Is there such a thing as a family of orthogonal functions defined over an interval $x\in[a,b]$, where the output is the interval $f(x)=y\in[c,d]$ (e.g. for simplicity $[0,1] \rightarrow [0,1]$).
I can use a polynomial with nth degree, defined over the 2 x-edges and midpoints within that rectangle. Then it is assured that at least these points are within the intervals. However, other points along the polynomial may still excess the y-interval.
Is there a general solution of orthogonal functions that I can add, while never exceeding the y-interval?