SET 3
MATH 543: ORTHOGONAL POLYNOMIALS
1. Generating Functions: For the classical orthogonal polynomials Cn (x)
we have seen so far there exists a generating function g(x, t) for each dened
by
an tn Cn (x),
g(x, t) =
(1)
n=0
where an ,s are some real numb
SET 4
MATH 543: GENERALIZED FUNCTIONS
1. A function g(x) that is dierentiable everywhere any number of times
is called a good function if it and its derivatives vanish as |x| faster
1
than any power of |x| . A function f (x) that is dierentiable everywher
SET 2
MATH 543: ORTHOGONAL FAMILIES AND BASIS
1. Prove that the Fourier coecients of any |f > L2 (a, b) form a Hilbert
w
space. First prove that the space of such coecients form an inner product
space then prove that this inner product space is complete.
SET 6
MATH 543: FROBENIUS METHOD
References: Hildebrand and Sadri Hassan.
The following theorem summarizes the Frobenius method: (Proved in Class
please see your lecture notes and also DK)
Theorem 1. Suppose that the dierential equation Lu = 0, where L is
SET 1
MATH 543: INNER PRODUCT SPACE
Some preliminaries from Haaser and Sullivan:
LINEAR SPACES
Denition 1: A linear (vector) space X over a eld F is a set of elements
together with a function , called addition, from X X into X and a function
, called scal
MATH 543
METHODS OF APPLIED MATHEMATICS I
First Midterm Exam
November 09, 2004
Thursday 18.00-20.00, SAZ-18
QUESTIONS: Choose any three out of the following four problems
1a. Prove that any orthonormal family |ei >, i = 1, 2 in L2 (a, b) is
w
linearly ind
MATH 543
METHODS OF APPLIED MATHEMATICS I
First Homework Set
For September 28, 2009
QUESTIONS
1. In an inner product space, we dene the norm of an element x of a vector
space X to be |x| = < x, x > (we use the notation of the lecture notes).
Prove the fol