Videos and presentation materials from other INI events are also available.
Event  When  Speaker  Title  Presentation Material 

ASC 
10th January 2019 15:00 to 16:00 
Peter Binev  Greedy algorithms in reduced modeling  
ASC 
18th January 2019 11:00 to 12:00 
Andreas Seeger 
A maximal function for families of Hilbert transforms along homogeneous curves
Let H(u) be the Hilbert transform along the parabola (t; ut2) where u 2 R. For a set U of positive numbers consider the maximal function HUf = supfH(u)f : u 2 Ug. We obtain (essentially) optimal results for the Lp operator norm of HU when 2 < p < 1. The results are proved for families of Hilbert transforms along more general non at homogeneous curves. Joint work with Shaoming Guo, Joris Roos and PoLam Yung. 

ASC 
24th January 2019 15:00 to 16:00 
Elijah Liflyand 
The Fourier transform of a function of bounded variation: symmetry and asymmetry
New relations between the Fourier transform of a function of
bounded variation and the Hilbert transform of its
derivative are revealed. The main result is an asymptotic
formula for the cosine Fourier transform.
Such relations have previously been known only for the sine
Fourier transform. To prove the mentioned result,
not only a different space is considered but also a new way
of proving such theorems is applied.
Interrelations of various function spaces are studied in
this context. The obtained results are used for proving
new estimates for the Fourier transform of a radial function
and completely new results on the integrability of trigonometric series.


ASC 
29th January 2019 15:00 to 16:00 
Mikhail Tyaglov 
Hurwitz stable and selfinterlacing orthogonal polynomials.
In this survey talk,
we consider a few examples of Hurwitz stable and their dual (the socalled)
selfinterlacing polynomials and discuss a number of interrelations between
theory of orthogonal polynomials, moment theory, and the root distributions of
polynomials. Some specific relations between the Jacobi polynomials and
certain Hurwitz stable polynomials will be presented and an answer to a
question of Geno Nikolov on interlacing properties of the Jacobi polynomials
will be provided. We will also discuss the asymptotic behaviour of
sequences of Hurwitz stable polynomials. 

ASC 
11th February 2019 15:00 to 16:30 
Erich Novak 
Lecture 1: Some old and new results on InformationBased Complexity We give a short introduction to IBC and present some basic definitionsand a few results. The general question is: How many function values (or values of other functionals) of $f$ do we need to compute $S(f)$ up to an error $\epsilon$? Here $S(f)$ could be the integral or the maximum of $f$. 

ASC 
12th February 2019 15:00 to 16:30 
Ben Adcock 
Lecture 1: Overview and Theory
Lecture 1: Overview and Theory
In these lectures I will present an introduction to
compressed sensing and sparse approximation. The first lecture gives an
overview of compressed sensing and its standard theory. Next, I will
focus on two major areas of application. The second lecture considers
image reconstruction, and its application to medical and scientific imaging.
The third lecture considers highdimensional approximation via compressed
sensing, with application to parametric PDEs in Uncertainty Quantification.


ASC 
13th February 2019 15:00 to 16:30 
Erich Novak 
Lecture 2: Complexity results for integration.
We give a short introduction to IBC and present some basic
definitions and a few results. The general question is: How many function values (or values of other functionals) of $f$ do we need to compute $S(f)$ up to an error $\epsilon$? Here $S(f)$ could be the integral or the maximum of $f$. In particular we study the question: Which problems are tractable? When do we have the curse of dimension? In this second talk we discuss complexity results for numerical integration. In particular we present results for the star discrepancy, the curse of dimension for $C^k$ functions, and results for randomized algorithms 

ASC 
14th February 2019 15:00 to 16:30 
Ben Adcock 
Lecture 2: Compressive Imaging
In these lectures I will present an introduction to
compressed sensing and sparse approximation. The first lecture gives an
overview of compressed sensing and its standard theory. Next, I will
focus on two major areas of application. The second lecture considers
image reconstruction, and its application to medical and scientific imaging.
The third lecture considers highdimensional approximation via compressed
sensing, with application to parametric PDEs in Uncertainty Quantification.


ASC 
15th February 2019 15:00 to 16:30 
Ben Adcock 
Lecture 3: HighDimensional Polynomial Approximation
In these lectures I will present an introduction to
compressed sensing and sparse approximation. The first lecture gives an
overview of compressed sensing and its standard theory. Next, I will
focus on two major areas of application. The second lecture considers
image reconstruction, and its application to medical and scientific imaging.
The third lecture considers highdimensional approximation via compressed
sensing, with application to parametric PDEs in Uncertainty Quantification.


ASCW01 
18th February 2019 09:40 to 10:15 
Henryk Wozniakowski  Exponential tractability of weighted tensor product problems  
ASCW01 
18th February 2019 11:00 to 11:35 
Aicke Hinrichs 
Random sections of ellipsoids and the power of random information
We study the circumradius of the intersection of an $m$dimensional ellipsoid~$\mathcal E$ with half axes $\sigma_1\geq\dots\geq \sigma_m$ with random subspaces of codimension $n$. We find that, under certain assumptions on $\sigma$, this random radius $\mathcal{R}_n=\mathcal{R}_n(\sigma)$ is of the same order as the minimal such radius $\sigma_{n+1}$ with high probability. In other situations $\mathcal{R}_n$ is close to the maximum~$\sigma_1$. The random variable $\mathcal{R}_n$ naturally corresponds to the worstcase error of the best algorithm based on random information for $L_2$approximation of functions from a compactly embedded Hilbert space $H$ with unit ball $\mathcal E$. In particular, $\sigma_k$ is the $k$th largest singular value of the embedding $H\hookrightarrow L_2$. In this formulation, one can also consider the case $m=\infty$, and we prove that random information behaves very differently depending on whether $\sigma \in \ell_2$ or not. For $\sigma \notin \ell_2$ random information is completely useless. For $\sigma \in \ell_2$ the expected radius of random information tends to zero at least at rate $o(1/\sqrt{n})$ as $n\to\infty$. In the proofs we use a comparison result for Gaussian processes a la Gordon, exponential estimates for sums of chisquared random variables, and estimates for the extreme singular values of (structured) Gaussian random matrices. This is joint work with David Krieg, Erich Novak, Joscha Prochno and Mario Ullrich. 

