Let $u(t) = \Sigma\_{k=1}^n c\_k e^{\lambda\_k t} (c\_k \in \mathbb C, \lambda\_k \in \mathbb C) $ be an exponential polynomial of **order** $n$. 


Define $E\_n$ to be the collection of all exponential polynomial of order $n$, i.e.,


$$ E\_n:= \{ u : u(t) = \sum\_{k=1}^n c\_k e^{\lambda\_k t}, c\_k \in \mathbb C, \lambda\_k \in \mathbb C \}. $$ 
Notice, that two elememts of $E\_n$ may have different (disjoint) set of exponents. Only requirement for $u$ to be in $E\_n$ is that it has order at most $n$.


Let $\mathbf{P}\_n$ be the collection of all polynomials of **degree** at most $n$.


Consider a function $f = \sum\_{j=1}^{M} p\_{m\_j}(t) e^{\lambda\_j t}, p\_{m\_j} \in \mathbf{P}\_{m\_j}, \sum\_{j=1}^{M} (m\_j+1) \leq n$.


My question is, can one find a sequence $u\_m \in E\_n$ such that, $$\sup\_{x\in[0,1]}| f(x)- u\_m(x) |\rightarrow 0. $$


If possible, then how should one go about constructing such a sequence ? 



Approximation by exponential polynomials
This follows from the fact that the set of $n\times n$ matrices with simple spectrum is dense in the space of all $n\times n$ matrices ${\bf M}\_n(\mathbb C)$ (or that the set of polynomials of degree $n$ with simple roots is dense in the set of all complex polynomials
of degree $n$).


The function $f$ solves the Cauchy problem for a [linear ODE](http://en.wikipedia.org/wiki/Linear_differential_equation) with *constant* complex coefficients 
$$y^{(N)}+a\_1y^{(N-1)}+\dots+a\_Ny=0,$$
$$y^{(k)}(0)=f^{(k)}(0),\quad k=0,1\dots,N-1,$$
where $N=\sum\_{j=1}^{M} (m\_j+1) \leq n$. The exponents $\lambda\_j$, $j=1,\dots,M$ are the roots of the corresponding characteristic equation 
$$P(\lambda)=\lambda^{N}+a\_1\lambda^{N-1}+\dots+a\_N\lambda=0$$
with the multiplicities, respectively, $m\_j+1$, $j=1,\dots,M$. The coefficients $a\_1,\dots a\_N$ can be found from the relation
$$P(\lambda)=\prod\limits\_{j=1}^{M}(\lambda-\lambda\_j)^{m\_j+1}.$$


Now, a small generic perturbation of the ODE's coefficients will produce an ODE
$$y^{(N)}+a\_1^{\varepsilon}y^{(N-1)}+\dots+a\_N^{\varepsilon}y=0,$$
such that the roots of the corresponding characteristic equation are all simple. This implies that a solution to the latter ODE belongs to $E\_n$. Finally, we may use a standard result that solutions to linear ODEs depend continuously on the coefficients (in the topology of uniform convergence on finite time intervals). 


