We prove L estimates on the Hilbert transform along a measurable, non-vanishing, one-variable vector field in R. Aside from an L estimate following from a simple trick with Carleson’s theorem, these estimates were unknown previously. This paper is closely related to a recent paper of the first author ([2]).

The Hilbert transform is essentially the only singular operator in one dimension. This undoubtedly makes it one of the the most important linear operators in harmonic analysis. The Hilbert transform has had a profound bearing on several theoretical and physical problems across a wide range of disciplines; this includes problems in Fourier convergence, complex analysis, potential theory, modulat...

We construct a new method for approximating Hilbert transforms and their inverse throughout the complex plane. Both problems can be formulated as Riemann–Hilbert problems via Plemelj’s lemma. Using this framework, we rederive existing approaches for computing Hilbert transforms over the real line and unit interval, with the added benefit that we can compute the Hilbert transform in the complex ...

For almost two decades one of the most successful methods of demodulating fringe patterns was based on the Fourier transform implementation of the 1-D Hilbert transform /1/ . Recently, however, a method of fringe analysis based on an isotropic 2-D Hilbert transform has been proposed /2/ . An isotropic Hilbert transform had been previously considered topologically impossible by many researchers....

Hilbert transform of wavelets has been used to approximate functions in L2(R) . It is proved that Hilbert transform of wavelets with many vanishing moments does a good job in approximating smooth functions in L2(R) . We also prove that Hölder continuity of a function helps in the decay of wavelet coefficients and thereby helps in approximating it. Finally, we give a result that relates the Hilb...

This paper presents the application of Hilbert transform and artificial neural network (ANN) for power quality (PQ) disturbance classification. The input features of the ANN are extracted from the envelope of the disturbance signals by applying Hilbert transform (HT). The features obtained from the Hilbert transform are distinct, understandable and immune to noise. These features after normaliz...

We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power −2/3 (modulo some logarithmic factors). These results yield all known L bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint e...

Abstract This paper presents a general expression for a number-theoretic Hilbert transform (NHT). The transformations preserve the circulant nature of the discrete Hilbert transform (DHT) matrix together with alternating values in each row being zero and non-zero. Specific examples for 4point, 6-point, and 8-point NHT are provided. The NHT transformation can be used as a primitive to create cry...

This paper presents 10-point and 12-point versions of the recently introduced number theoretic Hilbert (NHT) transforms. Such transforms have applications in signal processing and scrambling. Polymorphic solutions with respect to different moduli for each of the two cases have been found. The multiplicity of solutions for the same moduli increases their applicability to cryptography.