Positive Kernels and Quantization

In the paper we investigate a method of quantization based on the concept of positive definite kernel on a principal $G$-bundle with compact structural group G. For G=U(1) our approach leads to Kostant-Souriau geometric quantization as well as to coherent state method of quantization. So, the theory proposed here can be treated as a generalization of both mentioned quantizations to the case of general compact group.