part in terms of a Fourier series, then some Fourier series can be expressed in part in terms of the greatest integer function. This simple concept helps us accelerate any Fourier series at any point except at its discontinuities by first locating and removing them. And in the vast majority of cases, we can accelerate Fourier series (if not express them exactly in terms of the greatest integer function) by expressing the Laurent terms of {$a_n$} and {$b_n$} exactly in terms of periodic Bernoulli polynomials. (My approach for accelerating Fourier series is not original.) I used this approach to express some complicated-looking inverse-Laplace transforms in exact form. For example, the inverse Laplace transform of $F(s)=\frac{\sinh(xs)}{s^2\cosh(s)}$ for all $x$ between 0 and 1 is $f(t)=x+\frac{8}{\pi^2}\sum_{n=1}^{\infty}\frac{(-1)^n}{(2n-1)^2}\sin(\frac{2n-1}{2}{\pi}x)\cos(\frac{2n-1}{2}{\pi}t)=$ $x-2p(\frac{x+t-1}{4})^2+2p(\frac{x+t+1}{4})^2-2p(\frac{x-t-1}{4})^2+2p(\frac{x-t+1}{4})^2$. So, if other types of oscillating series (such as certain forms of Mie series) can be expressed in terms of the greatest integer function, then we may be able to use the greatest integer function to accelerate them if not express them in exact form.

Update: A good answer to this question cannot be found in a book on infinite series nor will it come easily. My goal is to create a powerful, direct, and unified approach for accelerating many types of slowly converging oscillating infinite series which arise in various scientific disciplines. For example, Mie series arise in the study of refraction. Given the level of success I had with Fourier series, I believe it is worthwhile to determine what other oscillating series this approach can be extended to. If you find this question interesting, then I recommend collaboration. When you have a good answer, you will also have a discovery worthy of publication in a prestigious journal.

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Here are two classical sources that might contain some clues

G.H. Hardy: Divergent series, any edition

K. Knopp: Theory and application of infinite series, Dover

Knopp has a chapter on ways to boost the convergence rate of series. All of these are summability methods which are discussed at great length in Hardy's book. It turns out that Euler used these acceleration techniques to compute sums of series. In particular, according to Hardy, this is how Euler first "guessed" that

$$\sum_{n\geq 1} \frac{1}{n^2} =\frac{\pi^2}{6}.$$

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I appreciate your interest, but the answer you gave would have been better left as a comment. Many of us are tempted to be the first to answer a question to achieve self-actualization and earn points on MathOverflow (myself included), but for open problems such as this one, slow and steady wins the race. –  Ken Sep 22 '12 at 14:59
I prefer answers to comments for three technical reasons: I can easily detect my TeX errors, I can edit typos and errors, and I do not have an upper limit on the number of characters I can use. In any case, sumability techniques are frequently used in harmonic analysis, and Hardy's book seems to cover the most useful ones. In particular, there you can find a comparison of their strengths. I have to close here since I am reaching my allowed upper limit of characters. –  Liviu Nicolaescu Sep 22 '12 at 16:10