How can I express a family of polynomials fulfilling these constraints?

Question

Can I express somehow the family of polynomials $ p\in P \subset \mathbb P$ fulfilling

$$p(t) = \sum_{k=0}^N c_k t^k$$

1. $p'(0)=1$
2. $p(0)=0$
3. Monotone and smooth function $t\to f(t) \text{ for }\ t\in [0,255]$.
4. $p(255) = v_{max}$

?

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Own work: I am aware that we can create linear equation systems and conditions 1 and 2 boil down to : $$\cases{c_0 = 0\\c_1=1}$$ Condition $4$ will give us a linear equation added to the two above. But then condition $3$ are not so easy to express in the same way. Any ideas?

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For context: The application is for design of non linear remapping functions for run length coders. In other words we want to represent run lengths using $8$ bits but we have potentially a lot longer run lengths present.

In practice values anywhere in $[0,v_{max}]$

Answer 1

Smoothness is automatic for any polynomial, but as far as I know, there is no simple way to write down the monotonicity constraint as a constraint on the coefficients.

In practice, you might be able to work with a convenient subspace of all monotonic polynomials? For example, polynomials of the form

$$p(t) = \int_0^t q(s)^2 \,ds$$ $$q(t) = 1 + \sum_{k=1}^{\sqrt{N-1}} c_k t^k.$$