Tesis Resumen (ObservaciSeones)

7/23/2019 Tesis Resumen (ObservaciSeones)

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an − 1 + an +1 ≥ 2an nk− 1 < n < n k k > 0.

n = nk an 1 − 1 > M ≥ 1 ≥ (n 1 +1)2n 1

k = 1

an k − 1 + an k +1 ≥ nk − nk− 1 + 12k (nk − nk− 1)

+ nk+1 − nk − 1

2

2k (nk+1 − nk )

= 12k 1 +

1nk − nk − 1

+ 1 − 1

2(nk+1 − nk )

= 2 an k

+ 1

2k

1

nk − nk− 1−

1

2(nk+1 − nk )> 2a

n k.

( )

∞

n =1

an

n = ∞ ,

( )

∞

n =1

an cos(nt ),

( )

∞

n =1

an sen(nt ),

f ∈ L2(T ).

(1). s p(f ) 2 ≤ f 2 p ≥ 0.

(2). lım p→∞

f − s p(f ) 2 = 0. s p(f ) ( )



g, h = 1

2π π

− π

g(t)h(t)dt, en = eint , y s p = s p(f ) = p

n = − p

f (n)en .

f 22 = f, f , f (n) = f, e n = en , f , en , ek = 0 k = n

k = n,

s p, f = p

n = − p

f (n) en , f = p

n = − p

| f (n)|2,

f, s p = p

n = − p

f (n) f, e n = p

n = − p

| f (n)|2

s p, s p = p

n = − p

p

k= − p

f (n) f (k) en , ek = p

n = − p

| f (n)|2.

0 ≤ f − s p22 = f − s p, f − s p = f, f − f, s p − s p, f + s p, s p = f 2

2− p

n = − p

| f (n)|2

p ≥ 0

∞

n = −∞| f (n)|

2

≤ f 22.

( ) g ∈ L2(T ) g(n) = f (n)

n

g − s p 2 = g − s p(g) 2 → 0 p → ∞ .

f = g f − s p 2 = g − s p → 0.

f, g ∈ L2(T ).

(1). ∞n = −∞ f (n)g(n) = 1

2π π− π f (t)g(t)dt

(2). ∞n = −∞ | f (n)|2 = 1

2π π− π |f (t)|2dt.



s p(f ), s p(g) = p

n = − p

p

k= − p

f (n)g(k) en , ek = p

n = − p

f (n)g(n)

p ≥ 0,

| f, g − p

n = − p

f (n)g(n)| = | f, g − s p(f ), s p(g) |

≤ | f, g − s p(f ), g |+ | s p(f ), g − s p(f ), s p(g) |

≤ f − s p(f ) 2 g 2 + s p(f ) 2 g − s p(g) 2

≤ f − s p(f ) 2 g 2 + f 2 g − s p(g) 2 → 0 p → ∞ .

lım p→∞

p

n = − p

f (n)g(n) = f, g ,

f = g

( ) ( )

f → f L2(T ), L2(T ) l2(Z ).

f ∈ L2(T ),

f L2(T ).

f ∈ L2(T ),

lımn →∞

f − sn (f ) L 2 = 0.

L p(T )

1 ≤ p p = 2



f ∈ L1(T ) f

a0

2 +

∞

n =1

(ak cos(nx ) + bk sen(nx ));

f ∈ L1(T ), sn (f, x ) f (x)

x

L p(T ) 1 ≤ p,

f, s n (f, x ) f (x)

x,

sn (f, x ) f

(cn )∞n =0

sn =n

k=0

ck σn = 1n + 1

n

k=0

sk .

s ∈C lımn →∞

σn = s,

∞k=0 ck

C −∞

k=0

ck = s.



(cn (x))∞n =0 X = ∅

σn (x)

σn (x) → s(x) n → ∞ .

∞

k=0 ck (x)

(cn )∞n =0

A(r ) =∞

n =0

cn r n ,

0 ≤ r < 1. lımr ↑1

A(r ) = s ∈C ,

∞n =0 cn

A −

∞

n =0cn = s.

(cn (x))∞n =0

A(x, r ) =∞

n =0

cn (x)r n ,

x ∈ X 0 ≤ r < 1.

lımr ↑1

A(x, r ) = s(x) ∞n =0 cn (x)

(cn )∞n =0

X = ∅.

∞

n =0

cn .

( ) C ( )

lımn →∞

|| s − σn || u = 0. > 0 n0 ∈N

|| s − σn || u < 2

n ≥ n0.

r0 ∈ [0, 1)

(1 − r 0)2n 0

n =0

(n + 1) || s − σn || u < 2

.