ASCW01 
18th February 2019 11:40 to 12:15 
Yuri Malykhin  On some lower bounds for Kolmogorov widths  
ASCW01 
18th February 2019 13:40 to 14:15 
Jan Vybiral 
Approximation of Ridge Functions and Sparse Additive Models
The approximation of smooth multivariate functions is known to suffer the curse of dimension. We discuss approximation of structured multivariate functions, which take the form of a ridge, their sum, or of the socalled sparse additive models. We give also results about optimality of such algorithms.


ASCW01 
18th February 2019 14:20 to 14:55 
Alexander Litvak 
Order statistics and MallatZeitouni problem
Let $X$ be an $n$dimensional random centered Gaussian vector with independent but not necessarily identically distributed coordinates and let $T$ be an orthogonal transformation of $\mathbb{R}^n$. We show that the random vector $Y=T(X)$ satisfies $$\mathbb{E} \sum \limits_{j=1}^k j\mbox{}\min _{i\leq n}{X_{i}}^2 \leq C \mathbb{E} \sum\limits_{j=1}^k j\mbox{}\min _{i\leq n}{Y_{i}}^2$$ for all $k\leq n$, where ``$j\mbox{}\min$'' denotes the $j$th smallest component of the corresponding vector and $C>0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni regarding optimality of the KarhunenLo\`eve basis for the nonlinear reconstruction. We also show some relations for order statistics of random vectors (not only Gaussian), which are of independent interest. This is a joint work with Konstantin Tikhomirov. 

ASCW01 
18th February 2019 15:30 to 16:05 
Mario Ullrich 
Construction of highdimensional point sets with small dispersion
Based on deep results from coding theory, we present an deterministic algorithm that contructs a point set with dispersion at most $\eps$ in dimension $d$ of size $poly(1/\eps)*\log(d)$, which is optimal with respect to the dependence on $d$. The running time of the algorithms is, although superexponential in $1/\eps$, only polynomial in $d$.


ASCW01 
18th February 2019 16:10 to 16:45 
Dicussion  
ASCW01 
19th February 2019 09:00 to 09:35 
Winfried Sickel 
The Haar System and Smoothness Spaces built on Morrey Spaces
For some Nikol'skijBesov spaces $B^s_{p,q}$ the orthonormal Haar system can be used as an unconditional Schauder basis. Nowadays necessary and sufficient conditions with respect to $p,q$ and $s$ are known for this property. In recent years in a number of papers some modifications of Nikol'skijBesov spaces based on Morrey spaces have been investigated. In my talk I will concentrate on a version called Besovtype spaces and denoted by $B^{s,\tau}_{p,q}$. It will be my aim to discuss some necessary and some sufficient conditions on the parameters $p,q,s,\tau$ such that one can characterize these classes by means of the Haar system. This is joined work with Dachun Yang and Wen Yuan (Beijing Normal University).


ASCW01 
19th February 2019 09:40 to 10:15 
Dachun Yang 
Ball Average Characterizations of Function Spaces It is well known that function spaces play an important role in the study on various problems from analysis. In this talk, we present pointwise and ball average characterizations of function spaces including Sobolev spaces, Besov spaces and TriebelLizorkin spaces on the Euclidean spaces. These characterizations have the advantages so that they can be used as the definitions of these function spaces on metric measure spaces. Some open questions are also presented in this talk. 

ASCW01 
19th February 2019 11:00 to 11:35 
Wen Yuan 
Embedding and continuity envelopes of Besovtype spaces In this talk, we discuss about the sharp embedding properties between Besovtype spaces and TriebelLizorkintype and present some related necessary and sufficient conditions for these embedding. The corresponding continuity envelopes are also worked out. 

ASCW01 
19th February 2019 11:40 to 12:15 
Bin Han 
Directional Framelets with Low Redundancy and Directional Quasitight Framelets Edge singularities are ubiquitous and hold key information for many highdimensional problems. Consequently, directional representation systems are required to effectively capture edge singularities for highdimensional problems. However, the increased angular resolution often significantly increases the redundancy rates of a directional system. High redundancy rates lead to expensive computational costs and large storage requirement, which hinder the usefulness of such directional systems for problems in moderately high dimensions such as video processing. In this talk, we attack this problem by using directional tensor product complex tight framelets with mixed sampling factors. Such introduced directional system has good directionality with a very low redundancy rate $\frac{3^d1}{2^d1}$, e.g., the redundancy rates are $2$, $2\frac{2}{3}$, $3\frac{5}{7}$, $5\frac{1}{3}$ and $7\frac{25}{31}$ for dimension $d=1,\ldots,5$. Our numerical experiments on image/video denoising and inpainting show that the performance of our proposed directional system with low redundancy rate is comparable or better than several stateoftheart methods which have much higher redundancy rates. In the second part, we shall discuss our recent developments of directional quasitight framelets in high dimensions. This is a joint work with Chenzhe Diao, Zhenpeng Zhao and Xiaosheng Zhuang. 

ASCW01 
19th February 2019 13:40 to 14:15 
Clayton Webster 
Polynomial approximation via compressed sensing of highdimensional functions on lower sets
This talk will focus on compressed sensing approaches to sparse polynomial approximation of complex functions in high dimensions. Of particular interest is the parameterized PDE setting, where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we will present and analyze several procedures for exactly reconstructing a set of (jointly) sparse vectors, from incomplete measurements. These include novel weighted $\ell_1$ minimization, improved iterative hard thresholding, mixed convex relaxations, as well as nonconvex penalties. Theoretical recovery guarantees will also be presented based on improved bounds for the restricted isometry property, as well as unified null space properties that encompass all currently proposed nonconvex minimizations. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the described compressed sensing methods.


ASCW01 
19th February 2019 14:20 to 14:55 
Oscar Dominguez 
Characterizations of Besov spaces in terms of Kfunctionals
Besov spaces occur naturally in many fields of analysis. In this talk, we discuss various characterizations of Besov spaces in terms of different Kfunctionals. For instance, we present descriptions via oscillations, Bianchinitype norms and approximation methods. This is a joint work with S. Tikhonov (Barcelona).


