(1) f. (2) f. (Q n ; n Z >0 ) f Q j =sup{ f(x) Q j (x) ; x I} 0(n ) (4) f I. a n (x c) n (1) (2) Q 1 = P 1,Q j = P j P j 1 (j 2) (3) (4) (2) (3)

(1)

ίϯύΫτू߹্ͷ࿈ଓവ਺ͷ੒ۭؒ͢ͷ᜚ີ෦෼ू߹

ฏ੒ 26 ೥ 8 ݄ খᖒɹప http://www.ozawa.phys.waseda.ac.jp/index2.html

ίϯύΫτू߹্ͷ࿈ଓവ਺͕ଟ߲ࣜͷҰ༷ऩଋۃݶͱͯ͠ಛ௃෇͚ΒΕΔঢ়گʹ͍ͭͯ

వΊͯஔ͜͏ɻ

̍ɽ༗քด্۠ؒͷ࣮਺஋࿈ଓവ਺ɹ

ఆཧ̍ʢϫΠΤϧετϥεʣɹ༗քด۠ؒ I = [a, b]⊂ R ্ͷ࣮਺஋വ਺ f : I → R ʹର͠

࣍͸ಉ஋Ͱ͋Δɻ (1) f ͸࿈ଓͰ͋Δ

(2) f ͸Ұ༷࿈ଓͰ͋Δɻ

(3) f ͸ଟ߲ࣜͷྻͷ I ্ͷҰ༷ऩଋۃݶͰ͋Δɻଈͪɺଟ߲ࣜͷྻ

(P_n; n∈ Z>0) ͕ଘࡏ͠f − Pn∞ = sup{|f(x) − Pn(x)|; x ∈ I} → 0(n → ∞) ͱͳΔɻ (4) f ͸ଟ߲ࣜΛ߲ͱ͢Δແݶڃ਺ͷ I ্ͷҰ༷ऩଋۃݶͰ͋Δɻଈͪɺଟ߲ࣜͷྻ

(Q_n; n ∈ Z>0) ͕ଘࡏ͠f −

n j=1

Q_j∞ = sup{|f(x) −

n j=1

Q_j(x)|; x ∈ I} → 0(n → ∞) ͱͳΔɻ

மɹ (4) ͸ f ͕ I ͷҰͭͷ಺఺ c Λத৺ͱ͢Δ୯߲ࣜ a_n(x− c)ⁿΛ߲ͱ͢Δແݶڃ਺

∞ n=0

a_n(x− c)ⁿ ʢͷҰ༷ऩଋۃݶʣͱͯ͠ද͞ΕΔͱओு͍ͯ͠Δ༁Ͱ͸ͳ͍ɻ΋ͦ͠͏ͳ Β f ͸ IntI ্࣮ղੳͱͳΓ C^∞ͱͳͬͯ͠·͍ໃ६Ͱ͋Δɻ

ʢূ໌ʣɹίϯύΫτͳڑ཭্ۭؒͷ࿈ଓവ਺͸Ұ༷࿈ଓͰ͋Δ͔Β (1) ⇔ (2) ͕ै͍ɺଟ߲

ࣜ͸࿈ଓͰ͋Δ͔Β (3)⇒ (1) ͕ै͍ɺଟ߲ࣜͷ༗ݶ࿨͸ଟ߲ࣜͰ͋Δ͔Β (4) ⇒ (3) ͕ै͍

Q₁ = P₁, Q_j = P_j − Pj−1(j ≥ 2) ͱ͢Ε͹ (3) ⇒ (4) ͕ै͏ɻނʹ (2) ⇒ (3) Λࣔͤ͹ྑ͍ɻ

೚ҙͷ ε > 0 ʹର͠ δ > 0 ͕ଘࡏ͠|x − y| < δ ͳΔ೚ҙͷ x, y ∈ I ʹର͠ |f(x) − f(y)| < ε ͕

੒ཱͭɻͯ͞ 0 <|Δ| ≡ max

1≤j≤m(x_j − x_j−1) < δ ͳΔ I ͷ෼ׂ Δ : a = x₀ < x₁ <· · · < x_m = b ΛҰͭऔΓ I_j = [x_j−1, x_j],

f_m(x) = f (a) +

m−1

j=0

(d_j− d_j−1) max(x− x_j, 0), x∈ I, d_j = (f (x_j+1)− f(xj))/(x_j+1− xj), j ≥ 0; d−1 = 0

(2)

ͱஔ͘ɻ͜ͷͱ͖ x∈ Ikʹର͠౳ࣜ

f_m(x) = f (a) +

k−1 j=0

(d_j− d_j−1)(x− x_j)

= f (a) +

k−1 j=0

d_j(x− xj)−

k−2 j=0

d_j(x− xj+1)

= f (a) +

k−2 j=0

d_j(x_j+1− xj) + d_k−1(x− xk−1)

= f (x₀) +

k−2 j=0

(f (x_j+1)− f(x_j)) + d_k−1(x− x_k−1)

= f (x_k−1) + f (x_k)− f(xk−1)

x_k− x_k−1 (x− xk−1)

͕੒ཱͭɻ͜ΕΑΓ x∈ Ikʹର͠

f (x)− fm(x) = f (x)−

x− xk−1

x_k− x_k−1f (x_k) + x_k− x

x_k− x_k−1f (x_k−1)

= x− x_k−1

x_k− xk−1(f (x)− f(xk)) + x_k− x

x_k− xk−1(f (x)− f(xk−1)) ͱͳΔͷͰ

|f(x) − f_m(x)| ≤ sup

0≤θ≤1

(θ|f(x) − f(x_k)| + (1 − θ)|f(x) − f(x_k−1)|)

≤ sup

0≤θ≤1

(θ sup

ξ,η∈Ik

|f(ξ) − f(η)| + (1 − θ) sup

ξ,η∈Ik

|f(ξ) − f(η)|)

= sup

ξ,η∈Ik

|f(ξ) − f(η)|

ΑΓ

f − f_m_∞ = max

1≤k≤msup

x∈Ik

|f(x) − f_m(x)|

≤ max

1≤k≤m sup

ξ,η∈Ik

|f(ξ) − f(η)| ≤ ε

ΛಘΔɻͯ͞

max(x− x_j, 0) = 1

2((x− x_j) +|x − x_j|)

ΑΓ֤ j ʹର͠τxj| · | − Qⁿ_j∞ → 0(n → ∞) ͳΔଟ߲ࣜͷྻ (Qⁿ_j; n ∈ Z>0) ͕ଘࡏ͢Ε͹

ʢୠ͠ τ_x_j|x| = |x − xj|, x ∈ I ͱ͢Δʣ

P_n(x) = f (a) +

m−1

j=1

(d_j − d_j−1)1

2((x− x_j) + Qⁿ_j(x))

= f_m(x) +

m−1

j=1

(d_j− dj−1)(Qⁿ_j(x)− |x − xj|)

(3)

Ͱఆ·Δଟ߲ࣜ P_n͸

f − Pn∞ ≤ f − fm∞+

m−1

j=1

|dj − dj−1Qⁿ_j − τxj| · |∞

ΑΓ

lim sup

n→∞ f − P_n_∞≤ f − f_m_∞ ≤ ε

Λຬͨ͢ͷͰ (2)⇒ (3) ͕ࣔ͞ΕΔࣄͱͳΔɻͦ͜Ͱ࣍ͷิ୊ΛऔΓ্͛Δɻ

ิ୊̍ɹઈର஋ʹ෇ਵ͢Δവ਺ x→ |x| ͸༗քด۠ؒ [−1, 1] ্Ͱଟ߲ࣜͷҰ༷ऩଋۃݶͱ͠

ͯද͞ΕΔɻଈͪଟ߲ࣜͷྻ (P_n; n∈ Z_>0) ͕ଘࡏ͠

| · | − Pn∞ = sup{||x| − Pn(x)| ; |x| ≤ 1} → 0(n → ∞) ͱͳΔɻ

ิ୊̍Λࣔͤ͹ॆ෼Ͱ͋Δ͜ͱɿɹ্ͷ P_nʹର͠ Qⁿ_j(x) = (b− a)Pn_x−x_j

b−a

ٴͼ ξ = ^x−x_b−a^j ͱஔ͚͹|ξ| ≤ 1 Ͱ x = x_j + (b− a)ξ ͱͳΓ

τxj| · | − Qⁿ_j∞= sup{||x − xj| − Qⁿ_j(x)|; a ≤ x ≤ b}

≤ sup{||(b − a)ξ| − Qⁿ_j(x_j+ (b− a)ξ)|; |ξ| ≤ 1}

= (b− a) sup{ξ| − Pn(ξ)|; |ξ| ≤ 1} → 0 (n → ∞)

͕ै͏ɻ

ิ୊̍ͷূ໌ʢͦͷ̍ʣɿɹ|t| ≤ 1 ͳΔ t ʹର͠ (1 − t)^1/2ͷϚΫϩʔϦϯల։Λߟ͑Δɿ (1− t)^1/2 = 1−

∞ n=1

(2n− 3)!!