C x ∈ X

σn (x) → s(x) s(x) ∈C ,

sn (x)n

= [(n + 1) σn (x) − nσ n − 1(x)]

n → s(x) − s(x) = 0 ,



cn (x)n

= sn (x)

n −

n − 1n

sn − 1(x)n − 1

→ 0.

|cn (x)| ≤ n

n ≥ n1,

|A(x, r )| =∞

n =0

cn (c)r n ≤∞

n =0

|cn (x)|| r n | ≤n 1 − 1

n =0

|cn (x)|| r |n +∞

n = n 1

n |r |n ,

n |r |n ≥ 0 lımn →∞

n n |r |n = |r | < 1,

A(x, r ) x ∈ X

|r | < 1.

(1 − r )− 1 =∞

n =0

1r n

A(x, r ) = (1 − r )∞

n =0

sn (x)r n

A(x, r ) = (1 − r )2∞

n =0

(n + 1) σn (x)r n ,

x ∈ X |r | < 1.

s(x) = (1 − r )2∞

n =0

(n + 1) s(x)r n ,

s(x) − σn (x) = (1 − r )2∞

n =0

(n + 1) [s(x) − σn (x)] r n ,

x ∈ X |r | < 1.

x ∈ X r0 ≤ r < 1,



|s(x) − A(x, r )| ≤ (1 − r0)2n 0

n =0

(n + 1) || s − σn || u + (1 − r )2∞

n = n 0 +1

(n + 1) || s − σn || u r n

< 2

+ (1 − r )2∞

n = n 0 +1

(n + 1)2

r n

< 2

+ 2

(1 − r )2∞

n =0

(n + 1) r n

< 2

+ 2

= .

f ∈ L1(T ), Dn K n ( ) :

Dn (t) = sen (n + 1

2 )tsen( 1

2 t) , K n (t) =

sen2 ( n +12 )t

(n + 1) sen 2[(12 )t]

P r (t) =∞

k= −∞

r |k | eikt

n ≥ 0 0 ≤ r < 1 x, t ∈R .

(1). P r (t) = 1 − r 2

1 − 2r cos(t) + r2 ,

(2). sn (f, x ) = nk= − n f (k)eikt = 1

2π π− π f (x − t)Dn (t)dt,

(3). σn (f, x ) = 1n +1

nk=0 sk (f, x ) = 1

2π π− π f (x − t)K n (t)dt,

(4). α r (f, x ) = ∞k = −∞ r |k | f (k)eikt = 1

2π π− π f (x − t)P r (t)dt

t ∈R 0 ≤ r < 1.

P r (t) p

k= − p

r |k | eikt = p

k=0

r k eikt + p

k=1

r k e− ikt ,

p → ∞

∞

k=0

r k eikt +∞

k=1

r k e− ikt = 1

1 − re it + re− it

1 − re − it = 1 − re − it + re − it − r 2

1 − 2cos(t) + r2 = 1 − r 2

1 − 2r cos(t) + r2



Dn K n ( )

sn (f, x ) =n

k= − nf (k)eikx =

n

k= − n12π

π

− πf (u)e− iku due ikx

= 12π π

− πf (u)

n

k= − n

eik (x − u ) du

= 12π π

− πf (u)Dn (x − u)du

σn (f, x ) = 1n + 1

n

k=0sk (f, x ) = 1n + 1

n

k=012π

π

− πf (u)Dk (x − u)du

= 12π π

− πf (u)

1n + 1

n

k=0

Dk (x − u) du

= 12π π

− πf (u)K n (x − u)du,

t = x − u

p

p

k= − p

r |k | f (k)eikx = p

k= − p

r |k | 12π π

− πf (u)e− iku due ikx

= 12π

p

k= − p

r |k | π

− πf (u)eik (x − u )du

= 12π

π

− πf (u)

p

k= − p

r |k |eik (x − u ) du.

P r (t) [− π, π ]

0 ≤ r < 1

Dn (t) K n (t) P r (t)



K n P r n ≥ 0 0 ≤ r < 1

t ∈R ,

(1). 1

2π

π

− π K n (t)dt = 1

2π

π

− π P r (t)dt = 1,(2). K n (t) = K n (− t) P r (t) = P r (− t),

(3). 0 ≤ K n (t) ≤ n + 1

1− r1+ r ≤ P r (t) ≤ 1+ r

1− r ,

(4). 0 ≤ K n (t) ≤ π 2

(n +1) t 2 0 < t ≤ π.