ASCW01 
19th February 2019 15:30 to 16:05 
Wenrui Ye 
Local restriction theorem and maximal BochnerRiesz operator for the Dunkl transforms In this talk, I will mainly describe joint work with Dr. Feng Dai on the critical index for the almost everywhere convergence of the BochnerRiesz means in weighted Lpspaces with p=1 or p>2. Our results under the case p>2 are in full analogy with the classical result of M. Christ on estimates of the maximal BochnerRiesz means of Fourier integrals and the classical result of A. Carbery, José L. Rubio De Francia and L. Vega on a.e. convergence of Fourier integrals. Besides, I will also introduce several new results that are related to our main results, including: (i) local restriction theorem for the Dunkl transform which is significantly stronger than the global one, but more difficult to prove; (ii) the weighted Littlewood Paley inequality with Apweights in the Dunkl noncommutative setting; (iii) sharp local pointwise estimates of several important kernel functions. 

ASCW01 
19th February 2019 16:10 to 16:45 
Discussion  
ASCW01 
20th February 2019 09:00 to 09:35 
Holger Rauhut 
Recovery of functions of many variables via compressive sensing
The talk will report on the use of compressive sensing for the recovery of functions of many variables from sample values. We will cover trigonometric expansions as well as expansions in tensorized orthogonal polynomial systems and provide convergence rates in terms of the number of samples which avoid the curse of dimensionality. The technique can be used for the numerical solution of parametric operator equations.


ASCW01 
20th February 2019 09:40 to 10:15 
Dũng Dinh 
Dimensiondependence error estimates for sampling recovery on Smolyak grids We investigate dimensiondependence estimates of the approximation error for linear algorithms of sampling recovery on Smolyak grids parametrized by $m$, of periodic $d$variate functions from the space with LipschitzH\"older mixed smoothness $\alpha > 0$. For the subsets of the unit ball in this space of functions with homogeneous condition and of functions depending on $\nu$ active variables ($1 \le \nu \le d$), respectively, we prove some upper bounds and lower bounds (for $\alpha \le 2$) of the error of the optimal sampling recovery on Smolyak grids, explicit in $d$, $\nu$, $m$ when $d$ and $m$ may be large. This is a joint work with Mai Xuan Thao, Hong Duc University, Thanh Hoa, Vietnam. 

ASCW01 
20th February 2019 11:00 to 11:35 
Martin Buhmann 
Recent Results on Rational Approximation and Interpolation with Completely and Multiply Monotone Radial Basis Functions
We will report on new results about approximations to continuous functions of multiple variables. We shall use either approximation with interpolation or approximation by rational functions. For these kinds of approximations, radial basis functions are particularly attractive, as they provide regular, positive definite or conditionally positive definite approximations, independent of the spatial dimension and independent the distribution of the data points we wish to work with. These interpolants have very many applications for example in solving nonlinear partial differential equations by collocation. In this talk, we classify radial basis and other functions that are useful for such scattered data interpolation or for rational approximations from vector spaces spanned by translates of those basis functions (kernels); for this we study in particular multiply and/or completely monotone functions. We collect special properties of such monotone functions, generalise them and find larger classes than the well known monotone functions for multivariate interpolation. Furthermore, we discuss efficient ways to compute rational approximations using the same type of kernels.


ASCW01 
20th February 2019 11:40 to 12:15 
Lutz Kaemmerer 
Multiple Rank1 Lattices as Sampling Schemes for Approximation
The approximation of functions using sampling values
along single rank1 lattices leads to convergence rates of the approximation
errors that are far away from optimal ones in spaces of dominating mixed
smoothness.
A recently published idea that uses sampling
values along several rank1 lattices in order to reconstruct multivariate
trigonometric polynomials accompanied by fast methods for the construction of
these sampling schemes as well as available fast Fourier transform algorithms
motivates investigations on the approximation properties of the arising
sampling operators applied on functions of specific smoothness, in particular
functions of dominating mixed smoothness which naturally leads to hyperbolic
cross approximations.


ASCW01 
21st February 2019 09:00 to 09:35 
Thomas Kuehn 
Preasymptotic estimates for approximation of multivariate periodic Sobolev functions
Approximation of Sobolev functions is a topic with a long history and many applications in different branches of mathematics. The asymptotic order as $n\to\infty$ of the approximation numbers $a_n$ is wellknown for embeddings of isotropic Sobolev spaces and also for Sobolev spaces of dominating mixed smoothness. However, if the dimension $d$ of the underlying domain is very high, one has to wait exponentially long until the asymptotic rate becomes visible. Hence, for computational issues this rate is useless, what really matters is the preasymptotic range, say $n\le 2^d$. In the talk I will first give a short overview over this relatively new field. Then I will present some new preasymptotic estimates for $L_2$approximation of periodic Sobolev functions, which improve the previously known results. I will discuss the cases of isotropic and dominating mixed smoothness, and also $C^\infty$functions of Gevrey type. Clearly, on all these spaces there are many equivalent norms. It is an interesting effect that  in contrast to the asymptotic rates  the preasymptotic behaviour strongly depends on the chosen norm. 

ASCW01 
21st February 2019 09:40 to 10:15 
Konstantin Ryutin 
Best mterm approximation of the "stepfunction" and related problems
The main point of the talk is the problem of approximation of the stepfunction by $m$term trigonometric polynomials and some closely related problems: the approximate rank of a specific triangular matrix, the Kolmogorov width of BV functions. This problem has its origins in approximation theory (best sparse approximation and Kolmogorov widths) as well as in computer science (approximate rank of a matrix). There are different approaches and techniques: $\gamma_2$norm, random approximations, orthomassivity of a set.... I plan to show what can be achieved by these techniques.


ASCW01 
21st February 2019 11:00 to 11:35 
Michael Gnewuch 
Explicit error bounds for randomized Smolyak algorithms and an application to infinitedimensional integration
Smolyak's method, also known as hyperbolic cross approximation or sparse grid method, is a powerful %black box tool to tackle multivariate tensor product problems just with the help of efficient algorithms for the corresponding univariate problem. We provide upper and lower error bounds for randomized Smolyak algorithms with fully explicit dependence on the number of variables and the number of information evaluations used. The error criteria we consider are the worstcase root mean square error (the typical error criterion for randomized algorithms, often referred to as ``randomized error'') and the root mean square worstcase error (often referred to as ``worstcase error''). Randomized Smolyak algorithms can be used as building blocks for efficient methods, such as multilevel algorithms, multivariate decomposition methods or dimensionwise quadrature methods, to tackle successfully highdimensional or even infinitedimensional problems. As an example, we provide a very general and sharp result on infinitedimensional integration on weighted reproducing kernel Hilbert spaces and illustrate it for the special case of weighted Korobov spaces. We explain how this result can be extended, e.g., to spaces of functions whose smooth dependence on successive variables increases (``spaces of increasing smoothness'') and to the problem of L_2approximation (function recovery). 

