Sum of periodical functions

Let $f$ be a polynomial. Show that $f$ can be written as the sum of $\deg f +1$ periodical functions but not as the sum of less periodical functions.

Solution



Let $\mathcal{B}=\{e_a, \; a \in A\}$ be a Hamel Basis of  $\mathbb{R}$ over $\mathbb{Q}$. Every $x \in \mathbb{R}$ can be written uniquely as:

$$x =\sum_{a\in A} x_a e_a$$
where $x_a \in \mathbb{Q}$ for every $a \in A$ and the set $\{a \in A : x_a \neq 0\}$ is finite.

We observe that for every $x, y \in \mathbb{R}$ and every $a \in A$ holds:

$$(x+y)_{a}=x_a+y_a$$
Also, for $a, b \in A$ such that $a \neq b$ holds $b_a=0$. Hence for every $x \in \mathbb{R}$ and $a, b \in A$ such that $a \neq b$ holds $(x+b)_a=x_a$. In other words the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined as:

$$f_a(x)= x_ae_a$$
is periodic, with a period of every number $b \in A \setminus \{a\}$.

Let $P(x)$ be a polynomial of degree $n$ and $ x \in \mathbb{R}$ such that $\displaystyle x =\sum_{a \in A} x_a e_a$ as above. Then the number $P(x)$ is written as a finite $\mathbb{R}$ linear combination of terms that are of the form:

$$x_{a_1}, \; x_{a_2}, \dots, x_{a_n}, \; e_{a_1}, e_{a_2}, \dots, e_{a_n} \tag{1}$$

where $a_1, a_2, \dots , a_n \in A$.

Let $\widetilde {a_0},\widetilde {a_1}, \dots ,\widetilde {a_n}$ be random $n+1$ different elements of $A$. Then in every term of the form of $(1)$ we can see at most $n$ of

$$\widetilde {{a_0}},\widetilde {{a_1}}, \dots ,\widetilde {{a_n}}$$

Therefore we can write:

$$P\left( x \right) = \sum_{i = 0}^n f_i\left( x \right)$$

where $f_i(x)$ is the sum of the terms that are of the form $(1)$ in which it does appear $\widetilde {a_i}$.

Since each of the function $f_i$ is periodic with a period of $\widetilde {a_i}$ the result follows.


The exercise can also be found in mathematica.gr