This document summarizes some classical estimates in Fourier analysis and their analogues in nonlinear Fourier analysis. It discusses Carleson's theorem on convergence of Fourier series and Fourier transforms, Hausdorff-Young inequalities bounding Lp norms, and results on lacunary trigonometric series and products. Open questions are presented about extending these classical estimates to the nonlinear setting of the SU(1,1) Fourier transform and lacunary SU(1,1) trigonometric products.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russia)
TITLE: Dynamical Groups, Coherent States and Some of their Applications in Quantum Optics and Molecular Spectroscopy
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch)
TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect
The purpose of this work is to formulate and investigate a boundary integral method for the solution of the internal waves/Rayleigh-Taylor problem. This problem describes the evolution of the interface between two immiscible, inviscid, incompressible, irrotational fluids of different density in three dimensions. The motion of the interface and fluids is driven by the action of a gravity force, surface tension at the interface, elastic bending and/or a prescribed far-field pressure gradient. The interface is a generalized vortex sheet, and dipole density is interpreted as the (unnormalized) vortex sheet strength. Presence of the surface tension or elastic bending effects introduces high order derivatives into the evolution equations. This makes the considered problem stiff and the application of the standard explicit time-integration methods suffers strong time-step stability constraints.
The proposed numerical method employs a special interface parameterization that enables the use of an efficient implicit time-integration method via a small-scale decomposition. This approach allows one to capture the nonlinear growth of normal modes for the case of Rayleigh-Taylor instability with the heavier fluid on top.
Validation of the results is done by comparison of numeric solution to the analytic solution of the linearized problem for a short time. We check the energy and the interface mean height preservation. The developed model and numerical method can be efficiently applied to study the motion of internal waves for doubly periodic interfacial flows with surface tension and elastic bending stress at the interface.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russia)
TITLE: Dynamical Groups, Coherent States and Some of their Applications in Quantum Optics and Molecular Spectroscopy
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch)
TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect
The purpose of this work is to formulate and investigate a boundary integral method for the solution of the internal waves/Rayleigh-Taylor problem. This problem describes the evolution of the interface between two immiscible, inviscid, incompressible, irrotational fluids of different density in three dimensions. The motion of the interface and fluids is driven by the action of a gravity force, surface tension at the interface, elastic bending and/or a prescribed far-field pressure gradient. The interface is a generalized vortex sheet, and dipole density is interpreted as the (unnormalized) vortex sheet strength. Presence of the surface tension or elastic bending effects introduces high order derivatives into the evolution equations. This makes the considered problem stiff and the application of the standard explicit time-integration methods suffers strong time-step stability constraints.
The proposed numerical method employs a special interface parameterization that enables the use of an efficient implicit time-integration method via a small-scale decomposition. This approach allows one to capture the nonlinear growth of normal modes for the case of Rayleigh-Taylor instability with the heavier fluid on top.
Validation of the results is done by comparison of numeric solution to the analytic solution of the linearized problem for a short time. We check the energy and the interface mean height preservation. The developed model and numerical method can be efficiently applied to study the motion of internal waves for doubly periodic interfacial flows with surface tension and elastic bending stress at the interface.
Reinforcement learning: hidden theory, and new super-fast algorithms
Lecture presented at the Center for Systems and Control (CSC@USC) and Ming Hsieh Institute for Electrical Engineering,
February 21, 2018
Stochastic Approximation algorithms are used to approximate solutions to fixed point equations that involve expectations of functions with respect to possibly unknown distributions. The most famous examples today are TD- and Q-learning algorithms. The first half of this lecture will provide an overview of stochastic approximation, with a focus on optimizing the rate of convergence. A new approach to optimize the rate of convergence leads to the new Zap Q-learning algorithm. Analysis suggests that its transient behavior is a close match to a deterministic Newton-Raphson implementation, and numerical experiments confirm super fast convergence.
Based on
@article{devmey17a,
Title = {Fastest Convergence for {Q-learning}},
Author = {Devraj, Adithya M. and Meyn, Sean P.},
Journal = {NIPS 2017 and ArXiv e-prints},
Year = 2017}
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
The time scale Fibonacci sequences satisfy the Friedmann-Lema\^itre-Robertson-Walker (FLRW) dynamic equation on time scale, which are an exact solution of Einstein's field equations of general relativity for an expanding homogeneous and isotropic universe. We show that the equations of motion correspond to the one-dimensional motion of a particle of position $F(t)$ in an inverted harmonic potential. For the dynamic equations on time scale describing the Fibonacci numbers $F(t)$, we present the Lagrangian and Hamiltonian formalism. Identifying these with the equations that describe factor scales, we conclude that for a certain granulation, for both the continuous and the discrete universe, we have the same dynamics.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Body fluids_tonicity_dehydration_hypovolemia_hypervolemia.pptx
Scattering theory analogues of several classical estimates in Fourier analysis
1. Scattering theory analogues of several
classical estimates in Fourier analysis
Vjekoslav Kovaˇc (University of Zagreb)
Joint work with Jelena Rupˇci´c (University of Zagreb)
6th Croatian Mathematical Congress
Zagreb, June 15, 2016
2. Classical Fourier analysis
The trigonometric/Fourier series
a(t) =
+∞
n=−∞
Ane2πint
(at least formally)
Typically we take An = T f (u)e−2πinudu, T ≡ R/Z ≡ [0, 1)
Convergence? In which sense? Under which conditions?
