Dirichlet conditions for the convergence of Fourier series

Question

According to Wikipedia, Dirichlet conditions are

1. $f$ must be absolutely integrable over a period.
2. $f$ must have a finite number of extrema in any given bounded interval, i.e. there must be a finite number of maxima and minima in the interval.
3. $f$ must have a finite number of discontinuities in any given bounded interval, however the discontinuity cannot be infinite.

I find the second condition a little strange. By a "finite number of extrema," does it mean a finite number of (absolute? local?) extreme values of $f(x)$? Or a finite number of $x$ values at which $f(x)$ is an (absolute? local?) extremum? Are constant functions, or even square waves, for which we know the Fourier series converge to average values, excluded because they have "infinitely many extrema" and therefore do not satisfy the second condition?

Answer 1

In item 2, "a finite number of extrema" means a finite value of points $x$ at which an extremum is attained. You are right that this condition, as stated, appears to exclude the square wave.

But it did not have to be phrased so. What is actually used in the proof is that the interval $[0,T]$, on which $f$ is initially defined (I take $T$ to be its period) can be partitioned into finitely many subintervals on each of which $f$ is monotone. Monotonicity does not have to be strict; in particular the square wave qualifies.

And as mathworker21 said, the modern formulation of the Dirichlet condition is "the function has bounded variation", which is really the essence of what is going on. This is a weaker condition than having the domain split into finitely many intervals of monotonicity.

Answer 2

That's a case when the "sufficient" and "necessary" properties of statements come into play. Although the square wave function really doesn't satisfies the Dirichlet conditions (The definition of maximum is, according to Wikipedia

"If the domain X is a metric space then f is said to have a local (or relative) maximum point at the point x∗ if there exists some ε > 0 such that f(x∗) ≥ f(x) for all x in X within distance ε of x∗."

and every point in the upper parts of the wave does satisfies that, and the same is true for the lower parts of it), these conditions sufficient, but not necessary, so, there'll be functions out of those that will be still convergent. The square wave is such one. (It's a bad example if you ask me).