Positive C1 compact one-dimensional compander

Let $a<b$. Let $f,g:[a,b]\to (0,\mathrm{\infty })$ be $C^1$ probability densities, so ${\int }_a^{b}f={\int }_a^{b}g=1$. Write $G_g$ for Cumulative coordinate of a point density on a compact interval and $\varphi =G_g^{-1}$ for the $C^2$ inverse established by A positive C1 point density has a C2 cumulative inverse. For each integer $N\ge 1$, define boundaries

${\beta }_j:=\varphi (j\mathrm{/}N)(0\le j\le N),$

and reproduction points

$r_j:=\varphi ((j-\frac{1}{2})\mathrm{/}N)(1\le j\le N)\mathrm{.}$

The cells are $[{\beta }_{j-1},{\beta }_j)$ for $j<N$ and $[{\beta }_{N-1},b]$ for $j=N$. The compander quantizer maps every point in cell $j$ to $r_j$. It is a finite measurable quantizer in the sense of Finite measurable squared-error quantizer and nearest-center rule, with source law of density $f$ on $[a,b]$ and squared-error distortion

$D_N(f,g):={\sum }_{j=1}^N{\int }_{{\beta }_{j-1}}^{{\beta }_j}f(x)(x-r_j{)}^{2}dx\mathrm{.}$

Its cells are uniform in the cumulative coordinate of $g$; the definition does not assert that they are nearest-center cells in the original $x$ coordinate. Since the source is supported on the fixed compact interval and the cells cover it, there is no overload region in this model.