 
Summary: QUANTIZATION DIMENSION FUNCTIONS AND GIBBS MEASURES
MRINAL KANTI ROYCHOWDHURY
DEPT OF MATHEMATICS
THE UNIVERSITY OF TEXASPAN AMERICAN
Abstract: The term "quantization" in the title originates in the theory or signal pro
cessing. It was used by electrical engineers starting in the late 40's. Mathematicians started
working in this area almost in the last two decades. As a mathematical topic quantization
for probability distributions concerns the best approximation of a ddimensional probability
distribution P by a discrete probability with a given number of nsupporting points or in
other words, the best approximation of a ddimensional random vector X with distribution
P by a random vector Y with at most n values in its image. A given random vector X
can be approximated by a discrete random vector Y in infinitely many different ways. The
random vector Y which gives the error minimum is called the optimal quantizer of the ran
dom vector X and the corresponding error is called the optimal error. The image set of the
optimal quantizer is called the optimal set.
Goals of Quantization Theory:
Two main goals are as follows:
(1) Finding the exact configuration of the so called noptimal sets,
(2) Estimating the rate called Quantization dimension at which the specified measure
of the error goes to zero as n increases.