ASCW01 
21st February 2019 11:40 to 12:15 
Heping Wang 
Monte Carlo methods for $L_q$ approximation on periodic Sobolev spaces with mixed smoothness
In this talk we consider multivariate approximation of compact embeddings of periodic Sobolev spaces of dominating mixed smoothness into the $L_q,\ 2< q\leq \infty$ space by linear Monte Carlo methods that use arbitrary linear information. We construct linear Monte Carlo methods and obtain explicitindimension upper estimates. These estimates catch up with the rate of convergence.


ASCW01 
21st February 2019 13:40 to 14:15 
Robert J. Kunsch 
Optimal Confidence for Monte Carlo Integration of Smooth Functions
We study the complexity $n(\varepsilon,\delta)$ of approximating the integral of smooth functions at absolute precision $\varepsilon > 0$ with confidence level $1  \delta \in (0,1)$ using function evaluations as information within randomized algorithms. Methods that achieve optimal rates in terms of the root mean square error (RMSE) are not always optimal in terms of error at confidence, usually we need some nonlinearity in order to suppress outliers. Besides, there are numerical problems which can be solved in terms of error at confidence but no algorithm can guarantee a finite RMSE, see [1]. Hence, the new error criterion seems to be more general than the classical RMSE. The sharp order for multivariate functions from classical isotropic Sobolev spaces $W_p^r([0,1]^d)$ can be achieved via control variates, as long as the space is embedded in the space of continuous functions $C([0,1]^d)$. It turns out that the integrability index $p$ has an effect on the influence of the uncertainty $\delta$ to the complexity, with the limiting case $p = 1$ where deterministic methods cannot be improved by randomization. In general, higher smoothness reduces the effort we need to take in order to increase the confidence level. Determining the complexity $n(\varepsilon,\delta)$ is much more challenging for mixed smoothness spaces $\mathbf{W}_p^r([0,1]^d)$. While optimal rates are known for the classical RMSE (as long as $\mathbf{W}_p^r([0,1]^d)$ is embedded in $L_2([0,1]^d)$), see [2], basic modifications of the corresponding algorithms fail to match the theoretical lower bounds for approximating the integral with prescribed confidence. Joint work with Daniel Rudolf [1] R.J. Kunsch, E. Novak, D. Rudolf. Solvable integration problems and optimal sample size selection. To appear in Journal of Complexity. [2] M. Ullrich. A Monte Carlo method for integration of multivariate smooth functions. SIAM Journal on Numerical Analysis, 55(3):11881200, 2017. 

ASCW01 
21st February 2019 14:20 to 14:55 
Markus Weimar 
Optimal recovery using wavelet trees
This talk is concerned with the approximation of embeddings between Besovtype spaces defined on bounded multidimensional domains or (patchwise smooth) manifolds. We compare the quality of approximations of three different strategies based on wavelet expansions. For this purpose, sharp rates of convergence corresponding to classical uniform refinement, best $N$term, and best $N$term tree approximation will be presented. In particular, we will see that whenever the embedding of interest is compact, greedy tree approximation schemes are as powerful as abstract best $N$term approximation and that (for a large range of parameters) they can outperform uniform schemes based on a priori fixed (hence nonadaptively chosen) subspaces. This observation justifies the usage of adaptive nonlinear algorithms in computational practice, e.g., for the approximate solution of boundary integral equations arising from physical applications. If time permits, implications for the related concept of approximation spaces associated to the three approximation strategies will be discussed.


ASCW01 
21st February 2019 15:30 to 16:05 
Discussion  
ASCW01 
21st February 2019 16:10 to 16:45 
Discussion  
ASCW01 
22nd February 2019 09:00 to 09:35 
Jürgen Prestin 
Shiftinvariant Spaces of Multivariate Periodic Functions
One of the underlying ideas of multiresolution and wavelet analysis consists in the investigation of shiftinvariant function spaces. In this talk onedimensional shiftinvariant spaces of periodic functions are generalized to multivariate shiftinvariant spaces on nontensor product patterns. These patterns are generated from a regular integer matrix. The decomposition of these spaces into shiftinvariant subspaces can be discussed by the properties of these matrices. For these spaces we study different bases and their timefrequency localization. Of particular interest are multivariate orthogonal Dirichlet and de la Valle\'e Poussin kernels and the respective wavelets. This approach also leads to an adaptive multiresolution. Finally, with these methods we construct shearlets and show how we can detect jump discontinuities of given cartoonlike functions.


ASCW01 
22nd February 2019 09:40 to 10:15 
Bastian Bohn 
Least squares regression on sparse grids
In this talk, we first recapitulate the framework of least squares regression on certain sparse grid and hyperbolic cross spaces. The underlying numerical problem can be solved quite efficiently with stateoftheart algorithms. Analyzing its stability and convergence properties, we can derive the optimal coupling between the number of necessary data samples and the degrees of freedom in the ansatz space.Our analysis is based on the assumption that the leastsquares solution employs some kind of Sobolev regularity of dominating mixed smoothness, which is seldomly encountered for realworld applications. Therefore, we present possible extensions of the basic sparse grid least squares algorithm by introducing suitable apriori data transformations in the second part of the talk. These are tailored such that the resulting transformed problem suits the sparse grid structure. Coauthors: Michael Griebel (University of Bonn), Jens Oettershagen (University of Bonn), Christian Rieger (University of Bonn) 

ASCW01 
22nd February 2019 11:00 to 11:35 
Song Li 
Some Sparse Recovery Methods in Compressed Sensing
In this talk, I shall investigate some sparse recovery methods in Compressed Sensing. In particular, I will focus on RIP approach and DRIP approach. As a result, we confirmed a conjecture on RIP, which is related to Terence. Tao and Jean. Bourgain's works in this fields. Then, I will also investigate the relations between our works and statistics.


ASCW01 
22nd February 2019 11:40 to 12:15 
tba  
ASCW01 
22nd February 2019 13:40 to 16:45 
Discussion  
ASC 
25th February 2019 15:00 to 16:00 
Michael Lacey 
Discrete Spherical Averages
The
strongest inequalities concerning continuous spherical averages are phrased in
the language of $L^p$ improving inequalities. Replace the
continuous averages by discrete averages, that is average over lattice points on
a sphere. These inequalities then engage the continuous versions, the
HardyLittlewood circle method, and Kloosterman sums. We will report on
progress understanding these inequalities. Joint work with Robert Kesler, and
Dario Mena.