Dk (t)

|Dk (t)| =k

j = − k

eijt ≤ 2k + 1 ,

(n + 1) K n (t) ≤n

k=0

(2k + 1) = ( n + 1) 2.

sent2

≥ tπ

0 ≤ t ≤ π.

n = 0

P r (t) =∞

k= −∞

r |k |eikt = 1 +∞

k=1

r k e− ikt + eikt = 1 + 2∞

k=1

r k cos(kt ).

P r (t) R

12π π

− πP r (t)dt =

12π π

− πdt +

1π π

− πcos(kt )dt = 1.

(n + 1) K n (t) ≤n

k=0

(2k + 1) = ( n + 1) 2,

− 1 ≤ cos(t) ≤ 1

0 ≤ r < 1

(1 + r)2 = 1 + 2 r + r2 ≥ 1 − 2r cos(t) + r2



1 − 2r cos(t) + r2 ≥ 1 − 2r + r2 = (1 − r )2

1 − r1 + r ≤ P r (t) ≤ 1 + r1 − r .

sen( t2 ) ≥ t

π 0 ≤ t ≤ π,

t2

π 2 ≤ sen2( t2 ) 0 ≤ t ≤ π,

0 ≤ K n (t) ≤ 1

(n + 1) sen 2( t2 )

≤ π2

t2(n + 1).

f ∈ L1(T ),

(1). x ∈R f (x− ) = lım

t ↑xf (t) f (x+) = lım

t ↓xf (t)

lımn →∞

σn (f, x ) = f (x− ) + f (x+)

2 .

(2). f [a, b] ⊆R ,

σn (f, x ) =

n

k= − n1 −

|k|n + 1 f (k)eikx

f [a, b].

σn (f ) f

x ∈R , s(x) ∈C

σn (f, x ) − s(x) = 12π

π

− π [f (x − t) − s(x)] K n (t)dt

= 12π π

0[f (x − t) + f (x + t) − 2s(x)] K n (t)dt,

s(x) = f (x − )+ f (x +)2 . > 0 0 < δ < π

|f (x − t) + f (x + t) − 2s(x)| < 2



0 < t < δ. N ∈N

2π2

Nδ 2(|| f || 1 + |s(x)|) <

2

.

φ(x, t ) = f (x − t) + f (x + t) − 2s(x);

|σn (f, x ) − s(x)| ≤ 12π π

0|φ(x, t )K n (t)|dt.

( ) ( )

12π δ

0|φ(x, t )K n (t)|dt ≤

12π δ

0 2K n (t)dt ≤

2

.

( ) n ≥ N

π

δ|φ(x, t )K n (t)|dt ≤ π

δ|φ(x, t )|

π2

(n + 1) t2 dt

≤ π2

(n + 1) δ 2 π

δ[|f (x − t)| + |f (x + t)| + 2 |s(x)|]dt

≤ π2

(n + 1) δ 2(2π || f || 1 + 2π || f || 1 + 2π |s(x)|) < π ,

|σn (f, x ) − s(x)| <

n ≥ N.

f [a, b], s(x) = f (x)

x ∈ [a, b]. f [a, b],

0 < δ < π x ∈ [a, b] x ∈ [a, b] 0 < t < δ.

N ∈N |s(x)|

M = sup {|f (x)| : a ≤ x ≤ b} ,

N x ∈ [a, b]. |s(x)| M

x ∈ [a, b] n ≥ N.

( ) σn (f, x ) αr (f, x ) ( ).



|| f (t) − c| − | f (t) − cn || ≤ | c − cn | < 3

t,

1h x+ h

x|f (t) − c|dt −

1h x+ h

x|f (t) − cn |dt <

1h x+ h

x 3dt =

3

,

0 = h ∈R .

|| f (x) − cn | − | f (x) − c|| < 3

.

1h x+ h

x|f (t) − c|dt − | f (x) − c| ≤

1h x+ h

x|f (t) − c|dt −

1h x+ h

x|f (t) − cn |dt

+ || f (x) − c| − | f (x) − cn || +1h x+ h

x|f (t) − cn |dt − | f (x) − cn |

< 3

+ 3

+ 3

= .

0 < |h| < δ (h ∈R ).

x ∈ E 0 = h ∈R u = x + t

v = x − t,

0 ≤ 1h h

0|f (x + t) + f (x − t) − 2f (x)|dt

≤ 1h h

0|f (x + t) − f (x)|dt +

1h h

0|f (x − t) − f (x)|dt

= 1h

x+ h

x |f (u) − f (x)|du + 1− h

x− h

0 |f (v) − f (x)|dv, c = f (x).

f ( ). x ∈R

f ( )

f f



+ δ

1n

|φ(t)|K n (t)dt + π

δ|φ(t)|K n (t)dt.

n ≥ N. ( ),

1n

0|φ(t)|K n (t)dt ≤ 1

n

0|φ(t)|(n+1) dt = ( n+1)Φ

1n

≤ 2nΦ1n

< 213

,

0 < 1n < δ.

