[Turkmath:6848] RKHS Seminars will continue on December 10th, 2024! Online on Zoom!

[Omur Ugur]

Dear Friends,

We continue with the RKHS Seminars, now with an **expert** in the field, 
on **Tuesday, 10th of December** at 15:30 (Ankara, Turkey)!

The RKHS-Seminar Website is on https://iam.metu.edu.tr/rkhs-seminars. 
Please, do not miss the scheduled talk below!

Again, for a quick reminder, the event's page (Probability Error Bounds 
for Approximation of Functions in Reproducing Kernel Hilbert Spaces) is on:

- 
https://ougur.iam.metu.edu.tr/rkhs-seminars/2024/11/22/probability-error-bounds-for-approximation-of-functions-in-rkhs.html

Best wishes,
Omur

### RKHS-Seminars

Title: Probability Error Bounds for Approximation of Functions in 
Reproducing Kernel Hilbert Spaces

Speaker: Aurelian Gheondea (Institute of Mathematics of the Romanian 
Academy, Bucharest, and Bilkent University, Ankara)

Date / Time: Tuesday, December 10, 2024 / 15:30 (Ankara, Turkey)

Online on Zoom: 
https://us02web.zoom.us/j/88649160581?pwd=AwlZRvU8bsYO04EiM45VaC4pxFfFao.1

(in case needed; Meeting ID: 886 4916 0581 and Passcode: 313177)

Abstract: Better to read on 
https://ougur.iam.metu.edu.tr/rkhs-seminars/2024/11/22/probability-error-bounds-for-approximation-of-functions-in-rkhs.html 
if below is not clear enough.

Abstract: We find probability error bounds for approximations of 
functions \(f\) in
a separable reproducing kernel Hilbert space \(\mathcal{H}\)
with reproducing kernel \(K\) on a base space \(X\),
firstly in terms of finite linear combinations of functions of type 
\(K_{x_i}\) and
then in terms of the projection \(\pi_x^n\)
on \(\text{span}\left\{ K_{x_i} \right\}_{i=1}^{n}\),
or random sequences of points \(x = (x_i)_i\) in \(X\).
Given a probability measure \(P\), letting \(P_K\) be the measure 
defined by \(\text{d}P_K(x) = K(x, x)\text{d}P(x), \ x\in X\),
our approach is based on the nonexpansive operator</p>

\[L^2(X; P_K) \ni \lambda \mapsto L_{P,K} \lambda :=
   \int_X \lambda(x)\, K_x\, \text{d}P(x) \in \mathcal{H},\]

<p>where the integral exists in the Bochner sense.
Using this operator, we then define
a new reproducing kernel Hilbert space,
denoted by \(\mathcal{H}_P\), that is the operator range of \(L_{P,K}\).
Our main result establishes bounds,
in terms of the operator \(L_{P,K}\),
on the probability that the Hilbert space distance
between an arbitrary function \(f\) in \(\mathcal{H}\) and
linear combinations of functions of type \(K_{x_i}\),
for \((x_i)_i\), sampled independently from \(P\),
falls below a given threshold. For sequences
of points \((x_i)_i^\infty\) constituting a so-called
uniqueness set, the orthogonal projections \(\pi_x^n\)
to \(\text{span}\left\{ K_{x_i} \right\}_{i=1}^{n}\)
converge in the strong operator topology to the identity operator.
We prove that, under the assumption that \(\mathcal{H}_P\)
is dense in \(\mathcal{H}\),
any sequence of points sampled independently from \(P\) yields a
uniqueness set with probability \(1\).
This result improves on previous error bounds in
weaker norms, such as uniform or \(L^p\) norms,
which yield only convergence in probability
and not almost certain convergence.
Two examples that show the applicability of this
result to a uniform distribution on a compact interval
and to the Hardy space \(H^2(\mathbb{D})\) are presented as well.</p>

<p><em>Joint work with</em> <strong>Ata Deniz Aydın</strong>, ETH 
Zürich, Switzerland.</p>



#### Biography
Aurelian Gheondea studied mathematics and got his PhD at the University 
of Bucharest, Romania, with a thesis “Spectral Theory of Selfadjoint 
Operators in Krein Spaces” under the supervision of Ion Colojoară. He 
worked mainly at the Institute of Mathematics of the Romanian Academy in 
Bucharest and for twenty one years at the Bilkent University in Ankara. 
His fields of interest spans between Functional Analysis, Operator 
Theory and Operator Algebras, Matrix and Numerical Analysis, Hermitian 
Kernels, Quantum Operations, and Mathematical Modelling. He supervised 
eleven MSc students and one PhD student. He is the author of about 
eighty articles and three books. Currently he is a Senior Researcher at 
the Institute of Mathematics of the Romanian Academy in Bucharest and 
Professor Emeritus at the Bilkent University in Ankara.

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Dr. Omur Ugur                  | Middle East Technical University
http://users.metu.edu.tr/ougur | Institute of Applied Mathematics
Tel.: +90(312) 210 29 87       |             06800 Ankara, Turkey
Fax : +90(312) 210 29 85       |
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