@unpublished{Kreitmeier2009, author = {Kreitmeier, Wolfgang}, title = {Error bounds for high-resolution quantization with R{\´e}nyi - \&\#945; - entropy constraints}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-16647}, year = {2009}, abstract = {We consider the problem of optimal quantization with norm exponent r > 0 for Borel probabilities on Rd under constrained R{\´e}nyi-\&\#945;-entropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the special cases \&\#945; = 0 (memory-constrained quantization) and \&\#945; = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter \&\#945; \&\#8712; [1+r/d, \&\#8734;]. For \&\#945; \&\#8712; [0,1+r/d[ we specify the asymptotical order of the optimal quantization error under large entropy bound. The optimal quantization error decays exponentially fast with the entropy bound and the exact decay rate is determined for all \&\#945; \&\#8712; [0, \&\#8734;].}, subject = {Maßtheorie}, language = {en} }