π

δ|φ(t)|K n (t)dt ≤ π

δ|φ(t)|

a + 1

dt ≤ a + 1

Φ(π) < .

Φ(h) = h

0|φ(t)|dt

[1n , δ ]

G(t) = 1t 2

F (t) = h

0|φ(t)|dt,

( ),

δ

1n

|φ(t)|K n (t)dt ≤ δ

1n

|φ(t)| π2

(n + 1) t2 dt = π2

(n + 1) δ

1n

|φ(t)|1t2 dt

= π2

(n + 1) δ

0|φ(t)|dt

1δ 2

− n2 1n

0|φ(t)|dt + δ

1n

Φ(t) 2t3 dt

= π2

(n + 1)δ − 2Φ(δ ) − n2Φ

1n

+ 2 δ

1n

Φ(t)t− 3dt

< π2

(n + 1)δ − 1

13 +

2π2

(n + 1) δ

1n 13

t− 2dt

< π2

13 +

2π2

13(n + 1)(n − δ − 1) <

3π2

13 .

|σn (f, x ) − f (x)| < 12π

213

+ 3π2

13 + <

13

+ 613

+ 313

<

n ≥ N.



f ∈ L1(T ) s(f ) f R

g = f

x ∈R lım

n →∞sn (f, x ) = g(x) > 0

n0 ∈N |g(x) − sn (f, x )| <

2 n ≥ n0.

N > n 0

1N

n 0

k=0

(g(x) − sk (f, x )) < 2

.

n > N

|g(x) − σn (x)| =1

n + 1

n

k=0

(g(x) − sk (f, x ))

≤ 1n + 1

n 0

k=0

(g(x) − sk (f, x )) + 1n + 1

n

k= n 0 +1

|g(x) − sk (f, x )|

< 2

+ n − n0

n + 1 2 < .

( )

g(x) = lımn →∞

sn (f, x ) = lımn →∞

σn (f, x ) = f (x).

f ∈ C (T ),

D = re iθ : θ ∈R , 0 ≤ r ≤ 1

F (eiθ ) = f (θ) y F (re iθ ) = 12π π

− πf (θ − t)P r (t)dt si 0 ≤ r < 1.

|z | < 1

z = re iθ ( )

F (z ) =∞

k= −∞

r |k | f (k)eikθ =∞

k=0

f (k)z k +∞

k=1

f (− k)z k .

f |z | = |z | < 1.

|z | < 1 F



D 0 = {z ∈C : |z | < 1} .

f ∈ C (T ), ( ) [a, b] = [− π, π ]

(

)

lımr ↑1

F (re iθ ) = lımr ↑1

α r (f, θ ) = f (θ) = F (eiθ )

θ ∈R . > 0, 0 < δ < 1

1 − δ < r ≤ 1

|F (re iθ ) − F (eiθ )| < 2

θ ∈R .

f f

R , 0 < η ≤ δ

|F (eiθ ) − F (eiα )| = |f (θ) − f (α)| < 2

α, θ ∈R |α − θ| < η.

F D, α ∈R

> 0 δ η

|F (re iθ ) − F (eiα )| ≤ | F (re iθ ) − F (eiθ )| + |F (eiθ ) − F (eiα )| < 2

+ 2

=

1 − δ < r ≤ 1 |θ − α | < η.

f ∈ L1(T ) |f (x)| ≤ M < ∞ R

|| σn (f )|| u ≤ M n ≥ 0.

( ) ( )

|σn (f, x )| ≤ 12π π

− π|f (x − t)|K n (t)dt ≤

12π π

− πMK n (t)dt = M

x n.



L p(T ), 1 ≤ p

f R

t0

X 2π−

|| x|| ∞ = sup {|x(t)| : t ∈ [a, b]} .

X a = 0 b = 2π t0 = 0

T n = sn sn (x) t = 0 n

x



sn (x) = 1

2a0 +

n

m =1

am = 1

π 2π

0

x(t)1

2 +

n

m =1

cos(mt ) dt,

am = 1π 2π

0x(t)cos(mt )dt.

