We consider the Jacobi differential operator \(\begin{aligned} \textrm{L}_0(f)(x)=(1-x^2)f''(x)-\bigl [\alpha -\beta +(\alpha +\beta +2)x\bigr ]f'(x),\qquad \alpha ,\beta > -1, \end{aligned}\) and present two main contributions: Operational construction. We develop an operational approach for generating Jacobi polynomials. These polynomials are orthogonal over the interval \([-1,1]\) with respect to the weight function \((1-x)^{\alpha }(1+x)^{\beta }\) . This method provides a convenient spectral reformulation from which several classical properties of Jacobi polynomials can be recovered.
Beckner-type inequalities. Combining the spectral semigroup representation with the best polynomial approximation estimates in weighted Sobolev norms \(\mathcal {W}_2^r(\mu _0)\) , we prove a family of Beckner-type functional inequalities for the Jacobi measure that interpolate between Poincaré-type inequalities and higher-order spectral approximation bounds.
These results connect an operational viewpoint on Jacobi polynomials with spectral inequalities and approximation-theoretic estimates on \([-1,1]\) .
