Abhishek Halder

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: for , , Sobolev space $mathcal{H}^n [a,b]$ , Sobolev-type inequalities
     – and its properties: modulus of continuity, density of splines, density of polynomials in (Weierstrass Approximation Theorem), density of
         polynomials in for
     – 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 , 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 , distribution space $mathcal{D}^{prime}$ , Dirac distribution, integral representation, derivatives of distributions
– Green's funtions and their applications

Analysis for Applications I (MATH 641)