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Author Archives: ioannis parissis
Mat-1.C – Harmonic Analysis
For those of you who followed my Harmonic Analysis notes a couple of years back, here is an updated (and slightly more polished) version in pdf form, from an advanced course I gave at Aalto university: https://noppa.aalto.fi/noppa/kurssi/mat-1.c/harmonic_analysis
Posted in Mat-1.C - Harmonic Analysis, Teaching
Tagged BMO, Calderon-Zygmu, Calderon-Zygmund decomposition, dyadic, dyadic maximal function, Fourier Transform, Hardy-Littlewood maximal function, Harmonic Analysis, interpolation, lecture notes, Marcinkiewicz interpolation theorem, Riesz-Thorin interpolation theorem, Sobolev spaces, tempered distributions, weak L^p spaces
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Circle discrepancy for checkerboard measures
This week I am giving a talk at the Department of Mathematics and Systems Analysis of Aalto University where I will discuss results from a recent paper with Mihalis Kolountzakis. I will give a short introduction to different notions of … Continue reading
Posted in math.CA, math.NT, open problem, paper, seminar notes
Tagged checkerboard, co-author, coloring, discrepancy, paper
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DMat0101, Notes 8: Notes Littlewood-Paley inequalities and multipliers
In this final set of notes we will study the Littlewood-Paley decomposition and the Littlewood-Paley inequalities. These consist of very basic tools in analysis which allow us to decompose a function, on the frequency side, to pieces that have almost … Continue reading
DMat0101, Notes 7: General Calderón-Zygmund Operators
After having studied the Hilbert transform in detail we now move to the study of general Calderón-Zygmund operators, that is operators given formally as for an appropriate kernel . Let us quickly review what we used in order to show … Continue reading
DMat0101, Notes 6: Introduction to singular integral operators; the Hilbert transform
This week we come to the study of singular integral operators, that is operators of the form defined initially for `nice’ functions . Here we typically want to include the case where has a singularity close to the diagonal which … Continue reading
DMat0101, Notes 5: The Hardy-Littlewood maximal function
1. Averages and maximal operators This week we will be discussing the Hardy-Littlewood maximal function and some closely related maximal type operators. In order to have something concrete let us first of all define the averages of a locally integrable … Continue reading
DMat0101, Notes 4: The Fourier transform of the Schwartz class and tempered distributions
In this section we go back to the space of Schwartz functions and we define the Fourier transform in this set up. This will turn out to be extremely useful and flexible. The reason for this is the fact that … Continue reading
DMat0101, Notes 3: The Fourier transform on L^1
1. Definition and main properties. For , the Fourier transform of is the function Here denotes the inner product of and : Observe that this inner product in is compatible with the Euclidean norm since . It is easy to … Continue reading
Posted in math.CA, Teaching, Dmat0101 - Harmonic Analysis
Tagged Abel summability, approximation to the identity, Fourier Transform, Gauss kernel, Gauss-Weierstrass summability, Harmonic extension, Heat equation, integrable functions, inversion formula, Poisson kernel, Riemann-Lebesgue, summability method
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DMat0101, Notes 2: Convolution, Dense subspaces and interpolation of operators
1. Convolutions and approximations to the identity We restrict our attention to the Euclidean case . As we have seen the space is a vector space; linear combinations of functions in remain in the space. There is however a `product’ … Continue reading
DMat0101, Notes 1: Quick review of measure theory
0. About these notes The notes that will follow are meant to be a companion to the Harmonic Analysis course that I’m giving this semester at IST. These notes are inspired, influenced and sometimes shamelessly copied from books, lecture notes … Continue reading
