Math 118 :: Winter 2009 :: Homework 3 :: Due February 26
1. Show (rigorously) that the unit step function has TV norm equal to one on any interval
including the discontinuity jump.
2. (Uniform norm of interpolation.) The uniform norm of interpolation, for data {(xi , yi ), i =
0, . . . , N } and an interpolant p(x), is the smallest number CN such that
kp(x)k∞ ≤ CN max |yi |.
i
(1)
(The subscript N indicates that CN may depend on N .) The ∞ norm in the left-hand side is
the uniform norm taken over the interval [x0 , xN ]. Such an estimate is helpful to determine
the perturbative effect of noise in the data, on the interpolant.
(a) Determine CN for piecewise linear splines.
(b) Consider now a generic interpolation method, defined from data yj on an equispaced
grid xj = jh by a kernel k(x) such that
p(x) =
N
X
yj k(x − xj ).
j=1
(Disregard edge effects.) Justify that
CN = sup
N
X
|k(x − xj )|.
(2)
x∈[0,h] j=1
[Hint: in order to show that CN is the smallest possible constant in equation (1) above,
exhibit a choice of yj that makes (1) an equality.]
(c) For bandlimited interpolation on an unbounded equispaced grid, the kernel is given by
equation (2.8) in Trefethen’s book, see also equation (2.10). It’s the sinc function. Show
(rigorously) that for the sinc kernel, CN ≥ c log N + d for some c > 0 and d ∈ R, and
detail the constants c and d.
(d) For bandlimited interpolation on a periodic equispaced grid, the kernel is now given by
(3.7) in Trefethen’s book. It’s the periodized sinc function. How would you modify your
answer to (c) to treat the periodized sinc kernel?
(e) Give an estimate of the uniform interpolation norm for Chebyshev interpolation. [Hint:
use part (d)].
(f) (Bonus.) Give an estimate of the uniform interpolation norm for polynomial interpolation on an equispaced grid. [Hint: use potential theory. The more accurate your answer
the better bonus you get.]
3. Formulate a Plancherel theorem for expansion in Chebyshev polynomials, i.e., a result that
links the L2 norm in x ∈ [−1, 1] to the `2 norm in n ≥ 0, the index of the Chebyshev
polynomial. [Hint: in adapting the result from Fourier theory, watch for the Jacobian factor.
In addition, notice that the interval in x is [−1, 1] and not [−π, π].]
4. (Why quadratic splines are bad.) A quadratic spline is a piecewise quadratic interpolant,
imposed to have one continuous derivative at the knots (junction points) xi . For data {xi , yi },
it is given in the interval [xi , xi+1 ] by the formula
si (x) = yi + zi (x − xi ) +
zi+1 − zi
(x − xi )2 ,
2(xi+1 − xi )
where the zi obey the recurrence relation
zi+1 = −zi + 2
yi+1 − yi
.
xi+1 − xi
There is only one degree of freedom that the interpolation and continuity conditions do not
specify: the value of z0 .
Consider equispaced points xi = i for i ≥ 0, and corresponding data
y0 = 1,
y1 = 0,
yk = 1
for k ≥ 2.
Take z0 = 0.
(a) Compute zi , i ≥ 1.
(b) Plot the quadratic spline in the first four intervals, between x0 and x4 .
(c) What is the interpolation norm CN , in the sense of the previous exercise?
(d) Would you recommend quadratic splines as a practical interpolation method? Why not?
5. For computing the second derivative on an equispaced grid, one usually uses the finite difference formula
un+1 − 2un + un−1
.
wn =
h2
(a) Establish the order of approximation to the second derivative.
(b) Argue that applying centered differences twice does not give rise to the same formula.
Which one would you choose in practice and why? (Of course one always reserves the
right to revisit this decision later when faced with a particular problem, but a priori,
which method is better?)