Cesàro summability of Taylor series in higher order weighted Dirichlet type spaces

For a positive integer $m$ and a finite non-negative Borel measure $\mu$ on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces ${\mathcal{H}}_{\mu ,m}$. We show that if $\alpha >\frac{1}{2},$ then for any $f$ in ${\mathcal{H}}_{\mu ,m},$ the sequence of generalized Ces{à}ro sums $\{{\sigma }_n^{\alpha }[f]\}$ converges to $f$. We further show that if $\alpha =\frac{1}{2}$ then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer $m$.