Lower bounds for the truncated Hilbert transform (Q268236)

The paper investigates the following problem: Given two intervals \(I,J\subset \mathbb{R}\) and a real-valued function \(f\in L^2(I)\), when can \(f\) be reconstructed from knowing its Hilbert transform \(Hf\) on \(J\)? The solution depends on the relationship between \(I\) and \(J.\) There are four cases to consider: {\parindent=6mm \begin{itemize}\item[1.] The interval \(J\) covers the support \(I\) of \(f,\) that is \(I\subset J.\) \item[2.] The interval \(J\) is a subset of \(I\), that is \(J\subset I\). This case is known as the interior problem. \item[3.] The interval \(J\) does not intersect \(I\), that is \(I \bigcap J =\emptyset\). This case is known as the truncated Hilbert transform with a gap \item[4.] The interval \(J\) overlaps with \(I\). This case is known as the truncated Hilbert transform with overlap. NEWLINENEWLINE\end{itemize}} The paper deals with Cases 3 and 4, and consequently throughout this review we assume that \(I,J\) are as described in Cases 3 and 4.NEWLINENEWLINENow let \(f\in BV(I)\) be a function of bounded variation supported on \(I.\) It was shown before that there exists a positive function \(h:[0, \infty)\rightarrow \mathbb{R}^+ \) such that NEWLINE\[NEWLINE \left\| Hf\right\|_{L^2(J)}\geq h\left( \frac{|f|_{TV}}{\left\| f\right\|_{L^2(I)}} \right) \left\| f\right\|_{L^2(I)} ,NEWLINE\]NEWLINE where \(|. |_{TV}\) denotes the total variation of \(f.\) The authors conjecture that NEWLINE\[NEWLINE h(k)\geq c_1 e^{-c_2 k} , NEWLINE\]NEWLINE where \(c_1, c_2 >0\) are constants depending only on \(I\) and \(J.\)NEWLINENEWLINEBecause for weakly differentiable functions, we have NEWLINE\[NEWLINE \left| f\right|_{TV}=\int_I \left| f_x(x) \right| dx, NEWLINE\]NEWLINE we can identify the total variation of \(f\) with \(\left\| f_x \right\|_{L^1 (I)}.\) It is conjectured that NEWLINE\[NEWLINE \left\| Hf\right\|_{L^2(J)}\geq c_1 \exp\left( -c_2 \frac{\left\| f_x \right\|_{L^1(I)}}{\left\| f\right\|_{L^2(I)}} \right) \left\| f\right\|_{L^2(I)} .NEWLINE\]NEWLINENEWLINENEWLINEHowever, the authors' first result concerning that conjecture may be stated as follows: if \(f\) is weakly differentiable and \(f\in H^1(I),\) then NEWLINE\[NEWLINE \left\| Hf\right\|_{L^2(J)}\geq c_1 \exp\left( -c_2 \frac{\left\| f_x \right\|_{L^2(I)}}{\left\| f\right\|_{L^2(I)}} \right) \left\| f\right\|_{L^2(I)} ,NEWLINE\]NEWLINE where \(H^1\) is the Sobolev space \(W^{1,2}.\)NEWLINENEWLINEOther results of similar types are derived, such as NEWLINE\[NEWLINE \left\| Hf\right\|_{L^2(J)}\geq c_1 \exp\left( -c_2 \frac{| f|^2_{TV}}{\left\| f\right\|^2_{L^2(I)}} \right) \left\| f\right\|_{L^2(I)} ,NEWLINE\]NEWLINE where \(f\in W^{1,1}.\)NEWLINENEWLINEThe authors also discuss the following more general problem. Let \(I,J\subset \mathbb{R}\) be disjoint intervals and let \(T:L^2 (I)\rightarrow L^2(J)\) be an integral operator of convolution type, NEWLINE\[NEWLINE \left(Tf\right) (x)=\int_I K(x-y)f(y)dy ,NEWLINE\]NEWLINE for some kernel \(K,\) under what conditions does one expect an inequality of the type NEWLINE\[NEWLINE \left\|Tf\right\|_{L^2(J)}\geq h\left( \frac{| f|_{TV}}{\left\| f\right\|_{L^2(I)}}\right)\left\| f\right\|_{L^2(I)}NEWLINE\]NEWLINE to hold true for some positive function \(h:[0, \infty)\rightarrow \mathbb{R}^+ \) ?