Reference no: EM132329733

Question 1 - Define L²_w([0, 1]) = {f ∈ L²([0, 1]) : ₀∫¹|f(x)|²e^2/x dx < ∞}. C^∞ members of L²_w should vanish to infinite order at x = 0.

(a) Show that L²_w is a separable Hilbert space with the natural inner product:

(f, g) = ₀∫¹f(x)(g(x))^-e^2/xdx.

Define T : L²_w → L²_w by

T(f) = 1/x₀∫^xf(t)dt.

(b) Prove that T is bounded and compact (Recall that T is also bounded on L² but it isn't compact on L²!)

(c) provide that if f ∈ L¹([0, 1]), it may be that Tf ∉ L¹ but if f ∈ L¹, then Tf is in weak-L¹.

Question 2 - Assume that f ∈ C(R) satisfies f(x + 1) = f(x) for all x ∈ R. Suppose that α ∈ R\ Q. Then,

lim_n→∞ 1/N _n=1∑^Nf(nα) = ₀∫¹f(x)dx.

What if α α∈ Q?

Question 3 - Let H be a Hilbert space with orthonormal basis {e_i}_i∈N. Let c:= {c_k} be a sequence of non-negative real numbers and associated to it a set Q_c ⊂ H.

Q_c = {∑_k∈Nδ_ke_k ∈ H : |δ_k| ≤ c_k}.

Give a necessary and sufficient condition on c so that Q_c is compact.