ASC 
12th March 2019 15:00 to 16:30 
Elena Berdysheva 
Metric Approximation of SetValued Functions
We study approximation of setvalued functions (SVFs)  functions mapping a real interval to compact sets in Rd. In addition to the theoretical interest in this subject, it is relevant to various applications in elds where SVFs are used, such as economy, optimization, dynamical systems, control theory, game theory, dierential inclusions, geometric modeling. In particular, SVFs are relevant to the problem of the reconstruction of 3D objects from their parallel crosssections. The images (values) of the related SVF are the crosssections of the 3D object, and the graph of this SVF is the 3D object. Adaptations of classical samplebased approximation operators, in particular, of positive operators for approximation of SVFs with convex images were intensively studied by a number of authors. For example, R.A Vitale studied an adaptation of the classical Bernstein polynomial operator based on Minkowski linear combination of sets which converges to the convex hull of the image. Thus, the limit SVF is always a function with convex images, even if the original function is not. This eect is called convexication and is observed in various adaptations based on Minkowski linear combinations. Clearly such adaptations work for setvalued functions with convex images, but are useless for the approximation of SFVs with nonconvex images. Also the standard construction of an integral of setvalued functions  the Aumann integral  possesses the property of convexication. Dyn, Farkhi and Mokhov developed in a series of work a new approach that is free of convexication  the socalled metric linear combinations and the metric integral. Adaptations of classical approximation operators to continuous SFVs were studied by Dyn, Farkhi and Mokhov. Here, we develop methods for approximation of SFVs that are not necessarily contin uous. As the rst step, we consider SVFs of bounded variation in the Hausdor metric. In particular, we adapt to SVFs local operators such as the symmetric Schoenberg spline operator, the Bernstein polynomial operator and the Steklov function. Error bounds, obtained in the averaged Hausdor metric, provide rates of approximation similar to those for realvalued functions of bounded variation. Joint work with Nira Dyn, Elza Farkhi and Alona Mokhov (Tel Aviv University, Israel). 

ASC 
25th March 2019 11:00 to 12:00 
Dorothee Haroske 
Morrey sequence spaces
Morrey (function) spaces and, in particular, smoothness spaces of BesovMorrey or TriebelLizorkinMorrey type were studied in recent years quite intensively and systematically. Decomposition methods like atomic or wavelet characterisations require suitably adapted sequence spaces. This has been done to some extent already. However, based on some discussion at a conference in Poznan in 2017 we found that Morrey sequence spaces $m_{u,p}=m_{u,p}(\mathbb{Z}^d)$, $0
We consider some basic features, embedding properties, a predual, a corresponding version of Pitt's compactness theorem, and can further characterise the compactness of embeddings of related finite dimensional spaces in terms of their entropy numbers. This is joint work with Leszek Skrzypczak (Poznan). 

ASC 
8th April 2019 15:00 to 16:00 
Luz Roncal 
Hardytype inequalities for fractional powers of the DunklHermite operator
We prove Hardytype
inequalities for the conformally invariant fractional powers of the
DunklHermite operator. Consequently, we also obtain Hardy inequalities for
the fractional harmonic oscillator as well. The strategy is as follows: first, by introducing suitable polar coordinates, we reduce the problem to the Laguerre setting. Then, we push forward an argument developed by R. L. Frank, E. H. Lieb and R. Seiringer, initially developed in the Euclidean setting, to get a Hardy inequality for the fractionaltype Laguerre operator. Such argument is based on two facts: first, to get an integral representation for the corresponding fractional operator, and second, to write a proper ground state representation. This is joint work with \'O. Ciaurri (Universidad de La Rioja, Spain) and S. Thangavelu (Indian Institute of Science of Bangalore, India). 

ASC 
29th April 2019 14:00 to 16:00 
Yuan Xu 
Simultaneous approximation by polynomials
Least square polynomials in an
$L^2$ space are partial sums of the Fourier orthogonal expansions. If we were
to approximate functions and their derivatives simultaneously on a domain in
$R^d$ (as desired in spectral method), we would need to consider orthogonal
expansions in a Sobolev space, for which the orthogonality is defined with
respect to an inner product that contains derivatives. Since multiplication
operators are no longer selfadjoint under such an inner product, the
orthogonality is hard to understand and analyze. In the talk we will explain
what is known. 

ASC 
1st May 2019 13:00 to 14:45 
Marta Betcke 
Photoacoustic tomography with incomplete data
In photoacoustic
tomography, the acoustic propagation time across the specimen constitutes the
ultimate limit on sequential sampling frequency. Furthermore, the stateofthe
art PAT systems are still remote from realising this limit. Hence, for high
resolution imaging problems, the acquisition of a complete set of data can be
impractical or even not possible e.g. the underlying dynamics causes the object
to evolve faster than measurements can be acquired. To mitigate this problem we
revert to parallel data acquisition along with subsampling/compressed sensing
techniques. We consider different regularisation assumptions such as edge
sparsity, sparsity of image representation and wave field propagation in
Curvelet frame as well as learnt regularisation. We discuss the benefits and
limitations of the proposed approaches in PAT context 

ASC 
9th May 2019 14:00 to 15:00 
Nira Dyn 
Reconstruction of a 3D object from a finite number of its 1D parallel crosssections
The problem of
reconstruction of a 3D object from its parallel 2D cross sections has been considered by many researchers. In some previous works we suggested to regard the problem as an approximation of a setvalued function from a finite number of its samples, which are 2D sets. We used approximation methods for singlevalued functions by applying operations between sets instead of operations between numbers. Since 2D sets are much more complicated than 1D sets, we suggest here to regard 3D objects as bivariate functions with 1D sets as samples, and to use the analogue of piecewise linear interpolation on a triangulation as the approximation method. In this talk we present our method, and discuss the properties of the resulting interpolants, including continuity and approximation rates. Few examples will be presented. 

