Ряды Фурье и основы вейвлет-анализа. Фарков Ю.А. - 54 стр.

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|a
m,π(0)
| ≥ |a
m,π(1)
| ≥ ··· ≥ |a
m,π(2
m
−1)
|.
π {0, 1, . . . , 2
m
− 1}
ε > 0
ε
P
m
f ≈
s
X
k=0
a
m,π(k)
ϕ
m, k
, (25)
s
s = min {l | 0 ≤ l ≤ 2
m
− 1 |a
m,π(k)
| < ε k > l}.
f
{a
m,π(k)
| 0 ≤ k ≤ s } {d
j,k
| m −1 ≤ j ≤ n −1, 0 ≤ k ≤ 2
j
− 1 }.
f
a
m, k
ε
m
2
j
g = 2
j
g ≥ 2
m+1
c[1 . . . g]
g = g/2
2
j
i = 1 2
j
/2
c
0
[i] = (c[2i −1] + c[2i])/
√
2
c
0
[2
j
/2 + i] = (c[2i −1] − c[2i])/
√
2
c = c
0
g/2
íóìåðóþò â ïîðÿäêå óáûâàíèÿ èõ àáñîëþòíûõ âåëè÷èí:

                       |am,π(0) | ≥ |am,π(1) | ≥ · · · ≥ |am,π(2m −1) |.
Çäåñü π  áèåêòèâíîå îòîáðàæåíèå ìíîæåñòâà {0, 1, . . . , 2m − 1} íà ñåáÿ (ò.å.
íåêîòîðàÿ ïåðåñòàíîâêà ýòîãî ìíîæåñòâà). Ïîñëå ýòîãî çàäàþò ìàëîå ÷èñëî
ε > 0 è çàìåíÿþò íóëÿìè òå êîýôôèöèåíòû ðàçëîæåíèÿ (24), ìîäóëè êîòîðûõ
ìåíüøå ε.  ðåçóëüòàòå ïîëó÷àåòñÿ àïïðîêñèìàöèÿ:
                                          s
                                          X
                                 Pm f ≈         am,π(k) ϕm, k ,              (25)
                                          k=0

ãäå ÷èñëî s íàõîäèòñÿ èç óñëîâèÿ

            s = min { l | 0 ≤ l ≤ 2m − 1 è |am,π(k) | < ε äëÿ âñåõ k > l}.
Òàêèì îáðàçîì, ôóíêöèÿ f , çàäàííàÿ ôîðìóëîé (19), êîäèðóåòñÿ ñ ïîìîùüþ
íàáîðîâ êîýôôèöèåíòîâ

    { am,π(k) | 0 ≤ k ≤ s } è {dj,k | m − 1 ≤ j ≤ n − 1, 0 ≤ k ≤ 2j − 1 }.
Ïðèáëèæåííîå âîññòàíîâëåíèå ôóíêöèè f âíîâü îñóùåñòâëÿåòñÿ ñ ïîìîùüþ
ôîðìóë (23), íî ïðè ýòîì íà ïåðâîì øàãå ÷àñòü êîýôôèöèåíòîâ am, k çàìåíÿ-
åòñÿ íóëÿìè ïî óêàçàííîìó ε-êðèòåðèþ.
    çàêëþ÷åíèå ïàðàãðàôà ïðèâåäåì ïñåâäîêîäîâûå ïðîöåäóðû (ñì. [17,
ñ.35]), ðåàëèçóþùèå ïðÿìîå è îáðàòíîå äèñêðåòíûå ïðåîáðàçîâàíèÿ Õààðà.
Ïàðàìåòð m âûáèðàåòñÿ êàê â ôîðìóëàõ (21).
procedure Decomposition (c: array [1 . . . 2j ] of reals)
  g = 2j
  while g ≥ 2m+1 do
    DecompositionStep(c[1 . . . g])
    g = g/2
  end while
end procedure;

procedure DecompositionStep (c: array [1 . . . 2j ] of reals)
  for i = 1 to 2j /2 do   √
    c0 [i] = (c[2i − 1] + c[2i])/ 2      √
    c0 [2j /2 + i] = (c[2i − 1] − c[2i])/ 2
  end for
   c = c0
end procedure.
  Îòìåòèì, ÷òî â ïðîöåäóðå DecompositionStep àïïðîêñèìèðóþùèå êîýôôè-
öèåíòû íà êàæäîì øàãå ïîìåùàþòñÿ íà ïåðâûå g/2 ìåñò.

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