  We introduce three one-parameter semigroups of operators and determine their
spectra. Two of them are fractional integrals associated with the Askey-Wilson
operator. We also study these families as families of positive linear
approximation operators. Applications include connection relations and bilinear
formulas for the Askey-Wilson polynomials. We also introduce a
q-Gauss-Weierstrass transform and prove a representation and inversion theorem
for it.