3. Classical Fourier analysis
The trigonometric/Fourier series
a(t) =
+∞
n=−∞
Ane2πint
(at least formally)
Typically we take An = T f (u)e−2πinudu, T ≡ R/Z ≡ [0, 1)
Convergence? In which sense? Under which conditions?
The Fourier transform
f (ξ) :=
+∞
−∞
f (x)e−2πixξ
dx (at least formally)
Suppose that f is locally integrable and supported in [0, +∞)
d
dx a(x, ξ) = f (x)e−2πixξ
, a(0, ξ) = 0
=⇒ a(x, ξ) =
x
0 f (y)e−2πiyξdy = f 1[0,x](ξ), “a(+∞, ξ) = f (ξ)”
4. Nonlinear Fourier analysis
The SU(1,1) trigonometric product
aN(t) bN(t)
bN(t) aN(t)
=
N
n=0
An Bne2πint
Bne−2πint An
An > 0, Bn ∈ C, A2
n − |Bn|2 = 1
5. Nonlinear Fourier analysis
The SU(1,1) trigonometric product
aN(t) bN(t)
bN(t) aN(t)
=
N
n=0
An Bne2πint
Bne−2πint An
An > 0, Bn ∈ C, A2
n − |Bn|2 = 1
The SU(1,1) Fourier transform / the Dirac scattering transform
d
dx
a(x, ξ) b(x, ξ)
b(x, ξ) a(x, ξ)
=
a(x, ξ) b(x, ξ)
b(x, ξ) a(x, ξ)
0 f (x)e−2πixξ
f (x)e2πixξ 0
a(0, ξ) b(0, ξ)
b(0, ξ) a(0, ξ)
=
1 0
0 1
, “ f (ξ) =
a(+∞, ξ) b(+∞, ξ)
b(+∞, ξ) a(+∞, ξ)
”
Suppose that f is locally integrable and supported in [0, +∞)
=⇒ a(·, ξ) and b(·, ξ) exist as absolutely continuous solutions
6. Nonlinear Fourier analysis
SU(1, 1) :=
A B
B A
: A, B ∈ C, |A|2
− |B|2
= 1
aN(t) bN(t)
bN(t) aN(t)
SU(1,1)
=
N
n=0
An Bne2πint
Bne−2πint An
SU(1,1)
7. Nonlinear Fourier analysis
SU(1, 1) :=
A B
B A
: A, B ∈ C, |A|2
− |B|2
= 1
aN(t) bN(t)
bN(t) aN(t)
SU(1,1)
=
N
n=0
An Bne2πint
Bne−2πint An
SU(1,1)
su(1, 1) =
A B
B A
: A, B ∈ C, A ∈ iR
d
dx
a(x, ξ) b(x, ξ)
b(x, ξ) a(x, ξ)
SU(1,1)
=
a(x, ξ) b(x, ξ)
b(x, ξ) a(x, ξ)
SU(1,1)
0 f (x)e−2πixξ
f (x)e2πixξ 0
su(1,1)
This is not the linear Fourier transform on SU(1,1)!
8. Nonlinear Fourier analysis
In the scalar form:
∂x a(x, ξ) = f (x)e2πixξ
b(x, ξ), ∂x b(x, ξ) = f (x)e−2πixξ
a(x, ξ)
a(0, ξ) = 1, b(0, ξ) = 0
9. Nonlinear Fourier analysis
In the scalar form:
∂x a(x, ξ) = f (x)e2πixξ
b(x, ξ), ∂x b(x, ξ) = f (x)e−2πixξ
a(x, ξ)
a(0, ξ) = 1, b(0, ξ) = 0
Integral equations:
a(x, ξ) = 1 +
x
0
f (y)e2πiyξ
b(y, ξ)dy
b(x, ξ) =
x
0
f (y)e−2πiyξ
a(y, ξ)dy
10. Nonlinear Fourier analysis
In the scalar form:
∂x a(x, ξ) = f (x)e2πixξ
b(x, ξ), ∂x b(x, ξ) = f (x)e−2πixξ
a(x, ξ)
a(0, ξ) = 1, b(0, ξ) = 0
Integral equations:
a(x, ξ) = 1 +
x
0
f (y)e2πiyξ
b(y, ξ)dy
b(x, ξ) =
x
0
f (y)e−2πiyξ
a(y, ξ)dy
Picard’s iteration gives certain multilinear series expansions
Born’s approximation: b(x, ξ) ≈ f 1[0,x](ξ) when f L1
(R) 1
We care about the long term behavior and cannot linearize!
