What exactly is "Approximation Theory"? If I read the wikipedia-article I doesn't get much clearer. Why are "pure" mathematicians interested in it? I see a lot of people that do harmonic analysis also do approximation theory.
Approximation theory includes many subject areas of analysis, but the common idea is how well a target in a topological space (often a metric space) can be approximated by the points of a narrower subspace.
Some examples will illustrate the breadth of this topic. Given a real number x of a certain kind (e.g. algebraic), what are the rational numbers of bounded denominator that best approximate x, using the least absolute value of their difference as the objective?
Given a real function $f$ on $[0,1]$ of a certain kind (e.g. twice continuously differentiable), what are the polynomials of bounded degree that best approximate $f$? There are a variety of objectives that might be used, such as minimizing the square integral of the difference or minimizing the maximum difference.
Problems involving function approximation can be extended to higher dimensions, and the scope of approximating candidates can be varied endlessly (splines, trigonometric series, rational functions, etc.) and subject to many different restrictions (monotonicity, analyticity, symmetry, etc.).
In a strong sense all of analysis uses approximation theory.