Let $V: L^2(0,1) \to L^2(0,1)$ be the Volterra integration operator: $V(f)(x) := \int_0^x f(t) \, dt$.
Is there a universal function $C(L,\varepsilon) < \infty$ such that the following uniform version of this question holds? For any continuous function $f \in C([0,1])$ satisfying $1 \leq f \leq L$ on the whole segment $[0,1]$, there exist real numbers $a_0, a_1,\ldots, a_M$ such that $$ (1) \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \sum_{k=0}^M \frac{|a_k|}{k!} \leq C(L,\varepsilon) $$ and $$ (2) \quad \quad \quad \quad \quad \quad \Big\| 1 - \sum_{k =0}^M a_k V^k(f) \Big\|_{L^2(0,1)} \leq \varepsilon. $$
(The $a_k/k!$ terms in (1) are natural once one notes that the operator $V^k$ has norm $\leq 1/k!$.)
[A: Yes! As @fedja explained in the comments, a compactness argument applies non-constructively to show that (1) is an automatic consequence of (2). Arguing for contradiction, suppose to the contrary that there is a sequence $f_n$ with $1 \leq f_n \leq L$ that requires an arbitrarily large sum in (1). The set $\{ Vg \mid |g| \leq L \}$ is compact in $C[0,1]$, since it consists of continuously differentiable functions of a bounded $C^1$ norm. Hence, on passing to a subsequence, we may assume that $Vf_n \to f$ uniformly to a continuous function $f \in C[0,1]$, and since all $f_n \geq 1$, this limit function $f$ certainly fulfills a relation (2) with $\varepsilon$ replaced by $\varepsilon / 2$, by the qualitative results cited below the line. If now $n \to \infty$, this is in contradiction with our assumption that all relations (2) for $f_n$ must have a divergent sum in (1). ]
Q.: What could be said about $C(L,0.5)$ as a function of $L$?
The existence of some approximation (2), not necessarily satisfying the uniform condition (1), is known (following M.S. Brodskii) as the unicellularity of the Volterra integration operator, and has (it seems) essentially three fundamentally distinct proofs. One approach (Agmon, Kalisch, Donoghue, Sakhnovic...) is via zero distribution properties of entire functions; indeed the qualitative result (2) turns out to be equivalent to the Titchmarsh convolution theorem. Another proof (Sarason) derives the existence of a relation (2) from Beurling's characterization of the closed invariant subspaces of the unilateral shift of $\ell^2$. A third proof, due to Brodskii and Livsic, is purely functional analytic and based on their spectral resolution theory for abstract non-self adjoint completely continuous operators $V$ of a Hilbert space $H$ having a single point as their spectrum. (The total linear ordering property of the closed invariant subspaces of $V$ holds more generally for all such operators that, in addition, fulfill $\ker(V) \cap \ker(V^*) = \{0\}$ and $\dim\frac{V - V^*}{2i}(H) = 1$.)
To my surprise, I find that apparently none of these three existential schemes of proofs admits an obvious effective variant that says anything about the size of the coefficients $a_k$ used in the representation (2). I would like to know more about the constant $C(L,\varepsilon)$ of this putative statement, particularly as a function of $L$ for a fixed value $\varepsilon$ (is it $O(L)$? $o(L)$?). For a start, I have to understand if there are constructive techniques of functional analysis that would give this refinement, perhaps in combination with one of the existing solutions of (2).
