# On almost everywhere convergence of Malmquist-Takenaka series

@inproceedings{Mnatsakanyan2021OnAE, title={On almost everywhere convergence of Malmquist-Takenaka series}, author={Gevorg Mnatsakanyan}, year={2021} }

The Malmquist-Takenaka system is a perturbation of the classical trigonometric system, where powers of z are replaced by products of other Möbius transforms of the disc. The system is also inherently connected to the so-called nonlinear phase unwinding decomposition which has been in the center of some recent activity. We prove L bounds for the maximal partial sum operator of the Malmquist-Takenaka series under additional assumptions on the zeros of the Möbius transforms. We locate the problem…

## References

SHOWING 1-10 OF 23 REFERENCES

Pointwise Convergence of Fourier Series

- Mathematics
- 1980

In the early 19 century, J. Fourier was an impassioned advocate of the use of such sums, of course writing sines and cosines rather than complex exponentials. Euler, the Bernouillis, and others had…

The Polynomial Carleson operator

- MathematicsAnnals of Mathematics
- 2020

We prove affirmatively the one dimensional case of a conjecture of Stein regarding the $L^p$-boundedness of the Polynomial Carleson operator, for $1<p<\infty$.
The proof is based on two new ideas:…

OSCILLATORY INTEGRALS RELATED TO CARLESON'S THEOREM

- Mathematics
- 2001

now define the corresponding maximal operator by taking the sup over all the coefficients λ = (λα), that is with each λα ranging over R. What are the chances that such a wider result holds? There are…

A proof of boundedness of the Carleson operator

- Mathematics
- 2000

We give a simplified proof that the Carleson operator is of weak type (2, 2). This estimate is the main ingredient in the proof of Carleson’s theorem on almost everywhere convergence of Fourier…

On the convergence of Fourier series

- Mathematics
- 1968

Closely following the presentation of [Wer05], we begin by introducing the (formal) Fourier series of a given function f . Using an integral representation of the n-th partial sum sn(f, x), we are…

The (Weak-L2) Boundedness of the Quadratic Carleson Operator

- Mathematics
- 2007

We prove that the generalized Carleson operator with polynomial phase function of degree two is of weak type (2,2). For this, we introduce a new approach to the time-frequency analysis of the…

Carrier Frequencies, Holomorphy, and Unwinding

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2017

We prove that functions of intrinsic-mode type (a classical models for signals) behave essentially like holomorphic functions: adding a pure carrier frequency $e^{int}$ ensures that the…

Phase Unwinding, or Invariant Subspace Decompositions of Hardy Spaces

- MathematicsJournal of Fourier Analysis and Applications
- 2018

We consider orthogonal decompositions of invariant subspaces of Hardy spaces, these relate to the Blaschke based phase unwinding decompositions (Coifman and Steinerberger in J Fourier Anal Appl…

Maximal polynomial modulations of singular integrals

- MathematicsAdvances in Mathematics
- 2021

Let $K$ be a standard H\"older continuous Calder\'on--Zygmund kernel on $\mathbb{R}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by…

Multiscale decompositions of Hardy spaces

- Mathematics
- 2021

We would like to elaborate on a program of analysis pursued by Alex Grossmann and his collaborators on the analytic utilization of the phase of Hardy functions, as a multiscale signal processing…