Week of April 27, 2025

Tue Apr 29, 2025
3:00pm to 3:50pm - - Analysis
H. Turgay Kaptanglu - (Bilkent University, Ankara, Turkey)
Extremal Problems in Weighted Bergman-Besov Spaces through Bergman Projections

We extend the method of using Bergman projections for solving extremal
problems introduced by Ferguson to systematically compute extremal 
functions
in weighted Bergman-Besov spaces at arbitrary data points in the unit 
disc.
We include the case p=1 by first proving the existence of solutions to
a large class of extremal problems in this case.
We also develop expansions of analytic functions in terms of Mobius 
factors
similar to Taylor series to handle data points different from the 
origin.
Our method is especially suitable for Caratheodory-Fejer-type 
interpolation.

This is joint work with my students A. Balci and R. Ozbek.

Wed Apr 30, 2025
3:00pm to 3:50pm - 340P - Inverse Problems
Jesse Railo - (LUT University)
Recovering a first order perturbation of one-dimensional wave equation from white noise boundary data

We consider the following inverse problem: Suppose a (1 + 1)-dimensional wave equation on R+ with zero initial conditions is excited with a Neumann boundary data modelled as a white noise process. Given also the Dirichlet data at the same point, determine the unknown first order coefficient function of the system. The inverse problem is then solved by showing that correlations of the boundary data determine the Neumann-to-Dirichlet operator in the sense of distributions, which is known to uniquely identify the coefficient. The model has potential applications in acoustic measurements of internal cross-sections of fluid pipes such as pressurised water supply pipes and vocal tract shape determination. This talk is based on a joint-work with Emilia Blåsten, Antti Kujanpää and Tapio Helin (LUT), and Lauri Oksanen (Helsinki).

Thu May 1, 2025
4:00pm to 4:50pm - RH 306 - Colloquium
Rodolfo H. Torres - (University of California, Riverside)
Almost Orthogonality in Fourier Analysis: From Singular integrals, to Function Spaces, to Leibniz Rules for Fractional Derivatives

Fourier analysis has been an extraordinarily powerful mathematical tool since its development 200 years ago, and currently has a wide range of applications in diverse scientific fields including digital image processing, forensics, option pricing, cryptography, optics, oceanography, and protein structure analysis. Like a prism that decomposes a beam of light into a rainbow of colors, Fourier analysis transforms signals into a mathematical spectrum of basic wave components of different amplitudes and frequencies, from which many hidden properties in the data can be deciphered. At the abstract mathematical level signals are represented by functions and their filtering and other operations on them by operators. From a functional analytical point of view, these objects are studied by decomposing them into elementary building blocks, some of which have wavelike behavior too. Decomposition techniques such as atomic, molecular, wavelet and wave-packet expansions provide a multi-scale refinement of Fourier analysis and exploit a rather simple concept: "waves with very different frequencies are almost invisible to each other". Many of these useful techniques have been developed around the study of some particular operators called singular integral operators and,  recently, similar techniques have been pushed to the analysis of new multilinear operators that arise in the study of (para) product-like operations, null-forms, and other nonlinear functional expressions. In this talk we will present some of our contributions in the study of multilinear singular integrals and function spaces, and their applications to the development of the equivalent of the calculus Leibniz rule to the concept of fractional derivatives.