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MATH 619, Fall 2022, First Midterm

— Questions —

1.  By definition, we say a function u : R2  → R is a weak solution to the Wave Equation if u is continuous, and if

(⋆)                     u(x + h,t) + u(x − h,t) = u(x,t + h) + u(x,t − h)

holds for all x,t ∈ R and all h > 0.

We say u is a classical solution if u is C2 and if utt (x,t) = uxx (x,t) holds for all x,t ∈ R.

(a) Give an example of a weak solution that is not a classical solution, and explain your example.

(b) Prove, or give a counterexample to the following statement: if un   : R2   → R is a sequence of classical solutions to the Wave Equation, and if un converges uniformly to u : R2  → R then u is also a classical solution to the Wave Equation.

2.  Suppose f : R → C is a Riemann integrable 2π-periodic function.

(a) Show that if f is C1 then limk →±∞ kfˆk  = 0.

(b) Prove, or give a counterexample: if the Fourier coefficients of f satisfy |fˆk | ⩽  for all k  0, then f is C2 .

3.  Let n , n  ∈ C be two sequences of complex numbers for which

∞

工  |k | + |k | < ∞ .

k= −∞

Show that the series

∞

u(x,t) =   工  k eikx eikt + k eikx e−ikt

k= −∞

converges uniformly on R2 .

(b) Show that u(x,t) is a weak solution to the Wave Equation.