ASC 
21st May 2019 16:00 to 17:00 
Ronald DeVore  Rothschild Distinguished Visiting Fellow Lecture: Optimality of Algorithms for Approximation/Computation  
OFBW46 
23rd May 2019 10:00 to 10:10 
Jane Leeks, David Abrahams  Welcome and Introduction  
OFBW46 
23rd May 2019 10:10 to 10:20 
Anders Hansen  Introduction  
OFBW46 
23rd May 2019 10:20 to 11:00 
Alhussein Fawzi  Robustness and Geometry of Deep Neural Networks  
OFBW46 
23rd May 2019 11:20 to 12:00 
Hamza Fawzi  Fundamental Limitations on Adversarial Robustness  
OFBW46 
23rd May 2019 12:00 to 12:40 
Jennifer Boon  The Ethics of Algorithmic Decision Making  
OFBW46 
23rd May 2019 13:40 to 14:00 
Vegard Antun  On Instabilities of Deep Learning in Image Reconstruction  Part I  
OFBW46 
23rd May 2019 14:00 to 14:20 
Matthew Colbrook  On Instabilities of Deep Learning in Image Reconstruction  Part II  
OFBW46 
23rd May 2019 14:20 to 14:40 
Laura Thesing  A Stable Learning Framework  
OFBW46 
23rd May 2019 14:40 to 15:20 
Thomas Strohmer  Privacy Preserving Machine Learning: A Human Imperative?  
OFBW46 
23rd May 2019 15:40 to 16:20 
Neil Lawrence  Meta Modelling and Deploying Machine Learning Software  
OFBW46 
23rd May 2019 16:20 to 17:00 
Pearse Keane  The Moorfields  DeepMind Collaboration  Reinventing the Eye Exam using Deep Learning  
ASC 
7th June 2019 14:00 to 15:00 
Guergana Petrova 
Approximation via Deep Neural Networks
We will discuss the
approximation power of deep neural networks. In particular, we will present classes of functions which can be efficiently captured by neural networks where classical nonlinear methods fall short of the task. 

ASC 
12th June 2019 16:00 to 17:00 
Svitlana Mayboroda 
Kirk Distinguished Visiting Fellow Lecture: The hidden landscape of localization
Complexity of the geometry, randomness of the potential, and
many other irregularities of the system can cause powerful, albeit quite
different, manifestations of localization: a phenomenon of confinement of
waves, or eigenfunctions, to a small portion of the original domain. In the
present talk we show that behind a possibly disordered system there exists a
clear structure, referred to as a landscape function, which predicts the
location and shape of the localized eigenfunctions, a pattern of their
exponential decay, and delivers accurate bounds for the corresponding
eigenvalues in the range where, for instance, the classical Weyl law
notoriously fails. We will discuss main features of this structure universally
relevant for all elliptic operators, as well as specific applications to the
Schrodinger operator with random potential and to the PoissonSchrodinger
driftdiffusion system governing carrier distribution and transport in
semiconductor alloys.


ASC 
13th June 2019 15:00 to 16:00 
Maria del Carmen Reguera Rodriguez 
Sparse forms for BochnerRiesz operators
Sparse operators are
positive dyadic operators that have very nice boundedness properties. The L^p
bounds and weighted L^p bounds with sharp constant are easy to obtain for these
operators. In the recent years, it has been proven that singular integrals (cancellative
operators) can be pointwise controlled by sparse operators. This has made the
sharp weighted theory of singular integrals quite straightforward. The current
efforts focus in understanding the use of sparse operators to bound rougher
operators, such a oscillatory integrals. Following this direction, our goal in
this talk is to describe the control of BochnerRiesz operators by sparse
operators. 

ASCW03 
17th June 2019 09:50 to 10:40 
Akram Aldroubi 
Dynamical sampling and frames generated from powers of exponential operators
In this talk, I will give a brief review of the problem of frame generation from operator powers of exponentials acting on a set of vectors. I will discuss its relation to dynamical sampling, review some of the previous results and present several new ones.


ASCW03 
17th June 2019 11:10 to 12:00 
Denka Kutzarova 
Transportation cost spaces on finite metric spaces
Transportation cost spaces are studied by several groups of researchers, for different reasons and under different names. The term Lipschitzfree spaces is commonly used in Banach space theory. We prove that the transportation cost space on any finite metric space contains a large wellcomplemented subspace which is close to $\ell_1^n$. We show that transportation cost spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding dimensions. These classes contain wellknown families of diamond graphs and Laakso graphs. In the particular case of diamond graphs we prove that their cycle space is spanned by even levels of Haar functions. It is curious that the subspaces generated by all the even/odd levels of the Haar functions also appear in the study of quasigreedy basic sequences in $L_1[0,1]$. This research is joint with Stephen Dilworth and Mikhail Ostrovskii. 

ASCW03 
17th June 2019 13:30 to 14:20 
Albert Cohen 
Optimal sampling for approximation on general domains
We consider the approximation of an arbirary function
in any dimension from point samples. Approximants are picked from given or adaptively chosen finite dimensional spaces. Various recent works reveal that optimal
approximations can be constructed at minimal sampling budget by leastsquares methods with particular sampling measures. In this talk, we discuss strategies to construct these measures and their samples in the adaptive context and in general nontensorproduct multivariate domains. 

ASCW03 
17th June 2019 14:20 to 15:10 
Peter Binev 
High Dimensional Approximation via Sparse Occupancy Trees
Adaptive domain decomposition is often used in finite elements methods for solving partial differential equations in low space dimensions. The adaptive decisions are usually described by a tree. Assuming that can find the (approximate) error for approximating a function on each element of the partition, we have shown that a particular coarsetofine method provides a nearbest approximation. This result can be extended to approximating point clouds any space dimension provided that we have relevant information about the errors and can organize properly the data. Of course, this is subject to the curse of dimensionality and nothing can be done in the general case. In case the intrinsic dimensionality of the data is much smaller than the space dimension, one can define algorithms that defy the curse. This is usually done by assuming that the data domain is close to a low dimensional manifold and first approximating this manifold and then the function defined by it. A few years ago, together with Philipp Lamby, Wolfgang Dahmen, and Ron DeVore, we proposed a direct method (without specifically identifying any low dimensional set) that we called "sparse occupancy trees". The method defines a piecewise constant or linear approximation on general simplicial partitions. This talk considers an extension of this method to find a similar approximation on conforming simplicial partitions following an idea from a recent result together with Francesca Fierro and Andreas Veeser about nearbest approximation on conforming triangulations.