(2n)!! tⁿ= 1 +

∞ n=1

−1 2

1 2

· · ·

2n− 3 2

tⁿ n!

a_n= ^(2n−3)!!_(2n)!! ͱ͢Ε͹ ^aⁿ

an−1 = ^(2n−3)!!_(2n−5)!!^(2n−2)!!_(2n)!! = ²ⁿ⁻³_2n = 1− _2n³ Ͱ͋Δ͔ΒΨ΢εͷ൑ఆ๏ʹڌ Γӈลͷແݶڃ਺͸ [−1, 1] ্Ұ༷ʹઈରऩଋ͢Δɻt = 1 − x²ͱ͢Ε͹ [−1, 1] ্

|x| = 1 −

∞ n=1

(2n− 3)!!

(2n)!! (1− x²)ⁿ

͕੒ཱͪӈลͷແݶڃ਺͸ [−1, 1] ্Ұ༷ʹઈରऩଋ͢Δɻ

ิ୊̍ͷূ໌ʢͦͷ̎ʣɿɹଟ߲ࣜͷྻ (p_n; n ∈ Z≥0) Λ p₀(x) = 0, n ≥ 1 ʹର͠ pn(x) = p_n−1(x)+¹₂(x−pn−1(x)²) ͱؼೲతʹఆٛ͢Δɻ͜ͷͱ͖೚ҙͷ x∈ [−1, 1] ٴͼ೚ҙͷ n ∈ Z≥0

ʹର͠ෆ౳ࣜ

0≤ |x| − pn(x²)≤ 2|x|

2 + n|x|

͕੒ཱͭࣄΛ n ʹؔ͢Δؼೲ๏Ͱࣔͦ͏ɻn = 0 ͷ৔߹͸ޙ൒ͷෆ౳ࣜ͸౳ࣜͱͯ͠੒ཱͭɻ n− 1 ͷ৔߹ɺଈͪෆ౳ࣜ

0≤ |x| − p_n−1(x²)≤ 2|x|

2 + (n− 1)|x|

(4)

ΛԾఆ͢Δɻͯ͞

|x| − p_n(x²) = |x| − p_n−1(x²)−1

2(x²− p_n−1(x²)²)

= (|x| − p_n−1(x²))(1− 1

2(|x| + p_n−1(x²)))

= (|x| − pn−1(x²))(1− |x| + 1

2(|x| − pn−1(x²)))

ͷ࠷ޙͷ౳ࣜͷӈลͷ֤߲͸ؼೲ๏ͷԾఆͷલ൒ͷෆ౳ࣜΑΓશͯඇෛͱͳΔͷͰ|x| − p_n(x²)≥ 0 ͕ै͏ɻҰํɺؼೲ๏ͷԾఆͷޙ൒ͷෆ౳ࣜΑΓ

|x| − p_n(x²)≤ 2|x|

2 + (n− 1)|x| ·

1− |x| + |x|

2 + (n− 1)|x|

= 2|x|

2 + (n− 1)|x| ·2 + (n− 2)|x| − (n − 1)x² 2 + (n− 1)|x|

≤ 2|x|(2 + (n − 2)|x|)

(2 + (n− 1)|x|)² ≤ 2|x|(2 + (n − 2)|x|)

(2 + (n− 1)|x|)²− |x|² = 2|x|

2 + n|x|

͕ै͏ͷͰؼೲ๏͸׬݁͢Δɻͦ͜Ͱ P_n(x) = p_n(x²) ͱஔ͚͹| · | − Pn∞≤ 2/n ͱͳΓิ

୊͕̍ै͏ɻ

மɿɹଟ߲ࣜͷྻ (p_n; n ∈ Z≥0) ͸ฏํࠜ [0, 1] x → √

x∈ R ͷۙࣅྻͰ͋Γ (Φ(p))(x) = p(x) + ¹

2(x− p(x)²) Ͱఆ·Δࣸ૾

Φ : C([0, 1];R) → C([0, 1]; R) ͷෆಈ఺ x→√

x ʹऩଋ͢ΔϐΧϧͷஞ࣍ۙࣅྻͰ͋Δɻ

ิ୊̍ͷূ໌ʢͦͷ̏ʣɿɹ Kuhn ʹैͬͯ P_n(x) = x

1− 2(1 −_x+1

2

₂ⁿ

ͱ͢Ε͹ྑ͍ࣄ Λࣔͦ͏ɻҎԼɺϕϧψΠͷෆ౳ࣜ (1 + x)ⁿ ≥ 1 + nx(x ≥ −1, n ∈ Z>0) Λ༻͍Δɻ؆୯ͷ ҝ Q_n(x) = (1− xⁿ)²ⁿͱஔ͘ɻx∈ (0, 1] ʹର͠

|x| − P_n(x) = x− P_n(x) = x

1−

1− 2Q_n

x + 1 2

= 2xQ_n

x + 1 2

≥ 0

Ͱ͋Δ͔Βɺ౳ࣜ

x| − Pn(x)| = 2xQn

x + 1 2

͕ै͏ɻx∈ [−1, 0) ʹର͠

|x| − Pn(x) =−x − Pn(x) =−x

1 +

1− 2Qn

x + 1 2

= 2|x|

1− Qn

x + 1 2

≥ 0

(5)

Ͱ͋Δ͔Βɺ౳ࣜ

x| − Pn(x)| = 2|x|

1− 2Qn

x + 1 2

͕ै͏ɻ0 < δ < 1 ͳΔ೚ҙͷ δ ΛऔΔɻx ∈ [−1, −δ] ʹର͠ 0 ≤ _x+1

2

_n

≤ _1−δ

2

_n

Ͱ͋Δ

͔Β

1≥ Qn

x + 1 2

=

1−

x + 1 2

_n₂ⁿ

≥ 1 − 2ⁿ

x + 1 2

_n

≥ 1 − (1 − δ)ⁿ

ͱͳΓ

sup{||x| − P_n(x)|; x ∈ [−1, δ]} = sup{2|x|

1− Q_n

x + 1 2

; x∈ [−1, −δ]}

≤ 2(1 − δ)ⁿ

͕ै͏ɻx∈ [δ, 1] ʹର͠

0≤ Qn

1 + x 2

= 1 (1 + x)ⁿ

1−

1 + x 2

_n₂ⁿ

(1 + x)ⁿ

≤ 1

(1 + x)ⁿ

1−

1 + x 2

_n₂ⁿ 1 + 2ⁿ

1 + x 2

_n

≤ 1

(1 + x)ⁿ

1−

1 + x 2

_n₂ⁿ 1 +

1 + x 2

_n₂ⁿ

≤ 1

(1 + x)ⁿ

1−

1 + x 2

_2n₂ⁿ

≤ 1

(1 + x)ⁿ ≤ 1 (1 + δ)ⁿ ͱͳΓ

sup{||x| − Pn(x)|; x ∈ [δ, 1]} = sup{2xQn

1 + x 2

; x∈ [δ, 1]}

≤ 2(1 + δ)⁻ⁿ

͕ै͏ɻ·ͨ

sup{||x| − P_n(x)|; x ∈ [0, δ]} = sup{2xQ_n

1 + x 2

; x∈ [0, δ]} ≤ 2δ, sup{||x| − P_n(x)|; x ∈ [−δ, 0]} = sup{2|x|

1− Q_n

1 + x 2

; x∈ [−δ, 0]} ≤ 2δ ͱͳΔɻैͬͯ

| · | − P_n_∞≤ 4δ + 2(1 − δ)ⁿ+ 2(1 + δ)⁻ⁿ ΛಘΔɻ͜ΕΑΓ

lim sup

n→∞ | · | − P_n_∞≤ 4δ ΛಘΔɻδ > 0 ͸೚ҙނิ୊͕̍ै͏ɻ

(6)

