There is an intimate connection between analytic functions and the completeness of sets of complex exponentials $\{e^{i\lambda_n t}\}$. If, for example, the set $\{e^{i\lambda_n t}\}$ fails to be complete in $C[a,b ]$, then by virtue of the Riesz representation theorem, there is a function $\omega(t)$ of bounded variation on the interval $[a, b]$ that is not essentially a constant and for which $$\int_a^b e^{i\lambda_n t} d\omega(t)=0 \ \ \ (n=1,2,3,...)$$ If we let $f(z)$ be the Fourier-Stieltjes transform of $\omega(t)$, i.e., if $$f(z)=\int_a^b e^{iz t} d\omega(t)$$ then $f(z)$ is an entire function, not identically zero, and $f(z)$ vanishes at every $\lambda_n$.
Question. For the same reasons, and for the same Riesz representation theorem, can we conclude that $$f(z)=\int_a^b \operatorname{sinc}(z-t) d\omega(t)$$ is an entire function, not identically zero, that vanishes at every $\lambda_n$?