# Existence of $f \in L^2(\Bbb R^n)$ with $f=g_1$ on $E$ and $\mathscr{F}(f)=g_2$ on $F$

Question:
Suppose $$E,F$$ subsets of $$\Bbb R^n$$ have finite measure. Show that for any $$g_1,g_2 \in L^2(\Bbb R^n)$$ there exists $$f \in L^2(\Bbb R^n)$$ with $$f=g_1$$ on $$E$$ and $$\mathscr{F}(f)=g_2$$ on $$F$$,where $$\mathscr{F}(f)$$ is the Fourier transform of $$f$$.
My attempt:
This is a problem from Schlag's book Classical and multilinear harmonic analysis, chapter 10 problem 10.3. I try to prove it by contradiction and Amrein-Berthier thm but failed.
Any solution or hint is highly appreciated.

A-B thm: Let E,F be sets of finite measure in $$\Bbb R^n$$.Then $$\left\|f\right\|_{L^2(\Bbb R^n)} \leq C( \left\|f\right\|_{L^2(E^c)}+ \left\|{F}(f)\right\|_{L^2(F^c)} )$$ for some $$C=C(E,F,n)$$

• I looked up your 'Amrein-Berthier Theorem' but found it difficult to find the statement. So if you could at least state the statement or provide some link... Dec 21, 2020 at 9:51
• Here's a hint, admittedly with a lot to fill in: if $V$ is the subspace of $f \in L^2$ where $f = 0$ on $E$ and $W$ is the subspace of $f \in L^2$ where $\mathscr{F}(f) = 0$ on $F$, and $P_V$ and $P_W$ denote the projections onto these subspaces, use the fact that the norm of $P_V P_W$ is less than $1$ to show that $V + W = L^2$. Dec 26, 2020 at 22:43
• This has been posted on MO, with the notice "The question has been posted here but had no response." Dec 29, 2020 at 5:12
• @Jason Thanks for the hint.Can you explain why the norm is less than 1 is able to ensure V+W=L2 Dec 29, 2020 at 11:01
• @mathdogcmf Ack, I meant to write that $P_{V^{\perp}} P_{W^{\perp}}$ has norm less than $1$, not $P_V P_W$, apologies. My thinking was to do a series expansion based on the fact that $I = P_{W} + P_V P_{W^{\perp}} + P_{V^{\perp}} P_{W^{\perp}}$. Dec 29, 2020 at 20:26

Let $$V$$ be the closed subspace of $$L^2(\mathbb{R}^n)$$ consisting of those functions that vanish almost everywhere on $$E$$, let $$W \subseteq L^2$$ be the closed subspace of functions whose Fourier transforms vanish almost everywhere on $$F$$, and let $$V^{\perp}$$ and $$W^{\perp}$$ be their orthogonal complements.
It suffices to show that there exists $$f \in L^2$$ with $$f = g_1$$ on $$E$$ and $$\mathscr{F}(f) = 0$$ on $$F$$. Or, put another way, we'd like to show that an arbitrary coset of $$V$$ intersects $$W$$.
Now, if we knew that $$V + W = L^2$$, the result would follow from the (purely algebraic) isomorphism $$(V + W) / V \cong W / (V \cap W)$$. To see this, note that the natural isomorphism is given by $$h + (V \cap W) \mapsto h + V$$. So given an arbitrary coset $$g + V$$ ($$g \in L^2 = V + W$$), we could find a $$\tilde{g} \in W$$ so that $$\tilde{g} + V = g + V$$. In particular, $$\tilde{g} \in g + V$$, so this function demonstrates that the coset $$g + V$$ intersects $$W$$.
To finish, we show that $$V + W = L^2$$. We rely on the fact that the $$L^2 \to L^2$$ operator norm of $$P_{V^{\perp}} P_{W^{\perp}}$$ is strictly less than $$1$$, where $$P_{\cdot}$$ represents the orthogonal projection onto the relevant subspace. (This norm statement is connected to the Amrein-Berthier theorem, and follows from Theorem 10.4 and Lemma 10.5 in Muscalu and Schlag's book.) Using the fact that $$I = P_W + P_V P_{W^{\perp}} + P_{V^{\perp}} P_{W^{\perp}}$$, we can write an arbitrary $$f \in L^2$$ as $$f = g_1 + h_1 + P_{V^{\perp}} P_{W^{\perp}} f$$, where $$g_1 \in W$$ and $$h_1 \in V$$. Iterating this expansion on the remainder term and using the norm bound, we obtain convergent series $$\sum_j g_j$$ in $$W$$ and $$\sum_j h_j$$ in $$V$$ satisfying $$f = (\sum_j g_j) + (\sum_j h_j)$$. This completes the proof.