study of subspace generated by $f_k(x)=f(x+k)$ with f continuous, bounded..

Let $f:ℝ→ℝ$ be continuous, bounded function such that the space $\mathrm{lin}\{f_k(x)=f(x+k)∣k ∈\mathbb{N}\}$ is finite-dimensional.

Determine an expression of f.

I started with:

Let $P_f(X) = \sum_{k=0}^n p_k X^k$, the degree is n,and we define f on [0,n] but after I do not know how to continue.

Thank you in advance.

Since your space is finite dimensional, you have

$$\sum_{i=0}^n a_i f(x+i) = 0,$$ for all $x.$ Pick some random $x_0,$ and define $b_i=f(x_0 + i).$ The numbers $b_i$ satisfy a linear finite order recurrence with constant coefficients, so it is a linear combination of powers of roots of the characteristic polynomial of the recurrence (if there are multiple roots, there are summands of the form $i r^i,$ but we will get to that). Now, since the function is bounded, all of the roots of the characteristic polynomial have to have modulus $1$ (if modulus is smaller than one, the sequence will be unbounded going backwards), and, in fact, the case of $ir^i$ can't occur either, so we have a collection of distinct roots of absolute value $1.$ Since the coefficients of characteristic polynomial are real, the roots will occur in conjugate pairs, and this shows that the only functions which arise are trigonometric polynomials.