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We identify a class of composite membranes: fluid bilayers coupled to anelastic meshwork, that are such that the meshwork's energy is a function$F_\mathrm{el}[A_\xi]$ \textit{not} of the real microscopic membrane area $A$,but of a \textit{smoothed} membrane's area $A_\xi$, which corresponds to thearea of the membrane coarse-grained at the mesh size $\xi$. We show that themeshwork modifies the membrane tension $\sigma$ both below and above the scale$\xi$, inducing a tension-jump $\Delta\sigma=dF_\mathrm{el}/dA_\xi$. Thepredictions of our model account for the fluctuation spectrum of red bloodcells membranes coupled to their cytoskeleton. Our results indicate that thecytoskeleton might be under extensional stress, which would provide a means toregulate available membrane area. We also predict an observable tension jumpfor membranes decorated with polymer "brushes"

Fluctuation Spectrum of Fluid Membranes Coupled to an Elastic Meshwork: Jump of the Effective Surface Tension at the Mesh Size