2sen12

t n

m =1

cos(mt ) =n

m =1

2sen12

t cos(mt )

=n

m =1

− sen m − 12

t + sen m + 12

t

= − sen12

t + sen n + 12

t

sen 12 t

1 + 2n

m =1

cos(mt ) = sen n + 1

2 tsen 1

2 t = Dn (t),

Dn (t) sn (x)

sn (x) = 12π 2π

0x(t)Dn (t)dt Dn (t) =

sen n + 12 t

sen 12 t

.

sn

|sn (t)| ≤ 12π

sup |x(t)|

2π

0

|Dn (t)|dt = ||x|| ∞

2π

2π

0

|Dn (t)|dt.

sn x

|| x|| ∞ = 1,

|| sn || ≤ 12π 2π

0|Dn (t)|dt.

|Dn (t)| = y(t)Dn ,



y(t) = +1 t Dn (t) ≥ 0 y(t) = − 1

y > 0 y

x ||x|| ∞ = 1

12π

2π

0[x(t) − y(t)] Dn (t)dt < .

12π 2π

0x(t)Dn (t)dt − 2π

0y(t)Dn (t)dt = sn (x) −

12π 2π

0|Dn (t)|dt < .

> 0 ||x|| ∞ = 1

|| sn || = 12π

2π

0|Dn (t)|dt.

|| sn ||

Dn

sen12

t < t2

t ∈ (0, 2π]

v = n + 1

2 t

|| sn || = 12π 2π

0

sen n + 12 t

sen 12 t

dt

> 1π 2π

0

| sen n + 12 t |

t dt

= 1π

(2n +1) π

0

| sen(v)|v

dv

= 1π

2n

k=0 (k+1) π

kπ

| sen(v)|v

dv

≥ 1π

2n

k=0

1(k + 1) π (k+1) π

kπ| sen(v)|dv

= 2π2

2n

k=0

1k + 1

→ ∞ n → ∞ ,



|| sn || X

x ∈ X |sn (x)|

sn x

t = 0

(X, τ ) M ⊆ X.

M =

∞

i=1 Ai ,

Ai ( int (Ai ) = ∅ ).

(X, τ ) M ⊆ X.

M c = X \ M

X Y

A ⊆ L(X, Y ). supT ∈A {|| T (x)||} < ∞ x

supT ∈A {|| T ||} < ∞

Dn =T ∈A

{x ∈ X : ||T (x)|| ≤ n} .

n, Dn ⊆ X z ∈ Dn , (xn ) ⊂ Dn

lımn →∞

xn = z. T ∈ A, ||T (x j )|| ≤ n j

j → ∞

T ||T (z )|| ≤ n. Dn ⊆ Dn , Dn

n ∈N x0 ∈ Dn B(x0, r ) ⊂ Dn r > 0 int (Dn ) = ∅

n ∈N , X \ Dn n.

β =∞

n =1

X \ Dn ,



X x ∈ β T ∈ A ||T (x)|| = ∞ ,

n ∈N

int (Dn ) = ∅, B(x0, r ) ⊂ Dn r > 0 x0 ∈ Dn .

T ∈ A x ∈ X ||x|| ≤ r,

|| T (x)|| = ||T (x − x0) + T (x0)||

≤ || T (x − x0)|| + || T (x0)||

≤ n + n = 2n.

x ∈ X ||x|| = 1 ||rx || = r

|| T (rx )|| ≤ 2n, ||T (x)|| ≤ 2nr ,

supT ∈A

{|| T ||} < ∞ .

X Y {T j,k } ⊆ L(X, Y ). k, x ∈ X

sup {|| T j,k (x)|| : j ∈N} = ∞ . B ⊆ X

x ∈ B sup {|| T j,k (x) : j ∈N} = ∞ k.

x ∈ X

sup {|| T j,k (x) : j ∈N} < ∞ k X.

k ∈N

sup {|| T j,k || : j ∈N} < ∞ .

x ∈ X

Gδ F ⊂ C (T ), :

f ∈ F, f

R .

n f x ∈T

sn (f, x ) = 12π π

− πf (t)Dn (x − t)dt,



Dn (t) = n j = − n eijt

x ∈T

Θn : C (T ) → R Θn (f ) = sn (f, x ).

|| Θn || = 12π π

− π|Dn (t)|dt,

12π π

− π |Dn (t)|dt → ∞ n → ∞ .

sup {|| Θn || : n ∈N} = ∞ .

F x ⊂ C (T ) sup {|| Θn (f )|| : n ∈N} = ∞ f ∈ F x .

F x = Acx = X \ Ax , Ax

Ax =∞

i=1

Ai ,

Ai ∪∞i=1 Ai ⊂ ∪∞

i=1 Ai ,

∞

i=1

X \ Ai ⊆∞

i=1

X \ Ai = Acx = F x .

