We review the progress achieved in optical information processing during the last decade by applying fractional linear integral transforms. The fractional Fourier transform and its applications for phase retrieval, beam characterization, space-variant pattern recognition, adaptive filter design, encryption, watermarking, and so forth is discussed in detail. A general algorithm for the fractiona...

In this paper, the $$L^2$$ boundedness of Hilbert transform along variable flat curve $$(t,P(x_1)\gamma (t))$$ $$\begin{aligned} H_{P,\gamma }f(x_1,x_2):=\mathrm {p.\,v.}\int _{-\infty }^{\infty }f(x_1-t,x_2-P(x_1)\gamma (t))\,\frac{\text {d}t}{t},\quad \forall \, (x_1,x_2)\in {\mathbb {R}}^2, \end{aligned}$$ is studied, where P a real polynomial on $${\mathbb {R}}$$ . A new sufficient conditio...

An exact analytical image reconstruction method is presented for two-dimensional imaging. The method performs backprojection, the derivative and finite Hilbert transforms. This method can be applied to many imaging geometries. The backprojection procedure is imaging-geometry dependent, while the differentiation and the finite Hilbert transform procedures are identical for all imaging geometries...

It is shown that product BMO of S.-Y.A.Chang and R. Fefferman, defined on the space R1 ⊗ · · ·⊗Rdt , can be characterized by the multiparameter commutators of Riesz transforms. This extends a classical one-parameter result of R. Coifman, R. Rochberg, and G. Weiss [8], and at the same time extends the work of M. Lacey and S. Ferguson [12] and M. Lacey and E. Terwilleger [19], on multiparameter c...

This paper studies stable sheaves on abelian surfaces of Picard number one. Our main tools are semi-homogeneous sheaves and Fourier-Mukai transforms. We introduce the notion of semi-homogeneous presentation and investigate the behavior of stable sheaves under Fourier-Mukai transforms. As a consequence, an affirmative proof is given to the conjecture proposed by Mukai in the 1980s. This paper al...

We study invertibility and compactness of positive Toeplitz operators associated to a continuous Parseval frame on Hilbert space. As applications, we characterize affine Weyl-Heisenberg localization as well give uncertainty principles for the transforms.

Abstract We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and the...