30665  ELEMENTS OF REAL AND FOURIER ANALYSIS
Department of Decision Sciences
ELIA BRUE'
Synchronous Blended: Lezioni erogate in modalità sincrona in aula (max 1 ora per credito online sincrona)
Suggested background knowledge
Mission & Content Summary
MISSION
CONTENT SUMMARY
Complex formalism
 Complex plane, polar coordinates
 Exponential function, Euler formula
 Power series, complex derivative
Review of Measure Theory
 Measurable spaces
 Lebesgue integral
 Convergence Theorems (monotone, Fatou’s Lemma, dominated convergence)
 L^p spaces, convolution
Hilbert spaces
 Hilbert base, Parseval identity
 Riesz representation
 Complex Hilbert spaces
Fourier series
 Fourier decomposition of periodic functions
 convergence of Fourier series, derivatives, and regularity
 Applications: heat and wave equations, isoperimetric inequality
Fourier transform
 Parseval identity
 Fourier inversion formula
 Derivatives and regularity
 Examples and applications: Heisenberg uncertainty principle, Poisson summation formula, sampling theorem
Intended Learning Outcomes (ILO)
KNOWLEDGE AND UNDERSTANDING
 Develop a fundamental understanding of measure theory and integration.
 Understand the basic structure of Hilbert spaces, the use of scalar products, and of orthonormal systems and expansions.
 Acquire essential knoledge of functional spaces and functional analysis.
 Know the basic properties of periodic signals and the trigonometric basis, understand the meaning of Fourier expansion, reproduce basic series expansions, and describe their convergence properties.
 Understand Fourier transform, its inversion, and its link with Fourier series and Laplace transform.
APPLYING KNOWLEDGE AND UNDERSTANDING
 Manipulate complex numbers and power series in the complex plane.
 Compute Fourier series expansions of simple signals.
 Use Fourier series expansions for solving differential equations with periodic solutions.
 Estimate the behavior of the Fourier coefficients.
 Compute the Fourier/Laplace transform of simple signals.
 Interpret the Fourier transform and its behavior.
 Use simple relations between a function and its Fourier/Laplace transform.
Teaching methods
 Facetoface lectures
 Exercises (exercises, database, software etc.)
DETAILS
Students are assigned weekly exercises that are directly related to the concepts taught during the week. These exercises are meant to be solved independently by the students.
During the subsequent week's class, we will collaboratively solve and engage in discussions on the assigned exercises.
This approach promotes active learning, as students have the opportunity to engage in collaborative problemsolving. By discussing the exercises, students can clarify any misconceptions, deepen their understanding, and learn alternative approaches to problemsolving.
Assessment methods
Continuous assessment  Partial exams  General exam  


x  x 
ATTENDING AND NOT ATTENDING STUDENTS
 Partial exams consist of two written exams, one at the midpoint of the course and one at the end. The final grade is calculated as the average of these two scores.
 General exam: written exam at the end of the course that contributes to the overall assessment.
Each written exam comprises five exercises with multiple bullet points covering all the presented material.
A total of 34 points will be assigned, with scores above 34 receiving the maximum grade. The exam evaluates the acquisition of basic knowledge and problemsolving abilities. The exercises vary in difficulty, with the first three emphasizing fundamental concepts and accounting for 25 out of 34 points, while the last two require more problemsolving and critical thinking, accounting for the remaining points.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
 Lecture notes of the course
 Steven Krantz: A guide to complex variables. The Mathematical Association of America, 2008
 Elias Stein, Rami Shakarchi: Fourier Analysis, an introduction. Princeton University Press, 2002