The Finite Hilbert Transform in ℒ²

Abstract

It is well known that the finite HILBERT transform T is a NOETHER (FREDHOLM) operator when considered as a map from Y p into itself if 1 < p < 2 or 2 < p c co. When p = 2, the map T is not a NOETHER operator. We present two theorems which characterize the range of T in Y 2 and, as immediate consequences, give simple expressions for its inverse. IntroductionWe consider the finite HILBERT transform T over the open arc 1-1, l[. A well known result of M. RIESZ [12] tells us that the restriction Tp of T to the BANACH space Y p := YPa-1, 10 defines a continuous linear operator from Y p into Y p for every p €11, a[. Unless p = 2, the map T, is a NOETHER operator (for the definition of NOETHER operators, see section 2). The purpose of this paper is to investigate the operator T, : Y z + Y 2 whose range is a proper dense subspace of Y 2 . This we do in sections 3 and 4, after summarizing some basic results in section 2.Our principal result which characterizes the range of T, is given in Theorem 3.2. This is then followed by a simple inversion formula for T2 which is valid for all functions in the range of T2. These results are derived essentially by considering those operators T,, 1 < p < 2, whose restrictions to Y p are 7''.In section4, on the other hand, the properties of the operators T, , 2 < p c 00, are applied to the study of T2. In Theorem 4.2, the subspace of the range of T2, which consists of all functions f such that (1 -x2)-lI2 f is LEBESGUE integrable, is characterized and the restriction of T;' to that subspace is shown to have the same form as T i ' , 2 c p < m.From Theorem 4.2, the question naturally arises as to whether or not there exist functions 4 E Y 2 for which (1 -x2)-'"

Correction to: Inversion and extension of the finite Hilbert transform on $$(-\,1, 1)$$

“…We can now extend certain results obtained in [25], [32, §4.3] for the spaces L p with 1 ă p ă 2 to the larger family of r.i. spaces satisfying 1{2 ă α X ď α X ă 1.…”

“…To establish (3.8) let f P X Ď L p , with 1 ă p ă 2 as above. Then (3.8) above follows from the validity of (3.8) in L p ; see (2.6) on p.46 of [25].…”

“…In [17,Ch.11], [25], [32, §4.3] a detailed study of the inversion of the finite Hilbert transform was undertaken for T acting on the spaces L p whenever 1 ă p ă 2 and 2 ă p ă 8. We study here the extension of those results to a larger class of spaces, namely, the r.i. spaces.…”

Interior Tomography With Continuous Singular Value Decomposition

“…When g null, m ( t ) are generated in the range of the nite Hilbert transform (FHT), the original functions in the image space can be calculated by the following inverse nite Hilbert transform (iFHT) equation: $$f_{\mathrm{null},m}\left(t\right)=-\frac{w\left(t\right)}{\pi }{\int }_{a_1}^{a_4}\frac{g_{\mathrm{null},m}\left(t^{\prime }\right)}{w\left(t^{\prime )}\right(t^{\prime }-t)}d{t}^{\prime },t\in (a_2,a_3).$$ Here, $w\left(t\right)≔\sqrt{(t-a_1)(a_4-t)}$ is a weight function. Equation (23) is a special version of the iFHT that requires no line integral constant of f null, m ( t ) [22]. As f null, m ( t ) is unknown in our problem, using (23) avoids the calculation of the integral constant.…”

“…It also guarantees that f null, m ( t ) is in the null space, and the ROI region of the DBP data is not affected by the compensation. According to the reference, insuring that the range condition is satisfied requires [22] $$\{\begin{array}{cc}{\int }_{a_1}^{a_4}\hfill & \frac{g_{\mathrm{null},m}\left(t\right)}{w\left(t\right)}dt=0\hfill \\ \frac{w\left(t\right)}{\pi }\hfill & {\int }_{a_1}^{a_4}\frac{g_{\mathrm{null},m}\left(t^{\prime }\right)}{w\left(t^{\prime }\right)(t^{\prime }-t)}d{t}^{\prime }\in L^2\hfill \end{array}\phantom{true\}}.$$ The first equation in (24) can be ful lled by adjusting b m for each g null, m ( t ). This derivation is trivial and is omitted.…”