  We generalize the Beckner's type Poincar\'e inequality \cite{Beckner} to a
large class of probability measures on an abstract Wiener space of the form
$\mu\star\nu$, where $\mu$ is the reference Gaussian measure and $\nu$ is a
probability measure satisfying a certain integrability condition. As the
Beckner inequality interpolates between the Poincar\'e and logarithmic Sobolev
inequalities, we utilize a family of products for functions which interpolates
between the usual point-wise multiplication and the Wick product. Our approach
is based on the positivity of a quadratic form involving Wick powers and
integration with respect to those convolution measures. Our
dimension-independent results are compared with some very recent findings in
the literature. In addition, we prove that in the finite dimensional case the
class of densities of convolutions measures satisfies a point-wise covariance
inequality.
