FA 2017/6/20 Both sides of the answer sheet can be used.
1 Let f(t) be a tent function of t ∈ R defined by f (t) =
{
1 − | t | if | t | ≤ 1, 0 otherwise.
With help of Gaussian approximations, show that f is uniformly ap- proximated by entire functions.
2 Consider vectors
v =
1 1 1 1
, w =
1 i 1 i
and the subspace E = C v + C w of C4 spanned by them.
(i) Apply the Gram-Schmidt orthogonalization to { v, w } . (ii) Express the orthogonal projection to E as a 4 × 4 matrix.
3 Describe the Fr´ echet-Riesz theorem and explain the meaning of self- duality on Hilbert spaces.
Glossary:
uniform approximation
一様近似entire function
整関数subspace
部分空間Gram-Schmidt orthogonalization
グラム・シュミットの直交化orthogonal projection
正射影self-duality
自己双対性1
