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If I didn't miss anything, the dominated convergence argument turns out to work well. The following shows that $\widehat{G_k}$ can be dominated by a rapidly decreasing decaying function.

Denote $\psi=\widehat{\phi}, \psi_k=\widehat{\phi_k}$, then $\psi_k(y)=2^k\psi(2^ky), \int \psi_k=0$ and $\int |\psi_k|=\int |\psi|$, hence

$$|\widehat{G_k}(-x)| =|\widehat{\phi_k}\ast G(-x)| =|\int \psi_k(y)G(y+x)dy| =|\int \psi_k(y)[G(y+x)-G(x)]dy|$$ $$=|\int_{|y|\le |x|/2} \psi_k(y)[G(y+x)-G(x)]dy|+|\int_{|y|\ge |x|/2} \psi_k(y)[G(y+x)-G(x)]dy|$$ $$\le \frac{|x|}{2}\sup_{B(x,|x|/2)}|G'|\cdot \int |\psi|+C\int_{|y|\ge |x|/2}|\psi|dy$$

The last line is a function in $x$ decreasing decaying at any rate.

In view of the above argument, the function $\phi$ and $G$ can be replaced by any Schwartz functions where in addition $\phi$ vanishes at $0$, and we will still have $\|\widehat{G_k}\|_1 \rightarrow 0$. An estimate of the rate of convergence decay follows in the specified case.

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If I didn't miss anything, the dominated convergence argument turns out to work well. The following shows that $\widehat{G_k}$ can be dominated by a rapidly decreasing function.

Denote $\psi=\widehat{\phi}, \psi_k=\widehat{\phi_k}$, then $\psi_k(y)=2^k\psi(2^ky), \int \psi_k=0$ and $\int |\psi_k|=\int |\psi|$, hence

\begin{align} |\widehat{G_k}(-x)| &=|\widehat{\phi_k}\ast =|\widehat{\phi_k}\ast G(-x)| =|\int \psi_k(y)G(y+x)dy| =|\int \psi_k(y)[G(y+x)-G(x)]dy|\ &=|\int_{|y|\le psi_k(y)[G(y+x)-G(x)]dy|=|\int_{|y|\le |x|/2} \psi_k(y)[G(y+x)-G(x)]dy|+|\int_{|y|\ge |x|/2} \psi_k(y)[G(y+x)-G(x)]dy|\ &\le psi_k(y)[G(y+x)-G(x)]dy|\le \frac{|x|}{2}\sup_{B(x,|x|/2)}|G'|\cdot \int |\psi|+C\int_{|y|\ge |x|/2}|\psi|dy \end{align}$x|/2}|\psi|dy$$The last line is a function in$x$decreasing at any rate. In view of the above argument, the function$\phi$and$G$can be replaced by any Schwartz functions where in addition$\phi$vanishes at$0$, and we will still get have$\|\widehat{G_k}\|_1 \rightarrow 0$. An estimate of the rate of convergence will follow follows in the specified case. 1 If I didn't miss anything, the dominated convergence argument turns out to work well. The following shows that$\widehat{G_k}$can be dominated by a rapidly decreasing function. Denote$\psi=\widehat{\phi}, \psi_k=\widehat{\phi_k}$, then$\psi_k(y)=2^k\psi(2^ky), \int \psi_k=0$and$\int |\psi_k|=\int |\psi|$, hence$\begin{align} |\widehat{G_k}(-x)| &=|\widehat{\phi_k}\ast G(-x)| =|\int \psi_k(y)G(y+x)dy| =|\int \psi_k(y)[G(y+x)-G(x)]dy|\ &=|\int_{|y|\le |x|/2} \psi_k(y)[G(y+x)-G(x)]dy|+|\int_{|y|\ge |x|/2} \psi_k(y)[G(y+x)-G(x)]dy|\ &\le \frac{|x|}{2}\sup_{B(x,|x|/2)}|G'|\cdot \int |\psi|+C\int_{|y|\ge |x|/2}|\psi|dy \end{align}$The last line is a function in$x$decreasing at any rate. In view of the above argument, the function$\phi$and$G$can be replaced by any Schwartz functions where in addition$\phi$vanishes at$0$, and we will still get$\|\widehat{G_k}\|_1 \rightarrow 0\$. An estimate of the rate of convergence will follow in the specified case.