12. Motivation
Eigenproblem for the Dirac operator:
L :=
d
dx −¯f
f − d
dx
, i.e. L
ϕ
ψ
=
ϕ − ¯f ψ
f ϕ − ψ
L is skew-adjoint, so for ξ ∈ R we consider the eigenproblem:
L
ϕ(·, ξ)
ψ(·, ξ)
= −πiξ
ϕ(·, ξ)
ψ(·, ξ)
13. Motivation
Eigenproblem for the Dirac operator:
L :=
d
dx −¯f
f − d
dx
, i.e. L
ϕ
ψ
=
ϕ − ¯f ψ
f ϕ − ψ
L is skew-adjoint, so for ξ ∈ R we consider the eigenproblem:
L
ϕ(·, ξ)
ψ(·, ξ)
= −πiξ
ϕ(·, ξ)
ψ(·, ξ)
i.e. ∂x ϕ(x, ξ) + πiξϕ(x, ξ) = f (x)ψ(x, ξ)
∂x ψ(x, ξ) − πiξψ(x, ξ) = f (x)ϕ(x, ξ)
i.e. ∂x ϕ(x, ξ)eπixξ
a(x,ξ)
= f (x)e2πixξ
ψ(x, ξ)e−πixξ
b(x,ξ)
∂x ψ(x, ξ)e−πixξ
b(x,ξ)
= f (x)e−2πixξ
ϕ(x, ξ)eπixξ
a(x,ξ)
14. Carleson’s theorem
Classical/linear — Carleson (1966)
(An) ∈ 2
(Z) =⇒ lim
N→+∞
N
n=−N
Ane2πint
exists for a.e. t ∈ T
f ∈ L2
(R) =⇒ lim
R→+∞
+R
−R
f (x)e−2πixξ
dx exists for a.e. ξ ∈ R
15. Carleson’s theorem
Classical/linear — Carleson (1966)
(An) ∈ 2
(Z) =⇒ lim
N→+∞
N
n=−N
Ane2πint
exists for a.e. t ∈ T
f ∈ L2
(R) =⇒ lim
R→+∞
+R
−R
f (x)e−2πixξ
dx exists for a.e. ξ ∈ R
Nonlinear analogues — Open question
+∞
n=0
|Bn|2
< ∞
?
=⇒ lim
N→+∞
aN(t) bN(t)
bN(t) aN(t)
exists for a.e. t ∈ T
f ∈ L2
(R)
?
=⇒ lim
x→+∞
a(x, ξ) b(x, ξ)
b(x, ξ) a(x, ξ)
exists for a.e. ξ ∈ R
Even finiteness of supx∈[0,+∞) |a(x, ξ)| for a.e. ξ ∈ R is open
Muscalu, Tao, Thiele (2002): the Cantor group “toy-model”
16. Hausdorff-Young inequalities
Classical/linear — Young (1913), Hausdorff (1923)
1 ≤ p ≤ 2, 1/p + 1/p = 1 =⇒ f Lp
(R)
≤ f Lp
(R)
Babenko (1961), Beckner (1975)
1 < p < 2 =⇒ f Lp
(R)
≤ p1/2p
p
1/2p
<1
f Lp
(R)
17. Hausdorff-Young inequalities
Classical/linear — Young (1913), Hausdorff (1923)
1 ≤ p ≤ 2, 1/p + 1/p = 1 =⇒ f Lp
(R)
≤ f Lp
(R)
Babenko (1961), Beckner (1975)
1 < p < 2 =⇒ f Lp
(R)
≤ p1/2p
p
1/2p
<1
f Lp
(R)
Nonlinear analogues — Christ and Kiselev (2001)
1 ≤ p ≤ 2 =⇒ (log |a(+∞, ·)|2
)1/2
Lp
(R)
≤ Cp f Lp
(R)
p = 1 trivial by Gronwall’s lemma
p = 2 an identity (with C2 = 1) by the contour integration
Open question: Does Cp stay bounded as p ↑ 2?