ASCW03 
17th June 2019 15:40 to 16:30 
Claire Boyer 
Representer theorems and convex optimization
We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and elements of the extreme rays of the regularizer level sets. As a side result, we characterize the minimizers of the total gradient variation. As an ongoing work, we will also study the geometry of the total gradient variation ball. This is a joint work with Antonin Chambolle, Yohann De Castro, Vincent Duval, Frédéric de Gournay, and Pierre Weiss. 

ASCW03 
18th June 2019 09:00 to 09:50 
Simon Foucart 
Functions of Few Coordinate Variables: Sampling Schemes and Recovery Algorithms
I will revisit in this talk the task of approximating
multivariate functions that depend on only a few of their variables. The number
of samples required to achieve this task to a given accuracy has been
determined for Lipschitz functions several years ago. However, two questions of
practical interest
remain: can we provide an explicit sampling strategy and
can we efficiently produce approximants? I will (attempt to) answer these
questions under some additional assumptions on the target function. Firstly, if
it is known to be linear, then the problem is exactly similar to the standard
compressive sensing problem, and I will review some of recent contributions
there. Secondly, if the target function is quadratic, then the problem connects
to sparse phaseless recovery and to jointly lowrank and bisparse recovery, for
which some results and open questions will be presented. Finally, if the target
function is known to increase coordinatewise, then the problem reduces to group
testing, from which I will draw the soughtafter sampling schemes and recovery
algorithms.


ASCW03 
18th June 2019 09:50 to 10:40 
Alexander Olevskii 
Discrete translates in function spaces
Given a Banach function space on R^n, does there exist
a uniformly discrete set of translates of a single function, which spans
the space? I'll present a survey on the problem and discuss recent
results, joint with A.Ulanovskii.


ASCW03 
18th June 2019 11:10 to 12:00 
Olga Mula  Optimal algorithms for state estimation using reduced models  
ASCW03 
18th June 2019 13:30 to 14:20 
Joaquim OrtegaCerdà 
A sequence of wellconditioned polynomials We find an explicit sequence of polynomials of arbitrary degree with 

ASCW03 
18th June 2019 14:20 to 15:10 
Holger Rauhut 
Linear and onebit compressive sensing with subsampled random convolutions
Compressive sensing predicts that sparse vectors can recovered from incomplete linear measurements with efficient algorithms in a stable way. While many theoretical results work with Gaussian random measurement matrices, practical applications usually demand for structure. The talk covers the particular case of structured random measurements defined via convolution with a random vector and subsampling (deterministic or random as well). We will give an overview on the corresponding theory and will cover also recent results concerning recovery from onebit measurements arising in quantized compressive sensing. Based on joint works with Felix Krahmer, Shahar Mendelson, Sjoerd Dirksen and HansChristian Jung. 

ASCW03 
18th June 2019 15:40 to 16:30 
Lukas Herrmann 
QuasiMonte Carlo integration in uncertainty quantification of elliptic PDEs with logGaussian coefficients QuasiMonte Carlo (QMC) rules are suitable to overcome the curse of dimension in the numerical integration of highdimensional integrands. 

ASCW03 
19th June 2019 09:00 to 09:50 
Andreas Seeger 
Basis properties of the Haar system in various function spaces We present recent results on the Haar system in Besov and TriebelLizorkin spaces, with an emphasis on endpoint results. 

ASCW03 
19th June 2019 09:50 to 10:40 
Christoph Thiele 
On singular Brascamp Lieb integrals
We will give an overview over singular Brascamp Lieb integrals and discuss some recent results based on recent arxiv postings with Polona Durcik. 

ASCW03 
19th June 2019 11:10 to 12:00 
Hans Feichtinger 
Approximation of continuous problems in Fourier Analysis by finite dimensional ones: The setting of the Banach Gelfand Triple
When it comes to the
constructive realization of operators arising in Fourier Analysis, be it the Fourier transform itself, or some convolution operator, or more generally an (underspread) pseudodiferential operator it is natural to make use of sampled version of the ingredients. The theory around the Banach Gelfand Triple (S0,L2,SO') which is based on methods from Gabor and timefrequency analysis, combined with the relevant function spaces (Wiener amalgams and modulation spaces) allows to provide what we consider the appropriate setting and possibly the starting point for qualitative as well as later on more quantitative error estimates. 

ASCW03 
20th June 2019 09:00 to 09:50 
Sinan Güntürk 
Extracting bits from analog samples: in pursuit of optimality
Sampling theorems, linear or nonlinear, provide the
basis for obtaining digital representations of analog signals, but often it is
not obvious how to quantize these samples in order to achieve the best
ratedistortion trade off possible, especially in the presence of redundancy.
We will present a general approach called
"distributed beta encoding" which can achieve superior (and often
nearoptimal) ratedistortion performance in a wide variety of sampling
scenarios. These will include some classical problems in the linear setting
such as Fourier and Gabor sampling, and some others in the nonlinear setting,
such as compressive sampling, spectral superresolution, and phase retrieval.


ASCW03 
20th June 2019 09:50 to 10:40 
Peter Richtarik 
Stochastic QuasiGradient Methods: Variance Reduction via Jacobian Sketching
We develop a new family of variance reduced stochastic gradient descent methods for minimizing the average of a very large number of smooth functions. Our method—JacSketch—is motivated by novel de velopments in randomized numerical linear algebra, and operates by maintaining a stochastic estimate of a Jacobian matrix composed of the gradients of individual functions. In each iteration, JacSketch efficiently updates the Jacobian matrix by first obtaining a random linear measurement of the true Jacobian through (cheap) sketching, and then projecting the previous estimate onto the solution space of a linear matrix equa tion whose solutions are consistent with the measurement. The Jacobian estimate is then used to compute a variancereduced unbiased estimator of the gradient, followed by a stochastic gradient descent step. Our strategy is analogous to the way quasiNewton methods maintain an estimate of the Hessian, and hence our method can be seen as a stochastic q uasigradient method. Indeed, quasiNewton methods project the current Hessian estimate onto a solution space of a linear equation consistent with a certain linear (but nonrandom) measurement of the true Hessian. Our method can also be seen as stochastic gradient descent applied to a controlled stochastic optimization reformulation of the original problem, where the control comes from the Jacobian estimates. We prove that for smooth and strongly convex functions, JacSketch converges linearly with a meaningful rate dictated by a single convergence theorem which applies to general sketches. We also provide a refined convergence theorem which applies to a smaller class of sketches, featuring a novel proof technique based on a stochastic Lyapunov function. This enables us to obtain sharper complexity results for variants of JacSketch with importance sampling. By specializing our general approach to specific sketching strategies, JacSketch reduces to the celebrated stochastic average gradient (SAGA) method, and Coauthors: Robert Mansel Gower (Telecom ParisTech), Francis Bach (INRIA  ENS  PSL Research University) 

