30665 - ELEMENTS OF REAL AND FOURIER ANALYSIS

A sound knowledge of the main tools of calculus (limits, series, derivatives, integrals) and of linear algebra.

The course will provide the basic foundations of Fourier Analysis, introducing also a few elementary notion of complex analysis and some fundamental tools for signal theory. Complex analysis deals with complex functions of a complex variable and enlights remarkable links between complex differentiability, power series in the complex plane, and harmonic functions. Its main results are particularly important to understand power series (which lie at the core of functional calculus for linear operators), and the Laplace and the discrete Zeta and Fourier transforms. Fourier analysis is one of the most powerful tools to analyze functions and it is the basic building block of signal theory. Starting from Fourier series dealing with periodic signals, which can be interpreted as an orthonormal decomposition in Hilbert spaces, its range of applications is considerably expanded by the Fourier and the Laplace transforms, which cover the case of general signals. The course is meant to round up an adequate undergraduate preparation in Mathematical Analysis and to give students a hint on more advanced issues, that find surprising and remarkable applications in several theoretical and applied fields.

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

• 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.

• 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.

• Face-to-face lectures
• Exercises (exercises, database, software etc.)

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 problem-solving. By discussing the exercises, students can clarify any misconceptions, deepen their understanding, and learn alternative approaches to problem-solving.

• 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 problem-solving 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 problem-solving and critical thinking, accounting for the remaining points.

• 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