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, GramSchmidt procedure, projection operator
– Selfadjoint matrices and their properties, eigenvalue estimates: RayleighRitz maximum principle and CourantFischer 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: for , , Sobolev space , Sobolevtype inequalities
– and its properties: modulus of continuity, density of splines, density of polynomials in (Weierstrass Approximation Theorem), density of polynomials in for
– Hilbert spaces: leastsquare 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
– ShannonNyquist Sampling Theorem, sinc functions
– Mallat's multiresolution analysis (MRA), scaling function and wavelets, twoscale relation, Haar wavelets, Daubechies wavelets
– Finite elements, spline spaces and Bsplines , 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: HilbertSchmidt 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, HilbertSchmidt 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 , distribution space , Dirac distribution, integral representation, derivatives of distributions
– Green's funtions and their applications
