Analysis for Applications I (MATH 641)

by Dr. Francis J. Narcowich

  • Review of preliminary concepts
         – Normed linear spaces and inner product spaces
         – Subspaces, orthogonal complements, Gram-Schmidt procedure, projection operator
         – Self-adjoint matrices and their properties, eigenvalue estimates: Rayleigh-Ritz maximum principle and Courant-Fischer Minimax Theorem

  • Banach and Hilbert spaces
         – Convergent sequence, Cauchy sequence, completeness
         – Lebesgue measure and integral, measurable functions, Monotone and Dominated Convergence Theorems
         – Special (complete) spaces: ell^p, L^p for 1leq p < infty, C^k [a,b], Sobolev space mathcal{H}^n [a,b], Sobolev-type inequalities
         – C[a,b] and its properties: modulus of continuity, density of splines, density of polynomials in C[a,b] (Weierstrass Approximation Theorem), density of
             polynomials in L^p for 1leq p < infty
         – Hilbert spaces: least-square minimization (discrete and continuous), normal equations, Bessel's inequality, orthogonal polynomials and completeness

  • Approximation tools
         – Fourier series, application of Parseval's identity, Discrete Fourier Transform, FFT
         – Shannon-Nyquist Sampling Theorem, sinc functions
         – Mallat's multi-resolution analysis (MRA), scaling function and wavelets, two-scale relation, Haar wavelets, Daubechies wavelets
         – Finite elements, spline spaces mathcal{S}^h(k,r) and B-splines N_m(x), finite element method for boundary value problems

  • Linear operators and integral equations
         – Bounded operators: norms of linear operators, unbounded operators, continuous linear functionals, spaces associated with operators, projection theorem
         – Integral equations: Hilbert-Schmidt kernels, Fredholm kernel, Volterra Kernel
         – Riesz Representation Theorem, adjoint of operators, weak form of a boundary value problem
         – Compact operators: finite rank operators, approximation theorem, Hilbert-Schmidt operators, closed range theorem, Fredholm alternative theorem
         – Spectral theory of compact operators: eigenvalues and eigenspaces, completeness of eigenfunctions, application of eigenfunctions in solving integral equations

  • Distributions and applications
         – Test function space mathcal{D}, distribution space mathcal{D}^{prime}, Dirac delta distribution, integral representation, derivatives of distributions
         – Green's funtions and their applications