Let $f$ be a function and let $U$ be a closed and convex subset of $\mathbb{R}^m$, such that $$f \in F:= \{f \in L^{\infty}| f(t) \in U \,\,\text{for a.e}\,\, t \in [0,T]\}$$ Now let $Af$ be an approximation of $f$ such that $$t \mapsto Af(t):= \sum_{i=1}^N a_i \phi(t-ih),$$ where the basis function $\phi$ are nice function that decays rapidly and smooth and $a_i$s are the coefficients that are sampled from $f$ (like a quasi-interpolation maybe, not sure of the term).
Question 1: Can I conclude that $Af \in L^{\infty}([0,T];\mathbb{R}^m)$? I think yes: because these are defined over a compact domain and the functions $\phi$ are nice.
Question 2: What would happen if, for example, $f$ is Lipschitz continuous, then can I say $Af \in L^{\infty}([0,T];\mathbb{R}^m)$? again, I think the answer is positive.
Thank you, any help is appreciated.