Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 38 стр.

UptoLike

ВУЗ: 

РГГРУ | Москва

Составители: 

Рубрика: 

f, g ∈ L
1
(R) f ∗ g ∈ L
1
(R) f ∗ g = g ∗ f
f, g ∈ L
1
(R) g R f ∗g R
f, g, h ∈ L
1
(R) (f ∗ g) ∗ h = f ∗ (g ∗ h)
f ∈ L
1
(R) t
0
∈ R
lim
α→0
(f ∗ g
α
)(t
0
) = f(t
0
),
g
α
f, g ∈ S(R)
f R
supp
b
f ⊂ [−b, b] a < b < 3a
X
k∈Z
f(kT ) sinc (a(t − kT ),
T = π/a g
bg(ξ) =
b
f(ξ) +
b
f(ξ − 2a) +
b
f(ξ + 2a) ξ ∈ [−a, a]
bg(ξ) = 0 ξ /∈ [−3a, 3a]
f ∈ L
2
(R)
lim
m→∞
k
b
f −
b
f
m
k = 0,
b
f
m
f
m
(t) =
f(t), |t| ≤ m,
0, |t| > m.
B {N
m
}
N
1
= χ
[0,1)
, N
m
= N
m−1
∗ N
1
m ≥ 2.
B N
1
N
2
N
3
b
N
m
(ξ) = e
−imξ/2
sin(ξ/2)
ξ/2
m
.
ϕ(t) =
t, 0 ≤ t < 1,
2 − t, 1 ≤ t < 2,
0, t ∈ R \ [0, 2)
X
k∈Z
|bϕ(ξ + 2πk)|
2
=
1
3
+
2
3
cos
2
ξ
2
.
  4. Äîêàæèòå ñëåäóþùèå ñâîéñòâà:
       1) åñëè f, g ∈ L1 (R), òî f ∗ g ∈ L1 (R) è f ∗ g = g ∗ f ;
       2) åñëè f, g ∈ L1 (R) è g íåïðåðûâíà íà R, òî f ∗ g íåïðåðûâíà íà R;
       3) åñëè f, g, h ∈ L1 (R), òî (f ∗ g) ∗ h = f ∗ (g ∗ h);
       4) åñëè ôóíêöèÿ f ∈ L1 (R) íåïðåðûâíà â òî÷êå t0 ∈ R, òî
                                 lim (f ∗ gα )(t0 ) = f (t0 ),
                                 α→0

ãäå gα  ãàóññîâû ôóíêöèè, îïðåäåëåííûå ïî ôîðìóëå (18).
  5. Äîêàæèòå ïðàâèëà (Ï1)  (Ï4) äëÿ ôóíêöèé f, g ∈ S(R).
   6. Ïðåäïîëîæèì, ÷òî ôóíêöèÿ f íåïðåðûâíà íà R, óäîâëåòâîðÿåò óñëîâèþ
(8) è supp fb ⊂ [−b, b], ãäå a < b < 3a. Ïðîâåðüòå, ÷òî òîãäà êàðäèíàëüíûé
ðÿä                         X
                                    f (kT ) sinc (a(t − kT ),
                             k∈Z
ãäå T = π/a, ñõîäèòñÿ ê çíà÷åíèÿì ôóíêöèè g , òàêîé, ÷òî
            gb(ξ) = fb(ξ) + fb(ξ − 2a) + fb(ξ + 2a) äëÿ ξ ∈ [−a, a]
è gb(ξ) = 0 äëÿ ξ ∈
                  / [−3a, 3a].
  7. Ïóñòü f ∈ L2 (R). Äîêàæèòå, ÷òî
                                    lim k fb − fbm k = 0,
                                   m→∞

ãäå fbm  ïðåîáðàçîâàíèÿ Ôóðüå ñðåç-ôóíêöèé
                                        
                                            f (t),      | t| ≤ m,
                            fm (t) =
                                             0,         | t| > m.

  8. Ñèñòåìà íîðìàëèçîâàííûõ B -ñïëàéíîâ {Nm } îïðåäåëÿåòñÿ ôîðìóëàìè
                 N1 = χ[0,1) ,      Nm = Nm−1 ∗ N1 äëÿ m ≥ 2.
Ïîñòðîéòå ãðàôèêè B -ñïëàéíîâ N1 , N2 , N3 è äîêàæèòå ðàâåíñòâî
                                                                 m
                                         −imξ/2        sin(ξ/2)
                         N
                         bm (ξ) = e                                    .
                                                          ξ/2
  9. Ïîêàæèòå, ÷òî äëÿ ïðåîáðàçîâàíèÿ Ôóðüå ôóíêöèè
                                   
                                  t,
                                                    0 ≤ t < 1,
                        ϕ(t) =  2 − t,               1 ≤ t < 2,
                                0,                  t ∈ R \ [0, 2)
âûïîëíåíî ðàâåíñòâî
                                                                   
                      X                  1 2                       ξ
                             b + 2πk)|2 = + cos2
                            |ϕ(ξ                                      .
                                         3 3                       2
                      k∈Z


                                              38

Страницы