Integral quantizations with two basic examples

• Original approach to quantization based on (positive) operator-valued measures.
• Includes Berezin–Klauder–Toeplitz and Weyl–Wigner quantizations.
• Infinitely many such quantizations produce canonical commutation rule.
• Set of objects to be quantized is enlarged in order to include singular functions or distributions.
• Are given illuminating examples like quantum angle and affine or wavelet quantization.

The paper concerns integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also insist on the inherent probabilistic aspects of this classical–quantum map. The approach includes and generalizes coherent state quantization. Two applications based on group representation are carried out. The first one concerns the Weyl–Heisenberg group and the euclidean plane viewed as the corresponding phase space. We show that a world of quantizations exist, which yield the canonical commutation rule and the usual quantum spectrum of the harmonic oscillator. The second one concerns the affine group of the real line and gives rise to an interesting regularization of the dilation origin in the half-plane viewed as the corresponding phase space.