Applied Math Qualifier Syllabus May 2006
Course descriptions Math 641 and Math 642
- 641. Analysis for Applications I. (3-0). Credit 3. Review of
preliminary concepts; sequence and function spaces; normed linear
spaces, inner product spaces; spectral theory for compact operators;
fixed point theorems; applications to integral equations and the
calculus of variations. Prerequisites: MATH 447 and 640 or approval of
instructor.
- 642. Analysis for Applications II. (3-0). Credit 3. Distributions
and differential operators; transform theory; spectral theory for
unbounded self-adjoint operators; applications to partial differential
equations; asymptotics and perturbation theory. Prerequisite: MATH
641.
Sequence and function spaces
- Banach spaces. Lp, ℓp, C[a,b],
Ck[a,b]
- Weierstrass approximation theorem
- Hilbert spaces. L2, weighted L2,
ℓ2, Hk[a,b] (Sobolev space)
- Semi-normed vector spaces. D =
C0∞, S (Schwartz space)
- Continuous linear functionals. D′ (distributions),
S′ (tempered distributions)
Normed linear spaces and inner product spaces
- Definitions for Banach spaces, Hilbert spaces and semi-normed spaces.
- Basics of Lebesgue integration.
- Dual spaces. Lp has Lq as dual, 1/p+1/q =
1, 1 ≤ p < ∞ ; Hilbert spaces are self-dual (Riesz
representation theorem).
- Hölder, Minkowski, Schwarz, and triangle inequalities.
- Contraction mapping theorem, Neumann expansions.
- Compact and pre-compact sets relative to various norms
Hilbert space
- Completeness, separable vs. non-separable Hilbert spaces
- Norm, inner product, polarization identity
- Subspaces, closed subspaces, orthogonal complements, dense sets
- Projection theorem (decomposition theorem)
- Least squares, normal equations, best approximant, orthogonal projections
- Finite element method - 1D
- Splines linear, quadratic, cubic
- Gram-Schmidt
- Bessel's inequality, Parseval's identity
- Orthogonal and orthonormal sets, complete orthogonal (or
orthonormal) sets, orthogonal expansions
- Fourier series; pointwise, uniform, and L2 convergence
- Series expansions in orthogonal polynomials Legendre,
Hermite, etc.; L2 convergence
- Riesz representation theorem
- Bounded linear operators, norms, and adjoints
- Compact linear operators and spectral theory
- The operator limit of compact operators is compact.
- Hilbert-Schmidt integral kernels
- Fredholm alternative
- Integral equations
- Fredholm equations of the 1st and 2nd kind
- Volterra equations of the 1st and 2nd kind
- Galerkin approximation
Distributions
- D = C0∞ (test functions) and
its dual D′ (distributions)
- Generalized functions and delta sequences
- Weak solutions to equations
- Derivatives of distributions
- Linear differential operators and Green's functions; boundary
value problems
Calculus of variations
- Variational (Fréchet) derivative of nonlinear functionals
- Euler-Lagrange equations; natural boundary conditions; several
independent variables
- Variational formulation of Newtonian mechanics
- Lagrangians
- Legendre transformations
- Hamiltonians and Hamilton's equations
- Galerkin methods
- Minimax principle and estimating eigenvalues
Special functions
- Contour integration brief discussion
- Integral representation and properties of Gamma and Beta functions
- Bessel functions 1st and 2nd kind, as well as modified;
various integral representations; asymptotics
Unbounded linear operators and spectral theory
- Resolvent set, spectrum, and resolvent operator
- Discrete spectrum
- Continuous spectrum
- Residual spectrum
- Integral transforms derived via the spectral theorem and
resolution of the identity
- Fourier transform and properties
- Sampling theorem
- Uncertainty principle
- Fourier sine and cosine transforms, Mellin transform; inversion
formulas
- Laplace transform and the Bromwich inversion formula
Schwartz space and tempered distributions
- Schwartz space, S
- Definition
- Semi-norm and (equivalent) metric space topologies
- Theorem. S is dense in L2(R).
- Theorem. The Fourier transform of S is S.
- Various useful results: A polynomial × a Schwartz function
is a Schwartz function. One may also multiply by certain
C∞ functions and still have a Schwartz function.
Translates of Schwartz functions are Schwartz functions. Convolutions
of Schwartz functions are Schwartz functions.
- ``Useful'' form of Parseval's Theorem.
∫R f(u)g^(u)du = ∫R
f^(u)g(u)du (f^ and g^ are the Fourier transforms of f and g.)
- Tempered distributions, S′
: Four - Definition and notation
- Derivatives multiples of distributions
- The Fourier transform of a distribution is defined via Parseval's
identity,
∫R T(u)f^(u)du =
∫R T^(u)f(u)du
- Weak temperate convergence
- The Fourier transform takes weakly temperate convergent sequences
into weakly temperate convergent sequences.
- Differentiation takes weakly temperate convergent sequences into
weakly temperate convergent sequences.
- Theorem. The Fourier transform of S′ is
S′.
- Convolutions of distributions, the convolution theorem, and
convolution-type Fredholm equations
Partial differential equations
- Wave equation (1D)
- D'Alembert's solution
- Separation of variables; Helmholtz equation
- Solution via integral transforms
- Green's function
Asymptotics
- Asymptotic formulas, sequences, expansions; big "oh" and little
"oh" notation
- Methods of obtaining asymptotic formulas
- Integration by parts
- Laplace's method; Watson's lemma
- Steepest descent/saddle point method/Stationary phase
- Examples Stirling's formula; asymptotics of Bessel functions
References
- R. A. Adams, Sobolev Spaces, Academic Press, New
York, 1975.
- J. J. Benedetto, Harmonic Analysis and Applications,
CRC Press, Inc., Boca Raton, FL, 1997.
- N. G. de Bruijn, Asymptotic Methods in Analysis,
Dover Publications, New York, 1981.
- A. Erdélyi, Asymptotic Expansions, Dover
Publications, New York, 1956.
- G. B. Folland, Fourier Analysis and Its
Applications, Wadsworth & Brooks/Cole, Pacific Grove, CA,
1992.
- I. M. Gelfand and S. V. Fomin, Calculus of
Variations, Prentice-Hall, Englewood cliffs, NJ, 1963.
- J. P. Keener, Principles of Applied Mathematics:
Transformation and Approximation, Perseus books, Reading, MA,
1995.
- F. Riesz and B. Sz.-Nagy, Functional Analysis, Ungar
Publishing, New York, 1955.
- E. M. Stein, Introduction to Fourier Analysis on Euclidean
Spaces, Princeton University Press, Princeton, NJ, 1971.
- G. P. Tolstov, Fourier Series, Dover Publications,
New York, 1976.
- H. J. Wilcox and D. L. Meyers, An Introduction to Lebesgue
Integration and Fourier Series, Dover Publications, New York,
1994.
Updated 5/16/06 (fjn).