Лекции по курсу Математическая физика
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sin
[0,
π
2
]
sin(x) =
+∞
X
k=0
(−1)
k
x
2k+1
(2k + 1)!
.
n
∆
n
= max
x∈[0,
π
2
]
¡
|sin(x) − T
sin
(n, x)|
¢
≤ ε
ε
T
sin
[0,
π
2
]
∆
n
∆
1
= 1 S
1
(x) ≡ 0.5 ∆
1
= 0.5
|S
1
(x) −sin(x)|
T
sin
(2, x) = x
x =
π
2
∆
2
=
π
2
− 1 ≈ 0.5708
S
2
(x) = a + b · x
a =
1
2
sin
³
arccos
³
2
π
´´
−
1
π
arccos
³
2
π
´
, b =
2
π
.
0
π
2
0
π
2
sin(x) T
sin
(2, x)
0
1
0
π
2
S
2
(x)
−a
a
π
2
sin(x) − S
2
(x)
x
0
= 0 x
1
= arccos(b) x
2
= π/2 ∆
2
= a ≈ 0.1053
S
1
S
2
sin [0,
π
2
]
[α, β] f
n ∈ N S
n
n
∆
n
= max
[α,β]
¡
|S
n
(x) − f(x)|
¢
n f [α, β]
∆
n
→ 0
n → +∞
[α, β]
n + 1 x
0
, x
1
, . . . , x
n
, |S
n
(x
k
) −f(x
k
)| = ∆
n
S
n
(x
k
) − f(x
k
) x
0
, x
1
, . . . , x
n
[0,
π
2
].
∆
10
≈ 3.3 × 10
−11
∆
10
≈ 3.6 ×10
−6
0
π
2
sin(x) − S
10
(x)
[α, β],
kfk
C
= max
x∈[α,β]
¡
|f(x)|
¢
∆
n
= kf − S
n
k
C
.
[α, β]
hϕ, ψi =
β
Z
α
ϕ · ψ. (2.2.1)
kϕk =
p
hϕ, ϕi. (2.2.2)
hφ, ψi
r
=
β
Z
α
rφψ. (2.2.3)
r
r(x) ≡ 1
T > 0
h
−
T
2
,
T
2
i
ϕ
k
(x) = exp
³
i
2πk
T
x
´
; k ∈ Z.
δ
km
hϕ
k
, ϕ
m
i =
T/2
Z
−T/2
exp
³
i
2π(k − m)
T
x
´
dx = T · δ
km
.
exp
¡
i
2πk
T
x
¢
sin
¡
2πk
T
x
¢
cos
¡
2πk
T
x
¢
[0, 1]
r
0
(x) ≡ 1; r
k
(x) = sign
¡
sin(2
k
πx)
¢
, k ∈ N .
x
x
x
r
1
(x)
r
0
(x)
r
2
(x)
N Z
hr
k
, r
m
i =
1
Z
0
r
k
(x)r
m
(x)dx = δ
km
.
[−1, 1]
P
0
(x) ≡ 1; P
k
(x) =
1
2
k
· k!
·
¡
(x
2
− 1)
k
¢
(k)
, k ∈ N.
P
k
k
hP
k
, P
m
i =
1
Z
−1
P
k
(x)P
m
(x)dx =
2
2k + 1
· δ
km
.
[−1, 1]
T
k
(x) = cos
¡
k · arccos(x)
¢
, k = 0, 1, . . .
T
k
k
r(x) =
1
p
1 − x
2
.
t = arccos(x)
hT
k
, T
m
i
r
=
1
Z
−1
1
p
1 − x
2
T
k
(x)T
m
(x)dx =
π
Z
0
cos(kt)cos(mt)dt =
=
π
Z
0
1
2
¡
cos((k − m)t) + cos((k + m)t)
¢
dt.
k 6= m.
f [α, β]
(ϕ
k
)
[α, β].
n
f
n a
1
, . . . , a
n
°
°
°
f −
n
X
k=1
a
k
ϕ
k
°
°
°
≤
°
°
°
f −
n
X
k=1
b
k
ϕ
k
°
°
°
b
1
, . . . , b
n
.
°
°
°
f −
n
X
k=1
b
k
ϕ
k
°
°
°
2
=
D
f −
n
X
k=1
b
k
ϕ
k
, f −
n
X
k=1
b
k
ϕ
k
E
=
= hf, fi −
D
f,
n
X
k=1
b
k
ϕ
k
E
−
D
n
X
k=1
b
k
ϕ
k
, f
E
+
D
n
X
k=1
b
k
ϕ
k
,
n
X
k=1
b
k
ϕ
k
E
.
hϕ
k
, ϕ
m
i = 0 m 6= k
°
°
°
f −
n
X
k=1
b
k
ϕ
k
°
°
°
2
= kfk
2
−
n
X
k=1
b
k
hf, ϕ
k
i −
n
X
k=1
b
k
hϕ
k
, fi +
n
X
k=1
b
k
b
k
kϕ
k
k
2
.
b
f
k
=
hf, ϕ
k
i
kϕ
k
k
2
°
°
°
f −
n
X
k=1
b
k
ϕ
k
°
°
°
2
= kfk
2
+
n
X
k=1
¡
−b
k
b
f
k
− b
k
b
f
k
+ b
k
b
k
¢
kϕ
k
k
2
=
= kfk
2
−
n
X
k=1
|
b
f
k
|
2
kϕ
k
k
2
+
m
X
k=1
|
b
f
k
− b
k
|
2
kϕ
k
k
2
.
b
1
, . . . , b
n
b
k
=
b
f
k
k = 1, . . . , n
°
°
°
f −
n
X
k=1
b
f
k
ϕ
k
°
°
°
2
= kfk
2
−
n
X
k=1
|
b
f
k
|
2
kϕ
k
k
2
. (2.3.1)
b
f
k
=
hf, ϕ
k
i
kϕ
k
k
2
f ϕ
k
n
P
k=1
|
b
f
k
|
2
kϕ
k
k
2
n
X
k=1
|
b
f
k
|
2
kϕ
k
k
2
≤ kfk
2
, (2.3.2)
P
+∞
k=1
|
b
f
k
|
2
kϕ
k
k
2
(n = +∞)
+∞
X
k=1
|
b
f
k
|
2
kϕ
k
k
2
≤ kfk
2
.
f
+∞
X
k=1
|
b
f
k
|
2
kϕ
k
k
2
= kfk
2
lim
n=+∞
°
°
°
f −
n
X
k=1
b
f
k
ϕ
k
°
°
°
2
= 0.
+∞
X
k=1
b
f
k
ϕ
k
= f
+∞
X
k=1
hf, ϕ
k
i
kϕ
k
k
2
· ϕ
k
, (2.3.3)
f
(ϕ
k
) f
f
f
f
[α, β]
(ϕ
k
)
ϕ
m
f = ϕ
m
n
[α, β] (ϕ
k
)
[α, β]
f
f
(ϕ
k
)
ϕ
k
hf, ϕ
k
i = 0 k ∈ N
f
kfk = 0
cos(2πx) [0, 1]