I apologize if the post is incorrect.

I'm a sophomore student studying Electrical engineering. As a part of an introductory course on signal and linear systems, I'm required to learn Fourier and Laplace Transformation.

But the problem is I don't understand the concept of 'convergence' and it's really bugging me. It pops up throught the text.

So I want the suggestion to understand convergence from the beginning.

  • $\begingroup$ Your post is not "incorrect", but I just wonder if you didn't have any math course yet. The concept of convergence should be covered in a first year math course. This question is actually much too broad to be answered satisfactorily in such a forum. $\endgroup$ – Matt L. Dec 8 '19 at 16:47
  • $\begingroup$ I have completed calculus 1-3, course on ODE PDE, basic linear algebra. Convergence popped up here and there in calculus but I can't seem to relate to it now. $\endgroup$ – John Dec 8 '19 at 17:19

A first short shot, because I could waste hours of your time on that.

  1. The reality (whether it exists or not) looks, sounds, appears complicated
  2. As our senses are limited, and our brain is too slow, we need to shrink our perceptions, enough to understand most of what we need to grasp and share
  3. So with our limitations, we need (and are only capable of) approximations of the above reality, here some signal $s(t)$
  4. Depending on the time we have, the purpose, the extent, different kinds of approximations are required.

Convergence is the science of 'how approximations can be expected to be close enough to the reality'.

To draw a circle by hand, you only need to seize that $\pi \approx ~ 3$ (uniform 5% error). With a compas on a gridded sheet graph paper, perhaps $\pi \approx ~ 3.15$ is sufficient. For the Jet Propulsion Lab (JPL) "for interplanetary navigation, we use 3.141592653589793", see details in How Many Decimals of Pi Do We Really Need?

Convergence is about how more complex questions are answered (see for instance how can a function can be approximated by sines or cosines) :

  • can a calibration function be approximated by a parabola
  • if I want raise of 20 % in two years, can I have an increase of only $9.6 \%$ each of the two years (th answer is yes).

In more academic terms: suppose that there is a function $s(t)$. One has little knowledge about it, except that it wiggles. So you can choose a set of functions $\psi_\omega (t) $ depending on a parameter $\omega$, that wiggles like exponential sines. Can we approximate $s(t)$ with a certain combination of a finite number of $\psi_\omega (t)$? In the standard linear case, we seek a linear combination of the $\psi_\omega (t)$:

$$\sum a_\omega \psi_\omega (t)$$

The question rephrases as, for a given interval for $t$, and an increasing set $\Omega_\lambda$, $\lambda\in \Lambda$, of $\omega \in \Omega_\lambda$. The causal ideas, is: the bigger the number of terms (so the bigger the cardinal or the number of terms in $\Lambda$). In other terms, is

$$s(t)-\sum_{a_\omega \in \Omega_\lambda} a_\omega \psi_\omega (t)$$

very close to $0$, and how, depending on $t$ and $\Omega_\lambda$. Or: how fast does

$$\Delta\left(s(t)-\sum_{a_\omega \in \Omega_\lambda} a_\omega \psi_\omega (t)\right)$$

tends to zero, where $\Delta(\cdot)$ is a measure of closeness (a distance, a norm, a divergence).

  • $\begingroup$ I appreciate your comment for providing the intuition of convergence. But what would be an academic approach to understand it? $\endgroup$ – John Dec 8 '19 at 16:32
  • $\begingroup$ To help me write a follow-up, can you help formulate what would an academic approach to you? $\endgroup$ – Laurent Duval Dec 8 '19 at 17:09
  • $\begingroup$ To be specific, I want to understand the convergence of an integral- devolope a proper understanding and do some practice. $\endgroup$ – John Dec 8 '19 at 17:21
  • $\begingroup$ I have added a few details, can you comment on how it is clear, so that I can improve $\endgroup$ – Laurent Duval Dec 8 '19 at 18:51
  • 1
    $\begingroup$ @John: Since both Laplace and Fourier transform are defined by improper integrals, you should probably read up on these integrals. A possible starting point could be the corresponding wikipedia page. $\endgroup$ – Matt L. Dec 8 '19 at 18:55

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