I want to construct a function $f$ satisfying the following properties:
1) Holder continuous with exponent $\alpha \in (0,1)$ so that there exists $C > 0$ such that $$ |f(x+t) - f(x)| \le C|t|^{\alpha}, \forall x, t \in \mathrm{supp}(f) $$ The exponent should be tight.
2) $f$ has compact support (say) in $[-1,1]$, with $f(0) \ne 0$. Life is better for me if $f(0) = \left\|f \right\|_{\infty}$, but this is not a showstopper.
3) $f$ can be rapidly evaluated via arithmetic operations present in hardware, like adds/subtracts/multiplies/divides.
Is such a function possible? (Note: Daubechies wavelets come close to satisfying this criteria, but fast evaluation is currently not available. Also, the Holder exponent cannot be tuned.)