Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Past Paper of Mathematical Tripos which includes Combinatorics, Kruska Katona Theorem, Vertex Boundary, Harper’s Inequality, Usual Compression Operator, Construction of Cellular Homology, Intersection Pairing, Differentiable Map etc. Key important points are: Approximation Theory, Jackson Operator, Second Modulus of Smoothness, Bernstein Polynomial, Sequence of Polynomials, Integral Coefficients, Chebyshev Alternation Theorem, Polynomials of Degree
Typology: Exams
1 / 5
This page cannot be seen from the preview
Don't miss anything!
Monday, 4 June, 2012 9:00 am to 12:00 pm
Attempt no more than FOUR questions. There are SIX questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
Let jn be the Jackson operator, i.e., for a 2π-periodic function f from C(T),
jn(f, x) :=
∫ (^) π
−π
f (x − t)Jn(t) dt, Jn(t) :=
2 πn(2n^2 + 1)
sin^4 nt 2 sin^4 2 t
∫ (^) π
−π
Jn(t) dt = 1.
Prove that, for any f ∈ C(T), we have the estimate
‖jn(f ) − f ‖ 6 c ω 2 (f, (^) n^1 ) ,
where ω 2 (f, t) is the second modulus of smoothness of f.
Hence, derive that if f is twice continuously differentiable, then
En(f ) 6 c 1 n^2 ‖f ′′‖C(T).
a) For f ∈ C[0, 1], write down the definition of the Bernstein polynomial Bn(f ) of degree n, and prove that ‖Bn(f )‖∞ 6 ‖f ‖∞.
b) For a function f ∈ C[0, 1] that takes integer values at x = 0 and x = 1, and for the sequence of polynomials
B n∗(f, x) :=
∑^ n
k=
n k
f
k n
xk(1 − x)n−k^ ,
prove that ‖Bn(f ) − B∗ n(f )‖∞ → 0 as n → ∞. Here, ⌊t⌋ is the largest integer not bigger than t.
c) Hence show that a function f ∈ C[0, 1] is approximable by polynomials with integral coefficients if and only if f (0) and f (1) are integers.
Part III, Paper 70
Let (Ni) and (Mi) be the B-splines bases of degree k − 1 with L∞- and L 1 - normalization, respectively, defined on a knot sequence ∆ = (ti)ni=1+k ⊂ [0, 1].
Given f ∈ C[0, 1], let
PS (f ) := s∗^ =
∑^ n
j=
aj Nj
be the orthogonal projection of f onto S := span (Nj ) with respect to the ordinary inner product (f, g) =
0 f^ (x)g(x)^ dx. Then^ PS^ is also well defined as an operator from^ C[0,^ 1] onto C[0, 1].
a) Show that the max-norm of PS satisfies the inequality
‖PS ‖∞ 6 ‖G−^1 ‖ℓ∞ ,
where G = (gij ) is the Gram matrix with the elements gij = (Mi, Nj ).
b) For linear splines (k = 2) and equidistant ∆, with ti+1 − ti = h for all i, compute the entries of G.
c) Using the fact that G is totally positive, or otherwise, prove the estimate
‖G−^1 ‖ℓ∞ 6 3 , k = 2.
[You may use any appropriate theorems on the inverse of certain matrices if correctly stated.]
Part III, Paper 70
a) Define a multiresolution analysis of L 2 (R) with a generator φ and explain how it is related to existence of an orthonornal wavelet ψ.
b) Prove that the following properties of φ
n
anφ(2x − n), 2) {φ(· − n)}n∈Z is an orthonormal sequence
are equivalent to
1 ′) f (2t) = m(t)f (t) , m(t) = (^12)
n
ane−int,
|f (t + 2πk)|^2 ≡ 1 a.e.
where f is the Fourier transform of φ, i.e., f (t) = φ̂ (t) =
R φ(x)e
−ixt (^) dx.
c) Verify that conditions 1′) − 2 ′) are fulfilled for the function f = φ̂ defined as
f =
1 , t ∈ [−π, π) 0 , otherwise.
Using the inverse Fourier transform or otherwise, determine the corresponding generator φ.
Part III, Paper 70
