Given any finite direction set $\Omega$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_{\Omega}$ associated to this set. Our main result provides an essentially sharp uniform bound, depending only on $N$, for $L^2$ operator norm dimensions 3 and higher. The ingredients proof consist polynomial partitioning tools from incidence geometry almost-ort...

Wavelet transform is a powerful tool for analysing the problems arising in harmonic analysis, signal and image processing, sampling, filtering, so on. However, they seem to be inadequate representing those signals whose energy not well concentrated frequency domain. In pursuit of representations such signals, we propose novel time-frequency coined as quadratic-phase wave packet L2(R). The propo...

We introduce a generalization of the Fourier transform, denoted by FC , on the isotropic cone C associated to an indefinite quadratic form of signature (n1, n2) on R (n = n1 + n2: even). This transform is in some sense the unique and natural unitary operator on L2(C), as is the case with the Euclidean Fourier transform FRn on L 2(Rn). Inspired by recent developments of algebraic representation ...

In this paper, using a generalized translation operator, we prove theestimates for the generalized Fourier-Bessel transform in the space L2 on certainclasses of functions.

using a generalized spherical mean operator, we obtain the generalizationof titchmarsh's theorem for the dunkl transform for functions satisfyingthe lipschitz condition in l2(rd;wk), where wk is a weight function invariantunder the action of an associated reection groups.

As one of the major directions in applied and computational harmonic analysis, the classical theory of wavelets and framelets has been extensively investigated in the function setting, in particular, in the function space L2(R). A discrete wavelet transform is often regarded as a byproduct in wavelet analysis by decomposing and reconstructing functions in L2(R) via nested subspaces of L2(R) in ...

We establish an L2×L2 to L estimate for the bilinear Hilbert transform along a curve defined by a monomial. Our proof is closely related to multilinear oscillatory integrals.

We study the completeness properties of the set of wavelets in L2(R). It is well-known that this set is not closed in the unit ball of L2(R). However, if one considers the metric inherited as a subspace (in the Fourier transform side) of L2(R, dξ) ∩ L(R∗, dξ |ξ| ), we do obtain a complete metric space.

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...