  Communication of quantized information is frequently followed by a
computation. We consider situations of \emph{distributed functional scalar
quantization}: distributed scalar quantization of (possibly correlated) sources
followed by centralized computation of a function. Under smoothness conditions
on the sources and function, companding scalar quantizer designs are developed
to minimize mean-squared error (MSE) of the computed function as the quantizer
resolution is allowed to grow. Striking improvements over quantizers designed
without consideration of the function are possible and are larger in the
entropy-constrained setting than in the fixed-rate setting. As extensions to
the basic analysis, we characterize a large class of functions for which
regular quantization suffices, consider certain functions for which asymptotic
optimality is achieved without arbitrarily fine quantization, and allow limited
collaboration between source encoders. In the entropy-constrained setting, a
single bit per sample communicated between encoders can have an
arbitrarily-large effect on functional distortion. In contrast, such
communication has very little effect in the fixed-rate setting.
