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Choose $\varphi (x)=\int d^{4}p(2\pi {)}^{-4}{e}^{ip\cdot x}\tilde{\varphi }(p)$ in Euclidean signature. The inverse quadratic kernel is the smooth-cutoff scalar propagatorThe subscript records the explicit cutoff dependence implied by the condition $f_{\Lambda }(z)={\Lambda }^{2}z$ at small $z$. For $p^2+m^2\le {\Lambda }^2$, this reduces to $1/(p^2+m^2)$, the usual Euclidean scalar propagator. For $p^2+m^2\gg {\Lambda }^2$, the kernel $f_{\Lambda }$ grows rapidly and its inverse tends to zero.

High-momentum modes are strongly suppressed. For a finite smooth regulator they are not literally identically zero; that statement would require a sharp cutoff. The field variance carried by those Fourier transform modes is correspondingly negligible.

A smooth-cutoff scalar propagator is the inverse of a positive regulated quadratic kernel. It agrees with the unregulated scalar propagator at low momentum and decreases rapidly at high momentum. Smooth suppression is not identical to vanishing support. Its cutoff derivative is the line weight in an exact renormalization-group flow.