In this course, we re-consider Fourier series in the functional analysis framework, and we explain important inequalities and orthonormal system. We also explain definition and properties of Fourier transform which can be obtained as continuous limit of Fourier series. Lastly we consider several partial differential equations as applications of Fourier series and Fourier transform. This course is a continuation of Applied Analysis I.
The notions of Fourier series and Fourier transform are fundamental not only in mathematics but also in science, and are applicable to describe a wide variety of objects. On the other hand, these abstract notions are not easy to comprehend without suitable training. To that end, rigorous proofs will be given for most propositions, lemmas and theorems. Moreover we study the most fundamental PDEs such as wave, heat and Laplace's equations.
Students are expected to study basic concepts of complex analysis. More precisely, we study Fourier series, Fourier transform, these definitions, properties and method of these calculations. These have important roles in science and technology.
Bessel's inequality, Parseval's equality, Fourier transform, wave equation, heat equation, Laplace's equation
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
Before coming to class, students should read the course schedule and check what topics will be covered.
Required learning should be completed outside of the classroom for preparation and review purposes.
|Course schedule||Required learning|
|Class 1||Function spaces, Orthonormal system||Details will be provided during each class session|
|Class 2||Bessel's inequality, Parseval's equality|
|Class 3||Fourier 's integral formula|
|Class 4||Fourier transform and Fourier inverse transform|
|Class 5||properties of Fourier transform|
|Class 6||application to wave equations|
|Class 7||application to heat equations|
|Class 8||application to Laplace's equation, comprehension check-up|
Students' course scores are based on final exam 50% and midterm exam 50%.
Students are expected to have passed Calculus I/Recitation and Calculus II+ Recitation.