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 $\underline{order}$ $n$.

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 $\underline{degree}$ 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 ?

orderto clarify, but that word is extremely ambiguous anyway.... – Harry Gindi Jun 27 '10 at 15:56