Abstract. We consider the optimal quantization problem with Rényi-$\alpha $-entropy constraints for centered Gaussian measures on a separable Banach space. For $\alpha = \infty $ we can compute the optimal quantization error by a moment on a ball. For $\alpha\in {} ]1,\infty ]$ and large entropy bound we derive sharp asymptotics for the optimal quantization error in terms of the small ball probability of the Gaussian measure. We apply our results to several classes of Gaussian measures. The asymptotical order of the optimal quantization error for $\alpha > 1$ is different from the well-known cases $\alpha = 0$ and $\alpha = 1$.

AMS Subject Classification (1991): 60G15, 62E17, 94A17

Keyword(s): Gaussian measures, Rényi-$\alpha $-entropy, functional quantization, high-resolution quantization

Received February 18, 2010, and in revised form December 19, 2012. (Registered under 11/2010.)