X \ Ai F x Gδ

(x j )∞ j =1 ⊂ T

j ∈N F x j ⊂ C (T )

∞

j =1F x j ⊂ C (T )

Gδ

( ).



f ∈ L1(T ) 0 < δ ≤ π

(1). sn (f, x ) = 1π

δ

0 [f (x − t) + f (x + t)] sen( nt )t dt + n (x) n (x)

δ. R

n → ∞ .

(2). 1 = 2π δ

0sen( nt )

t dt + n , n δ

n → ∞ .

( )

Dn (t) = sen[(n + 1

2 )t]sen( t

2 ) = [sen(nt )] cot

t2

+ cos( nt ),

(

)

sn (f, x ) = 1π π

− πf (x − t)

12

cott2

sen(nt )dt + αn (x),

αn (x) = 12π π

− πf (x − t)cos(nt )dt.

lımt →0

12 cot t2 − 1t = 0 ,

lımt →0

12

cott2

− 1t

= lımt →0

− t2 sen t

2

t cos t2 + 2 sen t

2

= lımt →0

t cos t2 + 2 sen t

2

2t sen( t2 ) − 8 cos( t

2 )= 0 .

2π−

g1(t) = 12

cott2

− 1t

0 < |t | < π g1(0) = g1(π) = 0 L1(T )

π



sn (f, x ) = 1π π

− πf (x − t)

sen(nt )t

dt + β n (x) + αn (x),

β n (x) = 1π

π

− πf (x − t)g1(t)sen(nt )dt.

g2 2π− g2(t) = 1t

δ ≤ | t | < π

g2(t) = 0 |t | < δ t = π.

g2 L1(T ).

sn (f, x ) = 1π δ

− δf (x − t)

sen(nt )t

dt + γ n (x) + β n (x) + αn (x),

γ n (x) = 1π π

− πf (x − t)g2(t)sen(nt )dt.

n (x) = γ n (x) + β n (x) + αn (x).

( ) n → ∞ . f (t) = 1 t,

sn (f, x ) = 1 n x

f, g ∈ L1(T ) R : |g(t)| ≤ M < ∞

t ∈R

lım|n |→∞ π

− πf (x − t)g(t)eint dt = 0

x ∈R .

> 0 ( )

P (t) = p

j = − p

c j eijt ,

π

− π|f (u) − P (u)|du <

2M + 1

.

η =

1 + 4π p j = − p |c j |

,



( ) N ∈N

|g(k)| < η

k ∈Z

|k| ≥ N − p. |n | > N − p ≤ j ≤ p | j − n | ≥ | n | − | j | ≥ N − p,

π

− πeij (x − t )g(t)eint dt = eijx π

− πg(t)e− i ( j − n ) t dt = 2 π |g( j − n)| < 2πη.

π

− πP (x − t)g(t)eint dt ≤

p

j = − p

|c j | π

− πeij (x − t )g(t)eint dt

≤ 2πη p

j = − p

|c j | < 2

x |n | > N.

u = x − t f P 2π−

π

− πf (x − t)g(t)eint dt − π

− πP (x − t)g(t)eint dt

≤

π

− π|f (x − t) − P (x − t)| | g(t)eint |dt

≤ M π

− π|f (u) − P (u)|du <

2

.

π

− πf (x − t)g(t)eint dt ≤ π

− πf (x − t)g(t)eint dt − π

− πP (x − t)g(t)eint dt

+ π

− πP (x − t)g(t)eint dt <

2

+ 2

= .



f 1, f 2 ∈ L1(T ),

f 1(x) = f 2(x) x I ⊆R .

lımn →∞ |sn (f 1, x) − sn (f 2, x)| = 0 x ∈ I . J ⊆ I .

J ⊆ I 0 < δ < π x + t x − t I

x ∈ J 0 ≤ t ≤ δ.

f = f 1 − f 2, ( )

sn (f, x ) = sn (f 1, x) − sn (f 2, x) = n (x)

x ∈ J.

f ∈ L1(T ), 0 < δ ≤ π X ⊆R

φ(x, t ) = f (x + t) + f (x − t) − 2f (x) f X :

|f (x)| ≤ M < ∞

x ∈ X.

(1). lımn →∞

sn (f, x ) = f (x) x ∈ X,

(2). lımn →∞

1π δ

0

φ(x, t )t

sen(nt )dt = 0 x ∈ X.

f (x)

sn (f, x ) − f (x) = 1π δ

0

φ(x, t )t

sen(nt )dt + [ n (x) − n (x)].

f X, [ n (x) − n (x)]

X.



f ∈ L1(T ) x0 ∈R

0 < δ ≤ π

δ

0

|φ(x0, t )|t

dt < ∞ ,

φ ( ). lımn →∞

sn (f, x 0) = f (x0).

