MATH 727 Home Assignment 1
(1) Let f ∈ L1(0, 2π) be represented by the Fourier series f(x) = X n∈Z cneinx
with Fourier coefficients {cn}n∈Z. Show that cn → 0 as |n| → ∞ (the discrete analogue of the Riemann–Lebesgue Lemma).
(2) Consider the diffusion equation ut = uxx on the infinite line subject to the initial condition u|t=0 = f ∈ L2(R). By using Fourier transform and Plancherel’s Theorem, prove that ||u(·, t)||L2 ≤ ||f||L2 for all t ≥ 0. Find and justify the limit of ||u(·, t)||L2 as t → 0 and t → ∞?
(3) Given a function f such that √ 1 + x2f ∈ L2(R) and p 1 + ξ 2 ˆf ∈ L2(R), define the dilation transformation ft(x) := t−1/2f(x/t) for some t > 0. Compute the dispersion of ft and ˆft about the origin versus t and confirm the uncertainty principle for these dilated functions.
(4) Let f ∈ Lp(R), g ∈ Lq(R) for any p, q ∈ (1,∞) such that
1/r := 1/p + 1/q < 1.
Show that fg ∈ Lr(R).
(5) Let Ω be a bounded set in R. For any 1 ≤ p < q < ∞, show that Lq(Ω) is embedded into Lp(Ω) in the sense that there is a constant C that depends only on Ω such that
||u||Lp(Ω) ≤ C||u||Lq(Ω), for all u ∈ Lq(Ω).
(6) Fix f ∈ Lp(R) and g ∈ Lq(R) for some 1 ≤ p, q ≤ ∞ and define the dilations
fa(x) := f(ax), ga(x) := g(ax), a > 0.
Show that the generalized Young’s inequality
||fa ∗ ga||Lr ≤ ||fa||Lp ||ga||Lq cannot hold unless r is given by
1 + 1/r = 1/p + 1/q