Families of functions closed under integration

Question

What are some concrete families $\mathcal F$ of real functions that are closed under integration in the sense that for every $f \in \mathcal F$ there is $F \in \mathcal F$ such that $F'=f$?

Here are the simplest examples I know:

• The vector space generated by functions of the form $x^n$, that is, polynomial functions.

• The vector space generated by functions of the form $x^n e^{a x}$ for $a\in \mathbb R$, that is, sums of functions of the form $p(x) e^{a x}$, for $p$ a polynomial and different $a$.

• The vector space generated by functions of the form $x^n \log (a x)$ for $a\in \mathbb R$, that is, sums of functions of the form $p(x) \log (a x)$, for $p$ a polynomial and different $a$.

Are there any other examples?

Of course, the family of rational functions is not closed under integration.

The third example is interesting because, unlike the other two, it is not closed under differentiation.

In all these examples, the corresponding algebra is also closed under integration, that is, you can also consider products and still be able to find a primitive within the family. Are there any examples that are not algebras?

Answer 1

Another example (which is quite useful for a number of reasons) is the set of trigonometric polynomials

$$f(x) = \sum\limits_{k \in A} c_k e^{i k x}$$

where $A \subseteq \mathbb{Z}$ is finite. This is both closed under integration and differentiation, and in fact has some nice density properties related to Fourier series.

To make the functions strictly real valued, we would consider the (essentially identical) family

$$f(x) = \sum\limits_{k \in A} a_k \sin{kx} + b_k \cos{kx}$$

Answer 2

Two rather trivial examples:

• All polynomial functions divisible by $x^{2013}$ (every one has exactly one primitive in the class, and the class is not closed under differentiation).
• $\{\, x\mapsto ce^x \mid c\in \Bbb R\,\}$ (differentation or taking primitives is rather boring on this set). Since you did not ask for a vector space, you can also let $c$ range over a subset of$~\Bbb R$.