ASCW03 
20th June 2019 11:10 to 12:00 
Boris Kashin  On some theorems on the restriction of operator to coordinate subspace  
ASCW03 
20th June 2019 13:30 to 14:20 
Yuan Xu 
Orthogonal structure in and on quadratic surfaces
Orthogonal structure in and on quadratic
surfaces Text of abstract: Spherical harmonics are orthogonal polynomials on
the unit sphere. They are eigenfunctions of the LaplaceBeltrami operator on
the sphere and they satisfy an addition formula (a closed formula for their
reproducing kernel). In this talk, we consider orthogonal polynomials on
quadratic surfaces of revolution and inside the domain bounded by quadratic
surfaces. We will define orthogonal
polynomials on the surface of a cone that possess both characteristics of
spherical harmonics. In particular, the addition formula on the cone has a
onedimensional structure, which leads to a convolution structure on the cone
useful for studying Fourier orthogonal series. Furthermore, the same narrative
holds for orthogonal polynomials defined on the solid cones.


ASCW03 
20th June 2019 14:20 to 15:10 
Gitta Kutyniok 
Beating the Curse of Dimensionality: A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
Highdimensional parametric partial differential equations (PDEs) appear in various contexts including control and optimization problems, inverse problems, risk assessment, and uncertainty quantification. In most such scenarios the set of all admissible solutions associated with the parameter space is inherently low dimensional. This fact forms the foundation for the socalled reduced basis method. Recently, numerical experiments demonstrated the remarkable efficiency of using deep neural networks to solve parametric problems. In this talk, we will present a theoretical justification for this class of approaches. More precisely, we will derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric PDEs. In fact, without any knowledge of its concrete shape, we use the inherent lowdimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical approximation results. We use this lowdimensionality to guarantee the existence of a reduced basis. Then, for a large variety of parametric PDEs, we construct neural networks that yield approximations of the parametric maps not suffering from a curse of dimensionality and essentially only depending on the size of the reduced basis. This is joint work with Philipp Petersen (Oxford), Mones Raslan, and Reinhold Schneider. 

ASCW03 
21st June 2019 09:00 to 09:50 
Aicke Hinrichs 
How good is random information compared to optimal information?
We study approximation and integration problems and compare the quality of optimal information with the quality of random information. For some problems random information is almost optimal and for other problems random information is much worse than optimal information. We give a survey about known and new results. Parts of the talk are based on joint work with D. Krieg, E. Novak, J. Prochno and M. Ullrich.


ASCW03 
21st June 2019 09:50 to 10:40 
Karlheinz Groechenig 
Totally positive functions in sampling theory and timefrequency analysis
Totally positive functions play an important role in
approximation theory and statistics. In this talk I will present recent new
applications of totally positive functions (TPFs) in sampling theory and
timefrequency analysis.
(i) We study the sampling problem for shiftinvariant
spaces generated by a TPF. These spaces arise the span of the integer shifts of
a TPF and are often used as a
substitute for bandlimited functions. We give a complete
characterization of sampling sets
for a shiftinvariant space with a TPF generator of
Gaussian type in the style of Beurling.
(ii) A related problem is the question of Gabor frames,
i.e., the spanning properties of timefrequency shifts of a given function. It
is conjectured that the lattice shifts of a TPF generate a frame, if and only
if the density of the lattice exceeds 1.
At this time this conjecture has been proved
for two important subclasses of TPFs. For rational lattices it is true for arbitrary
TPFs. So far, TPFs seem to be the only
window functions for which the fine structure of the associated Gabor frames is tractable.
(iii) Yet another question in timefrequency analysis is
the existence of zeros of the Wigner distribution (or the radar ambiguity
function). So far all examples of zerofree ambiguity functions are related to
TPFs, e.g., the ambiguity function of the Gaussian is zero free.


ASCW03 
21st June 2019 11:10 to 12:00 
Feng Dai 
Integral norm discretization and related problems
In this talk, we will discuss the problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure. We study the problem for elements of finite dimensional spaces in a general setting, paying a special attention to the case of the multivariate trigonometric polynomials with frequencies from a finite set with fixed cardinality. Both new results and a survey of known results will be presented. This is a joint work with A. Prymak, V.N. Temlyakov and S. Tikhonov. 

ASCW03 
21st June 2019 13:30 to 14:20 
Milana Gataric 
Imaging through optical fibres
In this talk, I'll present some recent results on reconstruction of optical vectorfields using measurements acquired by an optical fibre characterised by a nonunitary integral transform with an unknown spatiallyvariant kernel. A new imaging framework will be introduced, which through regularisation is able to recover an optical vectorfield with respect to an arbitrary representation system potentially different from the one used for fibre calibration. In particular, this enables the recovery of an optical vectorfield with respect to a Fourier basis, which is shown to yield indicative features of increased scattering associated with tissue abnormalities. The effectiveness of this framework is demonstrated using biological tissue samples in an experimental setting where measurements are acquired by a fibre endoscope, and it is observed that indeed the recovered Fourier coefficients are useful in distinguishing healthy tissues from lesions in early stages of cancer. If time permits, I'll also briefly present a new method that enables recovery of nonunitary fibre transmission matrices necessary for minimally invasive optical imaging in inaccessible areas of the body.


ASCW03 
21st June 2019 14:20 to 15:10 
Geno Nikolov 
Markovtype inequalities and extreme zeros of orthogonal polynomials
The talk is centered around the problem of finding
(obtaining tight twosided bounds
for) the sharp constants in certain
MarkovBernstein type inequalities in weighted $L_2$ norms. It turns out that,
under certain assumptions, this problem is equivalent to the estimation of the
extreme zeros of orthogonal polynomials with respect to a measure supported on
$R_{+}$.
It will be shown how classical tools like the
EulerRayleigh method and Gershgorin circle theorem produce surprisingly good
bounds for the extreme zeros of the Jacobi, Gegenbauer and Laguerre
polynomials. The sharp constants in the $L_2$
Markov inequalities with the Laguerre and Gegenbauer weight functions
and in a discrete $\ell_2$ MarkovBernstein inequality are investigated using
the same tool.
