# Mean Square Error and Gibbs oscillations

While studying the convergence of Fourier transform, I got to know two conditions.

• $$\sum_{n=-\infty}^{\infty}|x(n)|<\infty$$
• $$\sum|x(n)|^{2} \leq [\sum|x(n)|]^{2}$$

While I was reading the text, I found this paragraph quite confusing. I didn't understood this.

If a sequence is not absolutely summable but has finite energy, one may employ a type of convergence in which the series converges so the mean square error is 0. The "attendant Gibbs oscillations" at a discontinuity are of practical significance in filter design.

I don't know what Gibbs oscillations mean here. So if someone could please explain the meaning of this paragraph (taken for DSP by Alan V. Oppenheim).

$$\sum_{n=-\infty}^{\infty}|x[n]|^2\lt\infty\tag{1}$$