K. (2010): confirmed in the Cantor group “toy-model”
18. Lacunary trigonometric series
1 ≤ m1 < m2 < m3 < · · · , q > 1, mj+1 ≥ qmj
Norm convergence — Zygmund (1920s)
N
j=1
Aj e2πimj t
Lp
t (T)
∼p,q
N
j=1
|Aj |2
1/2
, 0 < p < ∞
∞
j=1
Aj e2πimj t
converges in Lp
⇐⇒ (Aj ) ∈ 2
(N)
19. Lacunary trigonometric series
1 ≤ m1 < m2 < m3 < · · · , q > 1, mj+1 ≥ qmj
Norm convergence — Zygmund (1920s)
N
j=1
Aj e2πimj t
Lp
t (T)
∼p,q
N
j=1
|Aj |2
1/2
, 0 < p < ∞
∞
j=1
Aj e2πimj t
converges in Lp
⇐⇒ (Aj ) ∈ 2
(N)
Convergence a.e. — Kolmogorov (1924)
(Aj ) ∈ 2
(N) =⇒
∞
j=1
Aj e2πimj t
converges for a.e. t ∈ T
Converse of convergence a.e. — Zygmund (1930)
∞
j=1
Aj e2πimj t
conv. on a set of measure > 0 =⇒ (Aj ) ∈ 2
(N)
20. Lacunary SU(1,1) trigonometric products
ρ: SU(1, 1) × SU(1, 1) → [0, +∞)
ρ(G1, G2) := log 1 + G−1
1 G2 − I op
ρ is a complete metric on SU(1, 1)
21. Lacunary SU(1,1) trigonometric products
ρ: SU(1, 1) × SU(1, 1) → [0, +∞)
ρ(G1, G2) := log 1 + G−1
1 G2 − I op
ρ is a complete metric on SU(1, 1)
dp(g1, g2) :=
ρ(g1(t), g2(t)) Lp
t (T) for 1 ≤ p < ∞
ρ(g1(t), g2(t)) p
Lp
t (T)
for 0 < p < 1
Lp
(T, SU(1, 1)) := g : T → SU(1, 1) : dp(I, g) < +∞
dp is a complete metric on Lp
(T, SU(1, 1))
22. Lacunary SU(1,1) trigonometric products
1 ≤ m1 < m2 < m3 < · · · , mj+1 ≥ qmj
aN(t) bN(t)
bN(t) aN(t)
=
N
n=0
Aj Bj e2πimj t
Bj e−2πimj t Aj
23. Lacunary SU(1,1) trigonometric products
1 ≤ m1 < m2 < m3 < · · · , mj+1 ≥ qmj
aN(t) bN(t)
bN(t) aN(t)
=
N
n=0
Aj Bj e2πimj t
Bj e−2πimj t Aj
K. and Rupˇci´c (2016)
Assume q ≥ 2 and take 0 < p < ∞
lim
N→+∞
aN bN
bN aN
exists in dp ⇐⇒
∞
j=1
|Bj |2
< +∞
Recall ∞
j=1 |Bj |2 < +∞ ⇐⇒ ∞
j=1 log Aj < +∞
⇐⇒ ∞
j=1(A2
j + |Bj |2) < +∞
24. Lacunary SU(1,1) trigonometric products
K. and Rupˇci´c (2016)
Assume q ≥ 2
∞
j=1
|Bj |2
< +∞ =⇒ lim
N→+∞
aN(t) bN(t)
bN(t) aN(t)
exists for a.e. t ∈ T
25. Lacunary SU(1,1) trigonometric products
K. and Rupˇci´c (2016)
Assume q ≥ 2
∞
j=1
|Bj |2
< +∞ =⇒ lim
N→+∞
aN(t) bN(t)
bN(t) aN(t)
exists for a.e. t ∈ T
K. and Rupˇci´c (2016)
Assume q ≥ 3
lim
N→+∞
aN(t) bN(t)
bN(t) aN(t)
exists on a set of measure > 0
=⇒
∞
j=1
|Bj |2
< +∞
26. References
Introductory literature:
Tao, Thiele, Nonlinear Fourier Analysis, IAS/Park City
Graduate Summer School, unpublished lecture notes, 2003,
available at arXiv:1201.5129 [math.CA]
Tao, An introduction to the nonlinear Fourier transform,
unpublished note, 2002
Muscalu, Tao, Thiele, several papers, 2001–2007
Ablowitz, Kaup, Newell, Segur, The inverse scattering
transform — Fourier analysis for nonlinear problems,
Stud. Appl. Math. 53 (1974), 249–315
27. References
Introductory literature:
Tao, Thiele, Nonlinear Fourier Analysis, IAS/Park City
Graduate Summer School, unpublished lecture notes, 2003,
available at arXiv:1201.5129 [math.CA]
Tao, An introduction to the nonlinear Fourier transform,
unpublished note, 2002
Muscalu, Tao, Thiele, several papers, 2001–2007
Ablowitz, Kaup, Newell, Segur, The inverse scattering
transform — Fourier analysis for nonlinear problems,
Stud. Appl. Math. 53 (1974), 249–315
Thank you for your attention!