Fourier multipliers of approximate identities

Put $M={\sup }_{\epsilon }\mathrm{\parallel }{K}_{\epsilon }{\mathrm{\parallel }}_1$. Unit mass gives ${\widehat{K}}_{\epsilon }(\xi )-1=\int K_{\epsilon }(y)(e^{-2\pi iy\cdot \xi }-1)dy$. For fixed $\xi$, its modulus is bounded by $M{\sup }_{\mathrm{\mid }y\mathrm{\mid }<\delta }\mathrm{\mid }{e}^{-2\pi iy\cdot \xi }-1\mathrm{\mid }+2{\int }_{\mathrm{\mid }y\mathrm{\mid }\ge \delta }\mathrm{\mid }{K}_{\epsilon }(y)\mathrm{\mid }dy$. The latter tail tends to zero (bound it by the defining tail outside $\delta \mathrm{/}2$), and the first term tends to zero with $\delta$. This proves the pointwise multiplier limit.

F1 applies to every stated finite p and gives norm convergence, including p=1. By F2 and F3, $\mathrm{\parallel }\widehat{f}{\widehat{K}}_{\epsilon }-\widehat{f}{\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\parallel }\mathcal{F}(f\ast K_{\epsilon }-f){\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }f\ast K_{\epsilon }-f{\mathrm{\parallel }}_1\to 0$. Thus uniform transform convergence follows from norm approximation, not merely the pointwise multiplier limit. Countable choice is inherited from F1 and F2.