̎ɽϕϧϯγϡλΠϯଟ߲ࣜ

f ∈ C([0, 1]; R) ʹର͢Δ n ࣍ϕϧϯγϡλΠϯଟ߲ࣜ B_n(f ) Λ

(B_n(f ))(x) =

n k=0

f

k n

n k

x^k(1− x)^n−k

ͰఆΊΔɻ

ఆཧ̎ʢϕϧγϡλΠϯଟ߲ࣜʹґΔϫΠΤϧετϥεͷఆཧʣɹ༗քด۠ؒ I = [0, 1] ্ ͷ೚ҙͷ࿈ଓവ਺ f ∈ C(I; R) ͸ϕϧϯγϡλΠϯଟ߲ࣜͷྻ (Bn(f ); n∈ Z≥0) ͷ I ্ͷҰ

༷ऩଋۃݶͱͯ͠ද͞ΕΔɿ

f − Bn(f )∞ = sup{|f(x) − (Bn(f ))(x)|; x ∈ I} → 0 (n → ∞)

ʢূ໌ʣɹ࣍ͷิ୊Λ༻͍Δɻ

ิ୊̎ɹ࣍ͷ౳͕ࣜ੒ཱͭɿ (1)

n k=0

n k

x^k(1− x)^n−k = 1

(2)

n k=0

k

n k

x^k(1− x)^n−k = nx

(3)

n k=0

k²

n k

x^k(1− x)^n−k = n(n− 1)x²+ nx

ิ୊̎ͷূ໌ʢͦͷ̍ʣɹ (1) ͸ x ͱ 1− x ʹΑΔೋ߲ల։Ͱ͋Γⁿ

k=0

n k

x^k(1− x)^n−k = (x + (1− x))ⁿ = 1 ΑΓै͏ɻ(2) ͸

n k=0

k

n k

x^k(1− x)^n−k =

n k=1

k

n k

x^k(1− x)^n−k =

n k=1

k n!

k!(n− k)!x^k(1− x)^n−k

=

n k=1

n· (n − 1)!

(k− 1)!(n − k)!x^k(1− x)^n−k = nx

n k=1

(n− 1)!

(k− 1)!(n − k)!x^k−1(1− x)^n−k

= nx

n k=1

n− 1 k− 1

x^k−1(1− x)^{n−1−(k−1)} = nx

n−1 k=0

n− 1 k

x^k(1− x)^n−1−k

= nx(x + (1− x))ⁿ⁻¹ = nx

(7)

ΑΓै͏ɻಉ༷ʹ

n k=0

k(k− 1)

n k

x^k(1− x)^n−k =

n k=2

k(k− 1)

n k

x^k(1− x)^n−k

=

n k=2

k(k− 1) n!

k!(n− k)!x^k(1− x)^n−k = n(n− 1)x²ⁿ

k=2

(n− 2)!

(k− 2)!(n − k)!x^k−n(1− x)^n−k

= n(n− 1)x²

n k=2

n− 2 k− 2

x^k−2(1− x)^{n−2−(k−2)}= n(n− 1)x²

n−2 k=0

n− 2 k

x^k(1− x)^n−2−k

= n(n− 1)x²(x + (1− x))ⁿ⁻²= n(n− 1)x² ΛಘΔͷͰ (2) ͱลʑ଍͠߹ͤΔͱ (3) ͕ै͏ɻ

ิ୊̎ͷূ໌ʢͦͷ̎ʣɹ t∈ R ʹର͠

(xe^t+ (1− x))ⁿ =

n k=0

n k

e^ktx^k(1− x)^n−k Λߟ͑Δɻt Ͱ྆ลΛඍ෼ͯ͠

nxe^t(xe^t+ (1− x))ⁿ⁻¹ =

n k=0

k

n k

e^ktx^k(1− x)^n−k ΛಘΔͷͰ΋͏Ұճ྆ลΛඍ෼ͯ͠

n(n− 1)x²e^2t(xe^t+ (1− x))ⁿ⁻²+ nxe^t(xe^t+ (1− x))ⁿ⁻¹ =

n k=0

k²

n k

e^ktx^k(1− x)^n−k ΛಘΔɻ͜ΕΒͷ౳ࣜͰ t = 0 ͱஔ͍ͨ΋ͷ͸ิ୊̎ͷ౳ࣜʹ֎ͳΒͳ͍ɻ

ఆཧ̎ͷূ໌ɹ 0 < δ < 1 ͳΔ δ Λ೚ҙʹऔΓ

|f(x) − (B_n(f ))(x)| =

n k=0

f (x)− f

k n

n k

x^k(1− x)^n−k

≤ⁿ

k=0

f(x) − f k n

n k

x^k(1− x)^n−k

=

0≤k≤n

|x−k/n|<δ

+

0≤k≤n

|x−k/n|≥δ

f(x) − f k n

n k

x^k(1− x)^n−k

ͱධՁΛ෼ׂ͢Δɻ࠷ޙͷ౳ࣜͷӈลͷલ൒Λ

|x−k/n|<δ

sup{|f(ξ) − f(η)|; ξ, η ∈ I, |ξ − η| < δ}

n k

x^k(1− x)^n−k

≤ sup{|f(ξ) − f(η)|; ξ, η ∈ I, |ξ − η| < δ}ⁿ

k=0

n k

x^k(1− x)^n−k

= sup{|f(ξ) − f(η)|; ξ, η ∈ I, |ξ − η| < δ}

(8)

ͱධՁ͠ɺޙ൒Λ

2f∞

|x−k/n|≥δ

n k

x^k(1− x)^n−k

≤ 2f_∞ n²δ²

|nx−k|≥nδ

(nx− k)²

n k

x^k(1− x)^n−k

ͱධՁ͢Δɻิ୊̎ΑΓ

n k=0

(nx− x)²

n k

x^k(1− x)^n−k

=n²x²

n k=0

n k

x^k(1− x)^n−k− 2nx

n k=0

k

n k

x^k(1− x)^n−k+

n k=0

k²

n k

x^k(1− x)^n−k

=n²x²− 2n²x²+ n(n− 1)x²+ nx =−nx²+ nx = n 1

4 −

x−1

2

₂

≤ n 4 ΛಘΔɻ͜ΕΑΓධՁ

f − B_n(f )_∞≤ sup{|f(ξ) − f(η)|; ξ, η ∈ I, |ξ − η| < δ} + f_∞ 2nδ²

͕ै͍

lim sup

n→∞ f − B_n(f )_∞≤ sup{|f(ξ) − f(η)|; ξ, η ∈ I, |ξ − η| < δ}

ΛಘΔɻδ↓ 0 ͱ͢Ε͹ f ͷҰ༷࿈ଓੑΑΓఆཧ 2 ΛಘΔɻ

̏ɽϘʔϚϯɾίϩϑΩϯͷఆཧ

ϕϧγϡλΠϯଟ߲ࣜΛఆΊΔखଓ͖ f → Bn(f ) ͸࣍ͷ༷ʹଊ͑Δࣄ͕ग़དྷΔɿ (B1) ɹ B_n : C([0, 1];R) f → B_n(f )∈ C([0, 1]; R) ͸࣮ઢܕࣸ૾Ͱ͋Δɻ

(B2) ɹ B_n͸ਖ਼஋Ͱ͋Δɻଈͪ೚ҙͷඇෛ஋࿈ଓവ਺Λඇෛ஋࿈ଓവ਺ʹࣸ͢ɿ f ∈ C([0, 1]; R), f ≥ 0 ⇒ Bn(f )≥ 0

(B3) ɹ೚ҙͷೋ࣍വ਺ f ʹର͠B_n(f )− f_∞→ 0(n → ∞)

ఆٛٴͼิ୊ 2 ΑΓ (B1)(B2) ٴͼ (B3) ͸௚ͪʹै͏ɻ࣮ࡍิ୊ 3 ͸ e_k(x) ≡ x^kʹର͢Δ

౳ࣜ

B_n(e_k) = e_k, k = 0, 1 B_n(e₂) =

1− 1

n

e₂+ 1 ne₁

Λड़΂ͨ΋ͷͰ͋Δɻ্ͷࡾͭͷੑ࣭ʹண໨ͨ͠΋ͷ͕ϘʔϚϯɾίϩϑΩϯͷఆཧͰ͋

Δɻ֤ x ∈ I = [0, 1] ʹର͠ d_x(y) = |x − y| ͱஔ͍ͯఆ·Δവ਺ I y → d_x(y) ∈ R Λ d_x ∈ C(I; R) ͱ͢Δɻ

(9)

ఆཧ̏ʢϘʔϚϯɾίϩϑΩϯʣɹ༗քด۠ؒ I = [0, 1] ্ͷ࿈ଓവ਺શମͷ੒͢ϊϧϜۭ

ؒ C(I;R) ্ͷ࣮ઢܕਖ਼஋ࣸ૾ͷྻ (K_n; n∈ Z_≥0) ʹର࣍͠͸ಉ஋Ͱ͋Δɻ (1) ೚ҙͷೋ࣍ؔ਺ f ʹର͠K_n(f )− f_∞ → 0 (n → ∞)

(2) 0≤ k ≤ 2 ͳΔ೚ҙͷ੔਺ k ʹର͠ Kn(e_k)− ek∞→ 0 (n → ∞)

(3) sup{|(Kn(d²_x))(x)|; x ∈ I} → 0 (n → ∞) ׌ͭ Kn(e₀)− e0∞→ 0 (n → ∞) (4) ೚ҙͷ f ∈ C(I; R) ʹର͠ K_n(f )− f_∞→ 0 (n → ∞)

ʢূ໌ʣɹ (4) ⇒ (1) ⇔ (2) ͸໌Β͔Ͱ͋Δ͔Β (2) ⇒ (3) ⇒ (4) Λࣔͦ͏ɻ

(2) ⇒ (3)ɿɹ dx(y)² = |x − y|² = x²− 2xy + y²Ͱ͋Δ͔Β d²_x = x²e₀− 2xe1+ e₂͕੒ཱ

ͭɻK_nͷઢܕੑΑΓ

K_n(d²_x) = x²K_n(e₀)− 2xKn(e₁) + K_n(e₂) ଈͪ

(K_n(d²_x))(y) = x²(K_n(e₀))(y)− 2x(Kn(e₁))(y) + (K_n(e₂))(y) ΛಘΔɻಛʹ

(K_n(d²_x))(x) = x²(K_n(e₀))(x)− 2x(Kn(e₁))(x) + (K_n(e₂))(x)

= x²((K_n(e₀))(x)− 1) − 2x((Kn(e₁))(x)− x) + ((Kn(e₂))(x)− x²)

= x²(K_n(e₀)− e0)(x)− 2x(K_n(e₁)− e1)(x) + ((K_n(e₂)− e2)(x) ΑΓ

sup{|(Kn(d²_x))(x)|; x ∈ I}

≤ Kn(e₀)− e0∞+ 2Kn(e₁)− e1∞+Kn(e₂)− e2∞

ΛಘΔͷͰ (3) ͕ै͏ɻ

(3) ⇒ (4)ɿɹ |x − y| < δ ͳΔ೚ҙͷ x, y ∈ I ʹର͠

|f(x) − f(y)| ≤ sup{|f(ξ) − f(η)|; ξ, η ∈ I, |ξ − η| < δ}

Ͱ͋Γ|x − y| ≥ δ ͳΔ೚ҙͷ x, y ∈ I ʹର͠

|f(x) − f(y)| ≤ 2f∞≤ 2f∞

δ² |x − y|² Ͱ͋Δ͔Β೚ҙͷ x, y∈ I ʹର͠

|f(x) − f(y)| ≤ sup{|f(ξ) − f(η)|; ξ, η ∈ I, |ξ − η| < δ} + 2f∞

δ² |x − y|²

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͕੒ཱͭɻ͜ΕΛ y ͷവ਺ͱݟ၏ͤ͹ɺ೚ҙͷ x∈ I ʹର͠

|f(x)e0− f| ≤ sup{|f(ξ) − f(η)|; ξ, η ∈ I, |ξ − η| < δ}e0+2f_∞ δ² d²_x

͕੒ཱͭɻK_nͷઢܕੑͱਖ਼஋ੑΛ༻͍Δͱ

|(Kn(f ))(y)− f(x)(Kn(e₀))(y)|

=|(Kn(f − f(x)e0))(y)|

≤ (Kn(|f − f(x)e0|))(y)

≤ sup{|f(ξ) − f(η)|; ξ, η ∈ I, |ξ − η| < δ}Kn(e₀)(Y ) + 2f∞

δ² (K_n(d²_x))(y)

͕ै͏ɻ͜ΕΑΓ

(Kn(f )− f∞

≤ sup{|(K_n(f ))(x)− f(x)(K_n(e₀))(x)|; x ∈ I}

+ sup|f(x)(K_n(e₀))(x)− f(x)e0(x)|; x ∈ I}

≤ sup{|f(ξ) − f(η)|; ξ, η ∈ I, |ξ − η| < δ}K_n(e₀)_∞+ 2f∞

δ² sup{|(K_n(d²_x))(x)|; x ∈ I}

+f∞Kn(e₀)− e0∞

ΛಘΔͷͰ (3) ΑΓ lim sup

n→∞ Kn(f )− f∞≤ sup{|f(ξ) − f(η)|; ξ, η ∈ I, |ξ − η| < δ}

͕ै͏ɻf ͸Ұ༷࿈ଓނ δ↓ 0 ͱͯ͠ (4) ΛಘΔɻ

̐ɽϘʔϚϯɾίϩϑΩϯͷఆཧͷҰൠԽ

ϘʔϚϯɾίϩϑΩϯͷఆཧͷҰൠԽΛ Lomel´ı-Garc´ıa ʹैͬͯઆ໌͢Δɻ༗քด۠ؒ [0, 1]

ΛίϯύΫτۭؒʹҰൠԽ͠ɺڑ཭ͷࣗ৐ d²_xʹ૬౰͢Δ֓೦Λଋറവ਺ bounding function ʹҰൠԽͯ͠ߟ͑Δɻ͜ͷઅͰ͸ X ͸ίϯύΫτۭؒͱ͠ C(X;R) Λ X ্ͷ࿈ଓവ਺ͷ੒

͢ϊϧϜۭؒͱ͢Δɻ

ఆٛɹ f ∈ C(X; R) ʹର͢Δଋറവ਺ͱ͸௚ੵίϯύΫτۭؒ X × X ্ͷඇෛ஋࿈ଓവ਺

γ Ͱ γ⁻¹({0}) ⊂ Diag(f) ͳΔ΋ͷͱ͢Δɻ͜͜ʹ

Diag(f ) ={(x, y) ∈ X × X; f(x) = f(y)}

ͱ͢Δɻf ͷଋറവ਺ γ : X× X → R ٴͼ x ∈ X ʹର͠ γx ∈ C(X; R) ͕ γx(y) = γ(x, y) Ͱ ఆ·Δɻ

ఆཧ̐ʢϩϝϦɾΨϧγΞʣɹίϯύΫτۭؒ X ্ͷ࿈ଓവ਺શମͷ੒͢ϊϧϜۭؒ C(X :R)

্ͷ࣮ઢܕਖ਼஋ࣸ૾ͷྻ (L_n; n ∈ Z≥0) ʹର͠ɺ࣍͸ಉ஋Ͱ͋Δɻ

(11)

(1) ೚ҙͷ f ∈ C(X; R) ʹର͠ଋറവ਺ γ ∈ C(X × X; R) ͕ଘࡏ͠

sup{|(Ln(γ_x))(x)|; x ∈ X} → 0

׌ͭɹL_n(1)− 1_∞ → 0 (n → ∞)

͜͜ʹ 1 ͸ఆ਺വ਺Λද͢΋ͷͱ͢Δɻ

(2) ೚ҙͷ f ∈ C(X; R) ʹର͠ Ln(f )− f∞→ 0 (n → ∞) ఆཧ 4 ͷূ໌ʹ͸࣍ͷิ୊Λ༻͍Δɻ

ิ୊̏ɹ Y ΛίϯύΫτۭؒͱ͠ α ͱ β Λ Y ্ͷඇෛ஋࿈ଓവ਺Ͱ β⁻¹({0}) ⊂ α⁻¹({0}) Λຬͨ͢΋ͷͱ͢Δɻ͜ͷͱ͖೚ҙͷ ε > 0 ʹର͠ M (ε) > 0 ͕ଘࡏ͠ α≤ ε + M(ε)β Λຬ

ͨ͢ɻ

ʢূ໌ʣɹ༩͑ΒΕͨ ε > 0 ʹର͠ V_ε ≡ α⁻¹([0, ε)) = α⁻¹((−ε, ε)) ͸ Y ͷ։ू߹Ͱ͋Δ͔

Β Y\V_ε͸ίϯύΫτू߹ͱͳΔɻβ ͸ Y\V_ε্࠷খ஋ΛऔΔɻͦͷ఺ΛҰͭऔΓ x₀ ∈ Y \V_ε ͱ͢Δɻ͜ͷͱ͖ β(x₀) = 0 ͳΒԾఆΑΓ α(x₀) = 0 ͱͳΓ x₀ ∈ Vε͕ै͍ໃ६Λੜ͡Δɻ

ैͬͯ β(x₀) > 0 Ͱ͋Γ೚ҙͷ x ∈ Y \ Vε ʹର͠ β(x) ≥ β(x0) > 0 ͕੒ཱͭɻނʹ೚ҙͷ x∈ Y \V_εʹର͠ α(x)≤ α_∞≤ (α_∞/β(x₀))β(x) ͕੒ཱͭɻҰํɺ೚ҙͷ x∈ V_εʹର͠

α(x) < ε≤ ε + β(x) ͕੒ཱͭɻैͬͯ

M (ε) = max(α_∞/β(x₀), 1) = max(α_∞/min

x/∈Vε

β(x), 1) ͱஔ͚͹ྑ͍ɻ

ఆཧ̐ͷূ໌ɹ (1)⇒ (2) Λࣔͤ͹ॆ෼Ͱ͋ΔɻY = X×X ্ͷඇෛ஋࿈ଓവ਺ α ͕ α(x, y) =

|f(x) − f(y)| Ͱఆ·Δɻ͜ͷͱ͖ α⁻¹({0}) = Diag(f) Ͱ͋ΓɺఆٛΑΓ γ⁻¹({0}) ⊂ Diag(f) ͱͳ͍ͬͯΔͷͰɺิ୊ 3 ΑΓ೚ҙͷ ε > 0 ʹର͠ M (ε) > 0 ͕ଘࡏ͠

೚ҙͷ (x, y)∈ X × X ʹର͠ෆ౳ࣜ

|f(x) − f(y)| ≤ ε + M(ε)γ(x, y)

͕੒ཱͭɻ͜ΕΑΓ೚ҙͷ x∈ X ʹର͠ (Y ͷവ਺ͱͯ͠ͷෆ౳ࣜ)

|f(x) · 1 − f| ≤ ε · 1 + M(ε)γx

͕ै͏ɻL_n͸ਖ਼஋ઢܕͰ͋Δ͔Β೚ҙͷ y ∈ X ʹର͠ෆ౳ࣜ

|f(x)(L_n(1))(y)− (L_n(f ))(y)| ≤ ε(L_n(1))(y) + M (ε)(L_n(γ_x))(y)

͕੒ཱͭɻ͜ΕΑΓ

L_n(f )− f_∞≤ sup{|(L_n(f ))(x)− f(x)(L_n(1))(x)|; x ∈ X}

+ sup{|f(x)(L_n(1)− 1)(x)|; x ∈ X}

≤ εL_n(1)− 1_∞+ ε + M (ε) sup{|(L_n(γ_x))(x)|; x ∈ X}

+f∞Ln(1)− 1∞

(12)

ͱͳΔͷͰ

lim sup

n→∞ L_n(f )− f_∞ ≤ ε

͕ै͏ɻε > 0 ͸೚ҙͰ͋ͬͨͷͰ (2) ͕ಋ͔ΕΔɻ

̑ɽετʔϯɾϫΠΤϧετϥεͷఆཧ

ίϯύΫτۭؒ X ্ͷ࿈ଓവ਺શମͷ੒͢ϊϧϜۭؒ C(X;R) ʹԙ͍ͯଖͷ෦෼ू߹͕᜚

ີͰ͋Δҝͷॆ෼৚݅Λ༩͑ΔετʔϯɾϫΠΤϧετϥεͷఆཧΛ୅਺ algebra ٴͼ ଋ lattice ͷ֓೦ͷೋͭͷଆ໘͔ΒవΊͯஔ͜͏ɻ

ఆٛɹ C(X;R) ͷ෦෼ू߹ A ʹର͠

ɾA ͸୅਺Ͱ͋Δ ⇔

def A ͸ੵͷఆٛ͞ΕͨϕΫτϧۭؒͰ͋Δ

ʢଈͪ೚ҙͷ f, g∈ A ʹର͠ fg ∈ A ͕ఆٛ͞Ε͍ͯΔʣ ɾA ͸ଋͰ͋Δɻ⇔

def

ೋͭͷA ͷݩʹର֤͠఺ͰͷେখΛऔΔࣄʹґͬͯग़དྷΔ

ೋͭͷവ਺͸A ʹଐ͢ʢଈͪ೚ҙͷ f, g ∈ A ʹର͠

f∨ g, f ∧ g ∈ A ͱͳΔɻ͜͜ʹ (f ∨ g)(x) = max(f(x), g(x)), (f ∧ g)(x) = min(f(x), g(x))

ɾA ͸఺Λ෼཭͢Δ ⇔

def ૬ҟͳΔ೚ҙͷೋ఺ x, y ∈ X ʹର͠

f ∈ A ͕ଘࡏͯ͠ f(x) = f(y) ɾA ͸ೋ఺Λิؒ͢Δ ⇔

def ૬ҟͳΔ೚ҙͷೋ఺ x, y ∈ X ٴͼ೚ҙͷ α, β ∈ R ʹର͠

f ∈ A ͕ଘࡏ͠ f(x) = α, f(y) = β ɾA ͸ೋ఺Λۙࣅิؒ͢Δ ⇔

def ૬ҟͳΔ೚ҙͷೋ఺ x, y ∈ X ٴͼ೚ҙͷ α, β ∈ R, ε > 0 ʹର͠ f ∈ A ͕ଘࡏ͠ |f(x) − α| < ε, |f(y) − β| < ε

ఆཧ̑ʢ෦෼ଋʹର͢ΔετʔϯɾϫΠΤϧετϥεͷఆཧɹίϯύΫτۭؒ X ্ͷ࿈ଓ വ਺શମͷ੒͢ϊϧϜۭؒ C(X;R) ͷ෦෼ଋ A ʹର࣍͠͸ಉ஋Ͱ͋Δɻɹ

(1) ɹA ͸ C(X; R) Ͱ᜚ີͰ͋Δʢଈͪ ¯A = C(X; R))

(2) ɹ A ͸೚ҙͷ f ∈ C(X; R) ʹରͯ͠ੜ੒͞ΕΔೋ఺Λۙࣅิؒ͢Δɻଈͪ೚ҙͷ f ∈ C(X; R), ૬ҟͳΔ೚ҙͷೋ఺ x, y ∈ X, ೚ҙͷ ε > 0 ʹର͠ g ∈ A ͕ଘࡏ͠

|f(x) − g(x)| < ε, |f(y) − g(y)| < ε ɹ

ূ໌ɹ (1) ⇒ (2): ԾఆΑΓ೚ҙͷ f ∈ C(X; R) ʹର͠ g ∈ A ͕ଘࡏ͠ ||f − g∞ < ε ͱͳ Δ͔Β (2) ͕ै͏ɻ

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(2) ⇒ (1): ೚ҙͷ f ∈ C(X; R) ٴͼ ε > 0 ΛऔΔɻԾఆΑΓ೚ҙͷೋ఺ x, y ∈ X ʹର͠

g_x,y ∈ A ͕ଘࡏ͠ |f(x) − g_x,y(x)| < ε, |f(y) − g_x,y(y)| < ε ͕੒ཱͭɻҰ఺ x ∈ X Λݻఆ͠

U_x,y ≡ {ξ ∈ X; f(ξ) − gx,y(ξ) < ε} Λߟ͑ΔɻUx,y = (f− gx,y)⁻¹((−∞, ε)) ނ Ux,y͸ y ∈ Ux,y

ͳΔ X ͷ։ू߹Ͱ͋Γ (U_x,y; y ∈ X) ͸ X ͷ։ඃ෴ͱͳΔɻX ͷίϯύΫτੑʹΑΓ༗ݶू

߹ F (x)⊂ X ͕ଘࡏ͠ (U_x,y; y ∈ F (x)) ͸ X ͷ༗ݶඃ෴ͱͳΔɻ

ͯ͞ g_x = max{gx,y; y ∈ F (x)} ͱஔ͘ͱ gx ∈ A Ͱ͋Γ೚ҙͷ ξ ∈ X ʹର͠ f(ξ) − gx(ξ) <

ε ͕੒ཱͭɻͦ͜Ͱ U_x ≡ {ξ ∈ X; f(ξ) − gx(ξ) > −ε} = (f − gx)⁻¹((−ε, ∞)) ͱஔ͘ɻ f (x)− g_x(x) = f (x)− max

y∈F (x)g_x,y(x) =− min

y∈F (x)(f (x)− g_x,y(x))≥ − min

y∈F (x)|f(x) − g_x,y(x)| > −ε ΑΓ x∈ U_xͱͳΔͷͰ (U_x; x∈ X) ͸ X ͷ։ඃ෴ͱͳΔɻX ͷίϯύΫτੑΑΓ༗ݶू߹

F ⊂ X ͕ଘࡏ͠ (Ux; x ∈ F ) ͸ X ͷ༗ݶඃ෴ͱͳΔɻg = min{gx; x ∈ F } ͱஔ͘ͱ g ∈ A Ͱ͋Γ೚ҙͷ ξ ∈ X ʹର͠ −ε < f(ξ) − g(ξ) = f(ξ) − min

x∈Fg_x(ξ) = max

x∈F(f (ξ)− gx(ξ)) < ε ͱ ͳΓf − g∞ ≤ ε ͕ै͏ɻ

ఆཧ̒ʢଋͷಛ௃෇͚ʣɹ C(X;R) ͷ෦෼ۭؒ A ʹର࣍͠͸ಉ஋Ͱ͋Δɻ (1) ɹA ͸ଋͰ͋Δɻ

(2) ɹA ͸ઈର஋ʹब͍ͯด͍ͯ͡Δɻ͜͜ʹ f ∈ C(X; R) ʹର͠

|f| = f+− f−, f₊= f ∨ 0, f−= f ∧ 0 ͱ͢Δɻɹ

ʢূ໌ʣɹ (1) ⇒ (2)ɿ೚ҙʹ f ∈ A ΛऔΔɻ0 ∈ A Ͱ͋Δ͔Β (1) ΑΓ f± ∈ A ͱͳΓ

|f| ∈ A ͕ै͏ɻ

(2) ⇒ (1) : ɹ − (f ∧ 0) = (−f) ∨ 0 ΑΓ |f| = (f ∨ 0) + ((−f) ∨ 0) ͕ै͏ɻ͜ΕΑ Γ|f| ≥ f ׌ͭ |f| ≥ −f ΛಘΔͷͰ |f| ≥ f ∨ (−f) ͕ै͏ɻҰํ f ∨ (−f) ≥ f, −f, 0 ΑΓ f ∨ (−f) ≥ f ∨ 0, (−f) ∨ 0 ΛಘΔɻ͜͜Ͱ supp(f ∨ 0) ∩ supp((−f) ∨ 0) = ∅ ΑΓ f ∨ (−f) ≥ (f ∨ 0) + ((−f) ∨ 0) ͕ै͏ɻނʹ |f| = f ∨ (−f) ΛಘΔɻ͜ΕΑΓ |f − g| = (f − g) ∨ (g − f), (f + g) + |f − g| = (f + g) + ((f − g) ∨ (g − f)) = 2(f ∨ g) ΛಘΔ ͷͰ f ∨ g = ¹₂((f + g) + |f − g|) ∈ A ͕ै͏ɻ͜ΕΑΓ f ∧ g = −((−f) ∨ (−g)) =

1

2((−f − g) + | − f + g|) = ¹₂((f + g)− |f − g|) ∈ A ͕ै͏ɻ

ఆཧ̓ʢఆ਺ΛؚΉ୅਺ͷ఺෼཭ͷಛ௃෇͚ʣɹ 1∈ A ͳΔ୅਺ A ⊂ C(X; R) ʹର࣍͠͸

ಉ஋Ͱ͋Δɻɹ

(1) ɹA ͸఺Λ෼཭͢Δɻ (2) ɹA ͸ೋ఺Λิؒ͢Δɻ (3) ɹA ͸ೋ఺Λۙࣅิؒ͢Δɻ

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ʢূ໌ʣɹ (2) ⇒ (3) ⇒ (1) ͸໌Β͔Ͱ͋ΔͷͰ (1) ⇒ (2) Λࣔͤ͹ॆ෼Ͱ͋Δɻx = y ͳΔ

ೋ఺ x, y ∈ X ΛऔΔɻԾఆʹΑΓ f ∈ A ͕ଘࡏ͠ f(x) = f(y) ͱͳΔɻα, β ∈ R Λ೚ҙʹ ༩͑Δɻg ∈ C(X; R) ͕

g(ξ) = (α− β)f(ξ) − αf(y) + βf(x)

f (x)− f(y) , ξ ∈ X

Ͱఆ·ΔɻA ͸ 1 ΛؚΉ୅਺Ͱ͋Δ͔Β g ∈ A ͱͳΓ g(x) = α, g(y) = β Λຬͨ͢ɻ

ఆཧ̔ʢ෦෼୅਺ʹର͢ΔετʔϯɾϫΠΤϧετϥεͷఆཧʣɹίϯύΫτۭؒ X ্ͷ࿈

ଓവ਺શମͷ੒͢ϊϧϜۭؒ C(X;R) ͷ෦෼୅਺ A ʹର࣍͠͸ಉ஋Ͱ͋Δɻ (1) ɹA ͸ C(X; R) Ͱ᜚ີͰ͋Δʢଈͪ ¯A = C(X; R)ʣ

(2) ɹ A ͸೚ҙͷ f ∈ C(X; R) ʹରͯ͠ੜ੒͞ΕΔೋ఺Λۙࣅิؒ͢Δɻଈͪ೚ҙͷ f ∈ C(X; R), ૬ҟͳΔ೚ҙͷೋ఺ x, y ∈ X, ೚ҙͷ ε > 0 ʹର͠ g ∈ A ͕ଘࡏ͠

|f(x) − g(x)| < ε, |f(y) − g(y)| < ε

ʢূ໌ʣɹ (1) ⇒ (2)ɿఆཧ 5 ͷ (1) ⇒ (2) ͷূ໌ͱಉ͡Ͱ͋Δɻ

(2)⇒ (1)ɿ(2) ͸ A Λ ¯A ʹஔ͖׵͑ͯ΋੒ཱͭͷͰ ¯A ͕෦෼ଋΛࣔ͢ࣄΛࣔͤ͹ఆཧ 5 Α Γ (2)⇒ (1) ͕ै͏ɻA ͸ C(X; R) ͷ෦෼୅਺ނ ¯A ΋ͦ͏Ͱ͋Δɻఆཧ 6 ΑΓ ¯A ͕ઈର஋

ʹब͍ͯด͍ͯ͡ΔࣄΛࣔͤ͹ॆ෼Ͱ͋Δɻ೚ҙʹ f ∈ ¯A ٴͼ ε > 0 ΛऔΔɻิ୊ 1 ʹΑ Γ sup{||x| − P (x)|; |x| ≤ 1} < ε/(||f||∞+ 1) ͳΔଟ߲ࣜ P :R → R ͕ଘࡏ͢Δɻ͜ͷͱ͖

Q(f )≡ f_∞P (f /f_∞)∈ ¯A Ͱ͋Γ

|f| − Q(f)∞=f∞|f/f∞| − P (f/f∞)∞

=f∞sup{x| − P (x)|; |x| ≤ 1} < ε ͱͳΔ͔Β|f| ∈ ¯A ͕ै͏ɻ͜Ε͕ࣔ͢΂͖ࣄͰ͋ͬͨɻ

ఆཧ̕ʢఆ਺ΛؚΉ෦෼୅਺ʹର͢ΔετʔϯɾϫΠΤϧετϥεͷఆཧʣɹఆ਺ΛؚΈɺ

఺Λ෼཭͢Δ୅਺A ⊂ C(X; R) ͸᜚ີͰ͋Δɻ

ʢূ໌ͦͷ̍ʣɹఆཧ 7 ΑΓA ͸ೋ఺Λิؒ͢ΔͷͰఆཧ 8 ΑΓ ¯A = C(X; R) ͕ै͏ɻ

ʢূ໌ͦͷ̎ʣɹ্هఆཧ 8 ͷূ໌Ͱ͸ɺ୅਺A ͷดแ A ͕࠶ͼ୅਺Λ੒͢ࣄͱɺA ͕ઈ ର஋ʹब͍ͯด͍ͯ͡Δࣄ͕ຊ࣭తͰ͋ͬͨɻ͜ΕʹΑΓɺ୅਺ͷߏ଄͕ଋͷߏ଄ʹҠ২͞

Εɺશͯ͸ఆཧ 5 ʹશͯؼண͞ΕΔͱӠ͏࿦๏Λߏ੒ͨ͠ͷͰ͋ΔɻҎԼͰ͸ɺ͜ͷ࿦๏Λ

༻͍ͳ͍ Brosowski ͱ Deutsch ʹґΔ௚઀తͰॳ౳తͳূ໌Λղઆ͠Α͏ɻͦͷҝʹೋͭͷ

ิ୊Λ४උ͢Δɻ

ิ୊4 ɹ೚ҙͷ x0 ∈ X ͱͦͷ೚ҙͷ։ۙ๣ U ʹର͠ V ⊂ U ͳΔ x0ͷ։ۙ๣ V ͕ଘࡏ͠Ҏ Լͷੑ࣭Λຬͨ͢ɻ೚ҙͷ ε > 0 ʹର͠ ϕ_ε ∈ A ͕ଘࡏ͠

(15)

(i) ɹ೚ҙͷ x∈ X ʹର͠ 0 ≤ ϕε(x)≤ 1 (ii) ɹ೚ҙͷ x∈ V ʹର͠ 0 ≤ ϕ_ε(x)≤ ε

(iii) ɹ೚ҙͷ x∈ X\U ʹର͠ 1 − ε ≤ ϕε(x)≤ 1

ʢূ໌ʣɹ֤ x ∈ X\U ʹର͠ A ͷ఺෼཭ੑΑΓ f_x ∈ A ͕ଘࡏ͠ f_x(x) = f_x(x₀) ͱͳΔɻ g_x(ξ) = f_x(ξ) − fx(x₀), ξ ∈ X Ͱఆ·Δ gx : X ξ → gx(ξ) ∈ R ͸ A ʹଐ͠ gx(x) =

g_x(x₀) = 0 Λຬͨ͢ɻ͜ͷͱ͖ h_x ≡ g_x²/gx²_∞͸ h_x(x₀) = 0, h_x(x) > 0, 0 ≤ hx ≤ 1 ͳ ΔA ͷݩͰ͋Δɻͯ͞ U(x) = {ξ ∈ X; h_x(ξ) > 0} ͱஔ͘ɻx ∈ U(x) = h⁻¹_x ((0,∞)) ނ (U (x) ∩ (X\U); x ∈ X\U) ͸ X\U ͷ։ඃ෴ͱͳΔɻX\U ͷίϯύΫτੑΑΓ༗ݶू߹

{x_j ∈ X\U; 1 ≤ j ≤ m} ͕ଘࡏ͠ X\U ⊂

n j=1

U (x_j) ͱ͢Δࣄ͕ग़དྷΔɻ

ͯ͞ h = _m¹

m j=1

h_x_jͱஔ͘ɻ͜ͷͱ͖ h∈ A Ͱ͋Γ 0 ≤ h ≤ 1, h(x0) = 0 ׌ͭ೚ҙͷ ξ ∈ X\U ʹର͠ h(ξ) > 0 Ͱ͋Δࣄ͕ै͏ɻh ͸ίϯύΫτू߹ X\U ্࠷খ஋ΛऔΔͷͰ 0 < δ < 1 ͕ ଘࡏ͠೚ҙͷ ξ ∈ X\U ʹର͠ h(ξ) ≥ δ ͱͳΔɻͯ͞ V = {ξ ∈ X; h(ξ) < δ/2} ͱஔ͘ͱ V ͸ x₀ ∈ V ͳΔ։ू߹Ͱ X\U ⊂ X\V ΑΓ V ⊂ U Ͱ͋Δࣄ͕෼͔Δɻk = min{ ∈ Z; 1/δ < } ͱஔ͘ɻ͜ͷͱ͖ k− 1 ≤ 1/δ < k Ͱ͋Γ k ≤ 1 + 1/δ < 2/δ ΑΓ kδ/2 < 1 < kδ ͕ै͏ɻ

ͦ͜Ͱ n∈ Z>0ʹର͠ ψ_n: X → R Λ

ψ_n(ξ) = (1− h(ξ)ⁿ)^kⁿ, ξ ∈ X

Ͱఆٛ͢Δɻ1, h ∈ A ނ ψn ∈ A Ͱ͋Γ 0 ≤ h ≤ 1, h(x0) = 0 ނ 0≤ ψn ≤ 1, ψn(x₀) = 1 Ͱ

͋Δɻ೚ҙͷ ξ∈ V ʹର͠ 0 ≤ kh(ξ) ≤ kδ/2 < 1 ͱͳΔ͔ΒϕϧψΠͷෆ౳ࣜΑΓ sup{|1 − ψ_n(ξ)|; ξ ∈ V } = sup{1 − ψ_n(ξ); ξ∈ V }

≤ sup{(kh(ξ))ⁿ; ξ∈ V } ≤ (kδ/2)ⁿ→ 0(n → ∞)

͕ै͏ɻҰํɺ೚ҙͷ x ∈ X\U ʹର͠ kh(ξ) ≥ kδ > 1 ͱͳΔ͔Β࠶ͼϕϧψΠͷෆ౳ࣜ

ΑΓ

sup{|ψn(ξ)|; ξ ∈ X\U} = sup{ 1

(kh(ξ))ⁿψ_n(ξ)(kh(ξ))ⁿ; ξ∈ X\U}

≤ sup{ 1

(kh(ξ))ⁿ(1− h(ξ)ⁿ)^kⁿ(1 + (kh(ξ))ⁿ); ξ ∈ X\U}

≤(kδ)⁻ⁿsup{(1 − h(ξ)ⁿ)^kⁿ(1 + k(ξ))ⁿ)^kⁿ; ξ ∈ X\U}

≤(kδ)⁻ⁿsup{(1 − h(ξ)²ⁿ)^kⁿ; ξ∈ X\U} ≤ (kδ)⁻ⁿ→ 0(n → ∞)

͕ै͏ɻ༩͑ΒΕͨ ε > 0 ʹର͠ (kδ)⁻ⁿ < ε ͳΔ n ΛऔΓ ϕ_ε = 1− ψ_nͱஔ͚͹ิ୊ 4 ʹڍ

͛ͨੑ࣭͕શͯຬͨ͞ΕΔɻ

ิ୊̑ɹ M ͱ N Λޓ͍ʹૉͳ X ͷด෦෼ू߹ͱ͢Δɻ͜ͷͱ͖ 0 < ε < 1 ͳΔ೚ҙͷ ε ʹ ର͠ ψ_ε ∈ A ͕ଘࡏ࣍͠Λຬͨ͢ɻ

(16)

(i) ɹ೚ҙͷ x∈ X ʹର͠ 0 ≤ ψε(x)≤ 1 (ii) ɹ೚ҙͷ x∈ M ʹର͠ 1 − ε < ψ_ε(x)≤ 1 (iii) ɹ೚ҙͷ x∈ N ʹର͠ 0 ≤ ψε(x)≤ ε

ʢূ໌ʣɹ U = X\M ͱ֤ͯ͠ x ∈ N ⊂ X\M = U ʹରͯ͠ิ୊ 4 Ͱ༩͑ΒΕΔ x ͷ։ۙ๣

V (x)⊂ U ΛऔΔɻ(V (x); x ∈ N) ͸ N Λ෴͏ͷͰ༗ݶू߹ {xj ∈ N; 1 ≤ j ≤ m} ͕ଘࡏ͠

N ⊂

m j=1

V (x_j) ͱ͢Δࣄ͕ग़དྷΔɻิ୊ 4 ΑΓ֤ j ʹର͠ ϕ_j ∈ A ͕ଘࡏ͠ 0 ≤ εj ≤ 1, V (xj)

্ 0 ≤ ϕ_j < ε/m, M = T\U ্ 1 − ε/m < ϕ_j ≤ 1 ͕੒ཱͭɻͦ͜Ͱ ψ_ε =

m j=1

ϕ_j ͱஔ͘ɻ

͜ͷͱ͖ ψ_ε ∈ A Ͱ͋Γ N(⊂

m j=1

V (x_j)) ্ 0 ≤ ψε =

m j=1

ϕ_j =

m j=1

(ε/m) = ε ߋʹ M ্

ψ_ε =

m j=1

ϕ_j > (1− ε/m)^m ≥ 1 − ε ͕੒ཱͭɻ

Ҏ্ͷิ୊ 4 ͱ 5 Λ༻͍ͯఆཧ 9 ͷূ໌ʢͦͷ 2ʣΛ༩͑Α͏ɻॳΊʹ೚ҙʹ f ∈ C(X; R) ͱ ε > 0 Λ༩͓͑ͯ͘ɻε ͸ॆ෼খ͍͞΋ͷͱԾఆͯ͠ྑ͍ʢҎԼͷٞ࿦Ͱ͸ ε < 1/3 Ͱॆ෼ʣɻ

f ≥ 0 ͷ৔߹ɿɹ n ≥ ^f_ε^∞ + 1 ͳΔ੔਺ n ΛҰͭݻఆ͢Δɻ֤ j ∈ {0, 1, · · · , n} ʹର͠

M_j, N_jΛ

M_j ={x ∈ X; f(x) ≥

j + 1

3

ε} = f⁻¹

[

j +1

3

ε, ∞)

