Related papers: Inversion Theorem for Bilinear Hilbert Transform
We give direct and inverse theorems for the weighted approximation of functions with inner singularities by combinations of Bernstein polynomials.
An index transform, involving the square of Whittaker's function is introduced and investigated. The corresponding inversion formula is established. Particular cases cover index transforms of the Lebedev type with products of the modified…
We prove bounds for the truncated directional Hilbert transform in $L^p(\mathbb{R}^2)$ for any $1<p<\infty$ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not…
We prove an inequality on positive real numbers, that looks like a reverse to the well-known Hilbert inequality, and we use some unusual techniques from Fourier analysis to prove that this inequality is optimal.
We use Salem's method to prove that there is a lower bound for partial sums of series of bi-orthogonal vectors in a Hilbert space, or the dual vectors. This is applied to some lower bounds on $L^{1}$ norms for orthogonal expansions. There…
We prove a Montel theorem for Hilbert space valued functions, and a non-commutative version of this theorem, by composing with unitaries to achieve convergence.
We obtain direct and inverse approximation theorems of $2\pi$-periodic functions by Taylor--Abel--Poisson operators in the integral metric.
The paper offers versions of Hilbert's Irreducibility Theorem for the lifting of points in a cyclic subgroup of an algebraic group to a ramified cover. A version of Bertini Theorem in this context is also obtained.
We prove variation-norm estimates for the Walsh model of the truncated bilinear Hilbert transform, extending related results of Lacey, Thiele, and Demeter. The proof uses analysis on the Walsh phase plane and two new ingredients: (i) a…
In this paper, we show that Hilbert transforms along some curves are bounded on $L^p({\mathbb R}^n;X)$ for some $1<p<\infty$ and some UMD spaces $X$. In particular, we prove that the Hilbert transform along some curves are completely…
In this note, some refinements of Young inequality and its reverse for positive numbers are proved and using these inequalities some operator versions and Hilbert-Schmidt norm versions for matrices of these inequalities are obtained.
In this paper, we study the inversion formula for recovering a function from its windowed Fourier transform. We give a rigorous proof for an inversion formula which is known in engineering. We show that the integral involved in the formula…
A master formula of transformation formulas for bilinear sums of basic hypergeometric series is proposed. It is obtained from the author's previous results on a transformation formula for Milne's multivariate generalization of basic…
We prove an important property of the binomial transform: it converts multiplication by the discrete variable into a certain difference operator. We also consider the case of dividing by the discrete variable. The properties presented here…
The generalized Parseval equality for the Mellin transform is employed to prove the inversion theorem in L_2 with the respective inverse operator related to the Hartley transform on the nonnegative half-axis (the half-Hartley transform).…
Let $f$ be a function on the real line. The Fourier transform inversion theorem is proved under the assumption that $f$ is absolutely continuous such that $f$ and $f'$ are Lebesgue integrable. A function $g$ is defined by…
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…
The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show…
Discrete analogs of the index transforms, involving Bessel and the modified Bessel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being…