X = {x0} . g R

g(t) = t− 1φ(x0, t ) 0 < t < δ 0

( )

lımn →∞

1π ∞

−∞g(t)e− int dt = lım

n →∞

1π δ

0

φ(x0, t )t

e− int dt = 0.

lımn →∞

1π δ

0

φ(x0, t )t

sen(nt )dt = 0, x ∈ X.

( ).

f ∈ L1(T ) x0 ∈R

|f (x0 + t) − f (x0)| ≤ M |t |α 0 < |t | < δ

α δ lımn →∞

sn (f, x 0) = f (x0).

|φ(x0, t )| = |f (x0 + t) + f (x0 − t) − 2f (x0)| ≤ 2M

0 < t < δ, t− 1|φ(x0, t )| ≤ 2Mt α − 1 0 < t < δ

δ

0 tα − 1

dt = δ α

α < ∞ .

( ).

f ∈ L1(T ), x0 ∈R , f x0.

lımn →∞

sn (f, x 0) = f (x0).



f x0

lımt →0+

f (x0 + t) − f (x0)

t

= f + (x0)

lımt →0−

f (x0 + t) − f (x0)t

= f − (x0).

δ 1 > 0 δ 2 > 0

|f (x0 + t) − f (x0)| ≤ (1 + |f + (x0)|)t 0 < t < δ 1

|f (x0 + t) − f (x0)| ≤ (1 + |f − (x0)|)|t |

− δ 2 < t < 0. δ = mın {δ 1, δ 2} α = 1 M max |f + (x0)|, |f − (x0)|

|f (x0 + t) − f (x0)| ≤ M |t | 0 < |t | < δ,

( ).

[a, b] ⊂R a ≤ b [a, b]

P = {a = x0 ≤ x1 ≤ x2 ≤ · · · ≤ xn = b} .

Ba,b

[a, b]. f [a, b],

f [a, b] P = {a = x0 ≤ x1 ≤ x2 ≤ · · · ≤ xn = b}

V ba (f, P ) =n

k=1

|f (xk ) − f (xk− 1)| ∈ [0, ∞ ).

f [a, b]

V ba (f ) = supP ∈B a,b

V ba (f, P ) ∈ [0, ∞ ].

f [a, b] V ba (f ) < ∞ . ([a, b])

f ∈ L1(T ) V 2π0 f < ∞

f ∈ (T )

|nf (n)| ≤ 14

V 2π0 f n ∈Z .



k, n ∈Z n = 0 , u = t + kπ

n

f (n) = 12π

2π

0 f (u)e− inu dt = (− 1)k

2π 2π

0 f t + kπ

n e− int dt;

2 f (n) = (− 1)k

2π 2π

0f t +

kπn

− f t + (k − 1)π

ne− int dt.

2|n |

4nf (n) = (− 1)k

2π

2π

0

2|n |

k=1

f t + kπ

n− f t +

(k − 1)πn

e− ikt dt,

4|nf (n)| ≤ 12π 2π

0

2|n |

k=1

f t + kπ

n− f t +

(k − 1)πn

dt,

4|nf (n)| ≤ V 2π0 f.

∞

n =0

cn = s, A −∞

n =0

cn = s.

∞n =0 cn s M − ∞

n =0 cn s M

A −∞

n =0

cn = s ncn −→ 0,

∞

n =0

= s.



X = ∅ (an (x))∞

n =0 C > 0

na n (x) ≤ C, x ∈ X n ≥ 0. ∞

n =0 an (x)

x ∈ X

A(x, r ) =∞

n =0

an (x)r n ,

0 ≤ r < 1 lımr ↑1

A(x, r ) = s(x) x ∈ X.

> 0 0 < r 0 < 1 |A(x, r ) − s(x)| < x ∈ X

r 0 < r < 1.

k ∈N A(x, r k ) → s(x) X r ↑ 1

r1k0 < r < 1 r0 < r k < 1

P (r ) = mk=0 ck r k

P (0) = 0 P (1) = 1 ,

∞

n =0

an (x)P (r n ) =m

k=1

ck

∞

n =0

an (x)r nk =m

k=1

ck A(x, r k ) →m

k=1

ck s(x)

= s(x)P (1) = s(x),

X r ↑ 1.