, N_j ={x ∈ X; f(x) ≤

j− 1

3

ε} = f⁻¹

(−∞,

j− 1

3

ε]

ͰఆΊΔɻM_jٴͼ N_j͸ X ͷޓ͍ʹૉͳด෦෼ू߹Ͱ M₀ ⊃ M₁ ⊃ · · · ⊃ M_n =∅,

∅ = N0 ⊂ N1 ⊂ · · · · ⊂ Nn = X

ͳΔؔ܎Λຬͨ͢ɻิ୊ 5 ʹΑΓ֤ j ʹର͠ ψ_j ∈ A ͕ଘࡏ͠ X ্ 0 ≤ ψ_j ≤ 1, M_j ্ 1− ^ε_n < ψ_j ≤ 1, Nj ্ 0≤ ψj ≤ _n^ε Λຬͨ͢ɻͦ͜Ͱ g = ε

n j=0

ψ_j ͱஔ͘ɻͯ͞೚ҙʹ x ∈ X ΛऔΔɻ1≤ j ≤ n ͳΔ j ͕།Ұͭଘࡏ͠ x_j ∈ N_j\N_j−1͕੒ཱͭɻ͜ͷͱ͖

j− 4

3

ε =

(j− 1) − 1 3

ε < f (x)≤

j− 1

3

ε,

0≤ max

j ψ_j(x)≤ ε n

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͕੒ཱͭɻ͜ΕΑΓ

g(x) = ε

j−1 i=0

ψ_j(x) + ε

n i=j

ψ_j(x)≤ εj + ε(n − j + 1)ε

n ≤ (j + ε)ε, g(x)− f(x) ≤ (j + ε)ε −

j− 1

3

ε =

ε + 1

3

ε < ε

͕ै͏ɻҰํ x∈ N_j\N_j−1, j ≥ 1 ͳΒ͹ f(x) > ((j − 1) − ¹₃)ε = (j−⁴₃)ε ͕ै͏ɻ j ≥ 2 ͷ৔߹ x ∈

j−2

i=0

M_iͱͳΔͷͰ

g(x) = ε

n i=0

ψ_i(x)≥ ε

j−2 i=0

ψ_i(x)≥ ε

j−2 i=0

1− ε

n

= (j− 1) 1− ε

n

ε

͕ै͏ɻ࠷ӈล͸ j = 1 Ͱ 0 ͱͳΔͷͰ݁ہ x∈ Nj\Nj−1ʹର͠

g(x)≥ (j − 1) 1− ε

n

ε

͕੒ཱͭɻ͜ͷͱ͖ x∈ N_j ΑΓ

f (x)− g(x) ≤

j −1

3

ε− (j − 1)

1− 2

n

ε =

2

3+ j− 1 n ε

ε <

2 3+ ε

ε < ε

͕ै͏ɻނʹ

|f(x) − g(x)| < ε

͕੒ཱͭɻx∈ X ͸೚ҙͰ͋ͬͨͷͰ

f − g∞ ≤ ε

͕ै͏ɻ͜Ε͕ࣔ͢΂͖ࣄͰ͋ͬͨɻ

Ұൠͷ৔߹ɿɹ f ∈ C(X; R) ʹର͠ f + f∞͸લஈͷԾఆΛຬͨ͢ͷͰ g ∈ A ͕ଘࡏ͠

(f + f_∞)− g_∞ < ε Λຬͨ͢ɻ͜ͷͱ͖ g− f_∞∈ A ͕ٻΊΔ΋ͷͱͳΔɻ

̒ɽετʔϯɾϫΠΤϧετϥεͷఆཧͷෳૉ൛

͜ͷઅͰ͸ετʔϯɾϫΠΤϧετϥεͷఆཧΛෳૉ਺஋࿈ଓവ਺ʹ֦ு͢ΔࣄΛߟ͑Α͏ɻ ઌͣఆཧ 9 ͰR Λ C ʹஔ͖׵໋͑ͨ୊͸੒ཱ͠ͳ͍ࣄʹ஫ҙ͢Δɻ࣮ࡍɺίϯύΫτू߹

X = {z ∈ C; |z| ≤ 1} ্ͷ࿈ଓവ਺ f : X z → ¯z ∈ C ͕ෳૉ܎਺ (anj; j ∈ Jn)ʢ͜͜ʹ J_n ⊂ Z_≥0, J_n<∞, n ∈ Z_≥0ʣΛ࣋ͭଟ߲ࣜ p_n(x) =

j∈Jn

a_njz^jͷྻ (p_n; n∈ Z_≥0) ͷҰ༷ۃݶ Ͱ͋ͬͨͱ͢Δͱ f ͸ྖҬ D = IntX ={z ∈ C; |z| < 1} ্ͷਖ਼ଇവ਺ྻ (pn; n ∈ Z≥0) ͷ޿

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ٛҰ༷ऩଋۃݶͱͯ͠ D ্ਖ਼ଇͱͳΔ͕ɺ͜Ε͸ໃ६Ͱ͋Δɻ࣍ͷ༷ͳ௚઀తͳܭࢉʹجͮ

͍ͯ΋ໃ६Λಋ͘ࣄ͕ग़དྷΔɿ π

2 = 2π

₁

0

r³dr =

X

(x²+ y²)dxdy =

X

z ¯zdxdy =

X

f ¯f

=

X

(f − pn) ¯f +

X

p_nf =¯

X

(f − pn) ¯f

≤ f − pn∞

X|f| = f − pn∞

X

rdxdy = f − pn∞· 2π

₁

0

r²dr

= 2π

3 f − p_n_∞

ΑΓf − pn∞ ≥ 3/4 ͱͳΓໃ६ɻ͜͜ʹ

X

p_nf =¯

X

p_n(z)zdxdy =

j∈Jn

a_nj

₁

0

r^j+1

_2π

0

e^ijθe^iθdθdr = 0

Λ༻͍ͨɻ

ఆٛɹίϯύΫτۭؒ X ্ͷෳૉ਺஋࿈ଓവ਺શମͷ੒͢ϊϧϜۭؒ C(X;C) ͷ෦෼ू߹

A ʹର͠

ɾA ͸୅਺Ͱ͋Δ ⇔

def.A ͸ੵͷఆٛ͞ΕͨϕΫτϧۭؒͰ͋Δ ɾA ͸఺Λ෼཭͢Δ ⇔

def. ૬ҧͳΔ೚ҙͷೋ఺ x, y ∈ X ʹର͠ f ∈ A ͕ଘࡏͯ͠ f(x) = f(y)

ɾA ͸ෳૉڞ᫂ʹब͍ͯด͍ͯ͡Δ ⇔

def.೚ҙͷ f ∈ A ʹର͠ ¯f ∈ A

ఆཧ̍̌ʢఆ਺ΛؚΉ෦෼୅਺ʹର͢ΔετʔϯɾϫΠΤϧετϥεͷఆཧͷෳૉ൛ʣɹ

ෳૉఆ਺ΛؚΈɺ఺Λ෼཭͠ɺෳૉڞ᫂ʹब͍ͯด͍ͯ͡Δ୅਺A ⊂ C(X; C) ͸᜚ີͰ

͋Δɻ

ূ໌ɹ k = 0, 1, 2 ʹର͠AkΛ

A0 ={f ∈ A ; f(X) ⊂ R},

A1 ={u ∈ C(X; R); f ∈ A ͕ଘࡏͯ͠ u = Ref}, A2 ={u ∈ C(X; R); f ∈ A ͕ଘࡏͯ͠ v = Imf}

ͱஔ͘ɻఆٛΑΓA0 ⊂ A1ͱͳΔɻA ͸ෳૉϕΫτϧۭؒͰ͋Δ͔Β A1 =A2͕ै͍ɺߋ ʹA ͸ෳૉڞ᫂ʹब͍ͯด͍ͯ͡Δ͔Β A1 =A2 ⊂ A0͕ै͏ɻނʹA0 =A1 =A2ͱͳ ΔɻA ͸୅਺Ͱ͋Δ͔Β A0΋୅਺Λ੒͢ɻ૬ҧͳΔ೚ҙͷೋ఺ x, y ∈ X ʹର͠ f ∈ A ͕ ଘࡏ͠ f (x) = f(y) ͱͳΔ͔Β Ref(x) = Ref(y) ·ͨ͸ Imf(x) = Imf(y) ͕੒ཱͪ Ref ·

ͨ͸ Imf ʹΑͬͯೋ఺͸෼཭͞ΕΔɻA ͸ෳૉఆ਺ΛؚΉͷͰ A0 = A1 = A2͸࣮ఆ਺Λ