INTRODUCTION TO FOURIER ANALYSIS

Prerequisites & course information

Requisites: MATH 521 or graduate/professional standing or member of the Pre-Masters Mathematics (Visiting International) Program Course Designation: Core GenEd - Natural Science & Wellness Level - Advanced L&S Credit - L&S Liberal Arts and Science Grad 50% - Counts toward 50% graduate coursework requirement Repeatable for Credit: No Last Taught: Fall 2023 Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties relevant to the analysis of Fourier series, the Fourier transform, and their applications in mathematics (e.g., convergence, divergence, convolution, Lipschitz property, Schwartz space, etc.). Audience: Both Grad & Undergrad 2. Use the above definitions to prove if a mathematical construct does or does not have the condition of being a particular mathematical object or having a particular property used in Fourier analysis (e.g., that a sequence is equidistributed, that a set of functions is uniformly bounded, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems and results of elementary Fourier analysis (e.g., the Fourier series of an integrable function converges to that function on its domain, etc.), and recall the arguments for these results and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use the above theorems in the context of longer arguments by examining their premises (e.g., verifying integrability of a function in order to conclude that its Fourier series converges, etc.). Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, induction, etc.). Audience: Both Grad & Undergrad 6. Convey arguments in oral and written forms using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties relevant to the analysis of Fourier series, the Fourier transform, and their applications in mathematics (e.g., convergence, divergence, convolution, Lipschitz property, Schwartz space, etc.). Audience: Both Grad & Undergrad 2. Use the above definitions to prove if a mathematical construct does or does not have the condition of being a particular mathematical object or having a particular property used in Fourier analysis (e.g., that a sequence is equidistributed, that a set of functions is uniformly bounded, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems and results of elementary Fourier analysis (e.g., the Fourier series of an integrable function converges to that function on its domain, etc.), and recall the arguments for these results and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use the above theorems in the context of longer arguments by examining their premises (e.g., verifying integrability of a function in order to conclude that its Fourier series converges, etc.). Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, induction, etc.). Audience: Both Grad & Undergrad 6. Convey arguments in oral and written forms using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate