Mathematics: Linear Analysis

<p>The course introduces fundamental concepts, results, and methods in Fourier analysis and aims for students to develop an understanding of and ability to apply these to describe, analyze, and solve relevant problems related to linear analysis. Furthermore, the course aims to prepare students for further studies in mathematics, natural sciences, and engineering.</p><p>The course treats:</p> <ul> <li>Basic properties of Fourier series of functions in one variable. Exponential form and trigonometric form. Riemann-Lebesgue lemma. Elementary conditions for pointwise and uniform convergence. Gibbs phenomenon.</li> <li>Basic properties of convolutions of periodic functions in one variable. Interaction with Fourier series. Convolution kernels and their applications in summing Fourier series. Fejér’s theorem and Weierstrass approximation theorem.</li> <li>Linear spaces and examples of linear operators. Vector norm, inner product, and Cauchy-Schwarz inequality. Hilbert space, minimum distance to closed convex sets, projection theorem. Orthonormal systems, Bessel’s inequality, and Parseval’s identity. Completeness of the Fourier system.</li> <li>Basic properties of Fourier transforms of functions in one variable. Interaction with translation, modulation, scaling, differentiation and convolution. Laplace transforms. Elementary conditions for pointwise convergence of the inverse Fourier transform. Some information on Schwartz functions and Plancherel’s identity.</li> <li>Applications towards classical partial differential equations such as the heat equation, wave equation, and Dirichlet problem in simple domains. Method of separation of variables.</li> </ul>