This paper relates to a family of transforms introduced by A. Weil [Acta Math. 111, 143–211 (1964; Zbl 0203.03305)]. For \(n \geq 1\) these transforms are unitary representations of the metaplectic group \(Mp_{2n}(\mathbb R)\) on the Hilbert space \(L^2(\mathbb R^n)\). \(Mp_{2n}\) is the double cover of the symplectic group \(Sp_{2n}\) and its Weil representation only defines a projective representation of \(Sp_{2n}\). Furthermore, since \(Mp_{2n}\) is not a matrix group, the projective representation of \(Sp_{2n}\) is inherently ambiguous up to multiplication by \(\pm 1\) (its parametrization requires \(2\)-cocycles \(\sigma : Sp_{2n} \times Sp_{2n} \mapsto \{z \in \mathbb \mathbb C: |z| = 1\}\)).
In the early 1970s, these transforms appear to have been independently discovered by physicists M. Moshinsky and C. Quesne [J. Math. Phys. 12, 1772–1780 (1971; Zbl 0247.20051)] and J. Li and P. Hao [J. Zhejiang Univ., Sci. A 7, No. 7, 1168–1177 (2006; Zbl 1103.68894)], who applied them to quantum mechanics and optics. Subsequently engineers have applied them, with \(n = 1\) so \(Sp_2(\mathbb R) = SL(2,\mathbb R)\), to signal processing. K. Kou and R. Xu [“Windowed linear canonical transform and its applications”, Signal Process. 92, No. 1, 179–188 (2012)] defined the linear canonical transform (LCT) with parameter \( A = \left[ \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right] \in SL(2,\mathbb R) \) by \[ L_Af(u) := \begin{cases} \int_{-\infty}^{\infty} f(t)\, K_A(t,u) \, dt , \, b \neq 0 \\ \sqrt {d} \, e^{j(cd/2)u^2}\, f(du), \, b = 0. \end{cases}\tag{1} \] where the kernel of the integral transform is \[ K_A(t,u) := \frac{1}{\sqrt {j2\pi b}}\, e^{\frac{j}{2b}(at^2-2tu + du^2)} \tag{2} \] and \(j^2 = 1\). They observe that if \(A_1, A_2 \in SL(2,\mathbb R)\) then \[ L_{A_1}\, L_{A_2} = L_{A_1A_2}. \tag{3} \] For the reasons explained above (1), (2), and (3) are technically incorrect because the right sides should have the factor \(\pm 1\) to show their ambiguity. However, regardless of the choice of this factor, the following inversion formula follows directly from the inversion formula for the Fourier transform: \[ L_A^{-1}f(t) = \int_{-\infty}^{\infty} f(u) \, \overline {K_A(t,u)} \, \, du.\tag{4} \] For a window function \(g \in L^\infty(\mathbb R)\) they introduce the corresponding windowed LCT \[ V_g^{(A)}f(x,u) := L_A (f(t)\overline{g(t-x)})(u), \ \ A \in SL(2,\mathbb R).\tag{5} \] For \(g_1, g_2 \in L^2(\mathbb R) \cap L^\infty(\mathbb R)\) they derive the orthogonality equation \[ \langle V_{g_1}^{(A)}f_1,f_2\rangle_{L^2(\mathbb R^2)} \, = \, \langle f_1,f_2\rangle\, \langle g_2, g_1\rangle, \ \ A \in SL(2,\mathbb R)\tag{6} \] which implies \((V_{g_2}^{(A)})^\ast V_{g_1}^{(A)} = \,\, \langle g_2,g_1\rangle\), where \(\ast \) denotes adjoint, and give the inversion formula \[ (V_{g_1}^{(A)})^{-1}h(t) = \, \frac{1}{\langle g_1,g_2\rangle} \int_{-\infty}^{\infty} h(x,u) \, \overline {K_A(t,u) g_2(t-x)} \, du\, dx.\tag{7} \] Choosing \(g_2 = g_1\) it gives the (matched filter) estimate of \(f\) that maximizes the signal-to-noise ratio in the presence of additive Gaussian noise in \(h = V_{g_1}^{(A)}f\).
Equations (4) and (5) imply that (8) gives an inversion formula \[ f(x) = \frac{1}{\overline {g(x_0)}} \int_{\mathbb R} V_{g}^{(A)}f(x-x_0,u)\, \overline {K_A(x,u)}\, du, \tag{8} \] which requires far less computation than (7). In this paper Han and Sun study convergence properties of the integral in (8) that are important for computation.
Section 1 defines for fixed \(x_0\) \[ R_{u_1, u_2}f(x) := \int_{-u_1}^{u_2} V_{g}^{(A)}f(x-x_0,u)\, \overline {K_A(x,u)}\, du,\tag{9} \] and states their first main result, which is completely analogous to Theorem 1.1 in W. Sun’s paper [Math. Nachr. 285, No. 7, 914–921 (2012; Zbl 1250.42026)] for the windowed Fourier transform:
Theorem 1.1. If \(g\) is continuous, \(g, \widehat g \in L^1(\mathbb R)\), and \(f \in L^p(\mathbb R)\) for \(1 < p < \infty\), then \[ \lim_{u_1, u_2 \rightarrow \infty} \|R_{u_1, u_2}f - \overline {g(x_0)}\, f\|_p = 0\tag{10} \] and \[ \lim_{u \rightarrow \infty} (R_{u, u}f)(x) = \overline {g(x_0)}\, f(x), \, \text{a.e.}\tag{11} \] they define the Cesaro sum for \(h > 0\) \[ \sigma_hf(x) := \frac{1}{h}\int_0^h (R_{u,u}f)(x)\, du\tag{12} \] and state their second main result:
Theorem 1.2. If \(g\) is continuous, \(g, \widehat g \in L^1(\mathbb R)\), and \(f \in L^p(\mathbb R)\) for \(1 \leq p < \infty\), then \[ \lim_{h \rightarrow \infty} \|\sigma_h f - \overline {g(x_0)}\, f\|_p = 0\tag{13} \] and \[ \lim_{h \rightarrow \infty} (\sigma_h f)(x) = \overline {g(x_0)}\, f(x), \, \text{a.e.}.\tag{14} \] Section 2 proves Theorem 1.1 using methods similar to those used to prove Theorem 1.1 in [Sun, loc. cit.]. These include the following maximal inequality form of the Carleson-Hunt theorem [C. E. Kenig and P. A. Tomas, Stud. Math. 68, 79–83 (1980; Zbl 0442.42013); L. Carleson, Acta Math. 116, 135–157 (1966; Zbl 0144.06402); J. M. Loh et al., J. Am. Stat. Assoc. 98, No. 463, 522–532 (2003; Zbl 1039.85504)]:
For \(\nu > 0\) and \(1 < p < \infty\) the operator \[ (S_\nu f)(x) := \int_{\mathbb R} f(y) \frac{\sin \nu(x-y)}{\pi(x-y)}\, dy, \ \ f \in L^p(\mathbb R),\tag{15} \] is bounded on \(L^p(\mathbb R)\) and there exist constants \(C_p > 0\) such that \[ \|\sup_{\nu > 0} |(S_\nu f)(x)\|_p \leq C_p\|f\|_p, \ \ f \in L^p(\mathbb R). \tag{16} \] They also include the Fourier multiplier result of [J. Duoandikoetxea, Fourier analysis. Transl. from the Spanish and revised by David Cruz-Uribe. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0969.42001), Corollary 3.8]: If \(\varphi : \mathbb R \mapsto \mathbb R\) has bounded variation and \(\widehat Tf = \varphi \, \widehat f\) for \(f \in L^2(\mathbb R)\) then \(T\) can be extended to a bounded operator on \(L^p(\mathbb R)\) for all \(1 < p < \infty\) and there exist constants \(C_p > 0\) such that \[ \|Tf\|_p \leq C_p\, V_\varphi\, \|f\|_p,\tag{17} \] where \(V_\varphi\) is the total variation of \(\varphi\). In addition to these deep results, they use standard results, e.g. Fubini and Lebesgue dominated convergence theorems, and extensive computation to prove Theorem 1.1.
Section 3 proves Theorem 1.2 which the authors observe is implied by Theorem 1.1 if \(1 < p < \infty\) so they give a separate proof for \(1 \leq p < \infty\). They use [Duoandikoetxea, loc. cit., Corollaries 2.8 and 2.9] which prove that \[ D_Rf(x) := \frac{1}{R} \int_0^R S_\nu(x)\, d\nu\tag{18} \] satisfies \(\sup_{R > 0} |D_Rf(x)|\) is weak \((1,1)\) and strong \((p,p)\) for \(1 < p < \infty\). They also use Minkowski’s inequality and extensive computation. The paper is well-referenced. It derives one proposition and ten lemmas to make the long proofs more comprehensible.