Distributions 2 | Test Functions [dark version]

Distributions rely on a special class of smooth “test functions” that act like probes: they are nonzero only in a small region of space, yet are infinitely differentiable. That combination—C∞ smoothness plus compact support—creates a controlled setting for defining and analyzing distributions, where convergence and differentiation must behave well.

Test functions are defined as functions Φ: R^n → R (or complex-valued, without changing the core ideas). The space of test functions is denoted by C_c^∞(R^n), built from two key requirements. First, Φ must be smooth: it has derivatives of all orders. Second, Φ must vanish outside a bounded set—equivalently, its support is compact. The “support” of Φ is the smallest closed set outside of which Φ is identically zero; formally, it is the closure of the set of points where Φ(x) ≠ 0. If that support is bounded, it becomes compact, matching the compact-support condition used in C_c^∞(R^n).

Because these test functions form a vector space, sums and scalar multiples stay inside the same class: adding two smooth functions with compact support still yields a smooth function whose nonzero region remains bounded. This linear structure matters, but it’s not the whole story. Distributions require a specific notion of convergence later on, and simply putting a norm on the vector space is not enough to capture the convergence behavior needed for distribution theory. The transcript flags that a suitable topology or metric will be introduced in a later part of the series.

To make the definitions concrete, the transcript works through examples in n dimensions. The simplest test function is the zero function, which is smooth everywhere and has support nowhere (hence compact). A less trivial example is constructed using a “bump” function tied to the unit ball. Using the Euclidean norm ||x||, the function is set to 0 outside the region where ||x|| ≥ 1, while inside the unit ball it takes a smooth expression resembling exp(1/(1-||x||^2)) (the transcript also mentions choosing it squared in the sketch). In two dimensions, this produces a dome-like surface above the unit circle—an archetypal compactly supported smooth function.

The transcript then introduces notation for derivatives using multi-indices. A multi-index α = (α1, …, αn) records how many times to differentiate with respect to each coordinate. The operator D^α means ∂^{|α|} / (∂x1^{α1} … ∂xn^{αn}). A worked two-variable example clarifies the mechanism: if f(x1, x2) = 2 x1^2 x2^3 and α = (2,1), then D^α f corresponds to differentiating twice with respect to x1 and once with respect to x2, producing a new function after successive partial derivatives.

Finally, the smoothness condition is restated in this language: Φ is in C^∞(R^n) exactly when D^α Φ is continuous for every multi-index α. With the test-function space, support, and multi-index differentiation in place, the series is positioned to build more test functions and then define the special convergence notion required for distributions.