[0, 1]

φ(r ) = 0 0 ≤ r < 12

1

12 ≤ r ≤ 1.

x ∈ X 0 < r < 1

Φ(x, r ) =∞

n =0

an (x)φ(r n ) =N (r )

n =0

an (x),

N (r ) = m ax n ∈N : n ≤ − log(2)log( r ) . N (r )

[12 , 1]

N.



lımr ↑1

Φ(x, r ) = s(x),

X.

ψ(r ) =

− 11− r 0 ≤ r < 1

2

1r

12 ≤ r ≤ 1.

φ(r ) = r + r(1 − r )ψ(r ),

r ∈ [0, 1]. 0 < < C δ = 24C

f 1 f 2 [0, 1]

f 1(r ) + δ ≤ ψ(r ) ≤ f 2(r ) − δ,

0 ≤ r ≤ 1

1

0[f 2(r ) − f 1(r )] dr < 10δ.

( f 1 = ψ − δ [0, 12 ) ∪ [1

2 + δ,1] [12 , 1

2 + δ ]

f 2 = ψ + δ [0, 12 − δ ]∪ [1

2 , 1] [12 − δ, 1

2 ] )

f 1 f 2, Q1 Q2

|f j (r ) − Q j (r )| < δ,

0 ≤ r ≤ 1 j = 1, 2.

Q1(r ) < ψ (r ) < Q 2(r ),

0 ≤ r ≤ 1

1

0[Q2(r ) − Q1(r )] dr ≤ 1

0|Q2(r ) − Q1(r )|dr

≤ 1

0|Q2(r ) − f 2(r )|dr + 1

0|f 1(r ) − Q1(r )|dr + 1

0|f 2(r ) − f 1(r )|dr

< δ + δ + 10δ = 12δ.



1

0[Q2(r ) − Q1(r )] dr < 12δ.

P 1

P 2

Q

P j (r ) = r + r(1 − r )Q j (r ), Q(r ) = Q2(r ) − Q1(r ).

P 2(r ) − P 1(r ) = r (1 − r )Q(r );

P 1(r ) ≤ φ(r ) ≤ P 2(r, )

Q(r ) > 0, 1

0 Q(r )dr < 12δ = 2C ,

P 1(0) = P 2(0) = 0 , P 1(1) = P 2(1) = 1

0 ≤ r ≤ 1.

0 ≤ r ≤ 1 n ∈N

1 − r n = (1 − r )(1 + r + ... + r n − 1) ≤ n(1 − r ).

Q Q(r ) = qk=0 bk r k .

Φ(x, r ) −∞

n =0

an (x)P 1(r n ) =∞

n =1

an (x) [φ(r n ) − P 1(r n )]

≤∞

n =1

C n

[P 2(r n ) − P 1(r n )]

= C ∞

n =1

1n

(1 − r n )r n Q(r n )

≤ (1 − r )C ∞

n =1

r n Q(r n )

= (1 − r )C q

k=0

∞

n =1

r n (k+1)

= C q

k=0

bk(1 − r )r k+1

1 − r k+1 = Cg(r ),



g (0, 1).

∞

n =0

an (x)P 2(r n ) − Φ(x, r ) ≤ Cg(r ),

0 < r < 1 x ∈ X. lımr →1

(1 − r k+1 )(1 − r )

= k + 1 ,

lımr ↑1

g(r ) =q

k=0

bk

k + 1 = 1

0Q(r ) <

2C

.

0 < r 0 < 1 Cg(r ) < 2 r0 < r < 1.

∞

n =0

an (x)P 2(r n ) −

2 < Φ(x, t ) <

∞

n =0

an (x)P 1(r n ) +

2,

r 0 < r < 1 x ∈ X. P j j = 1, 2 0 < r j < 1

s(x) −∞

n =0

an (x)P j (r n ) < 2

,

r j < r < 1 x ∈ X r3 = m ax {r 0, r 1, r 2}

s(x) − < Φ(x, r ) < s (x) + , r 3 < r < 1 x ∈ X.

f ∈ L1(T ) a < b R

V ba f < ∞

a < x < b

(1).

∞k= −∞ f (k)eikx = lım

n →∞sn (f, x ) =

f (x+) + f (x− )2

.

(a, b)

J ⊆ (a, b)

(2). lımn →∞

sn (f, x ) = f (x)

g ∈ L1(T ) g = f [a, b] V 2π0 g < ∞

b − a ≥ 2π g = f b− a < 2π g = 0 (b, a + 2π)

( )

k[g(− k)e− ikx + g(k)eikx ] ≤ 12

V 2π0 g = C < ∞ ,



k ∈N x ∈R .

( )

∞k= −∞ g(k)eikx

g = f

X = J,

X = {x} .

( ) ∞k = −∞ g(k)eikx

g g = f [a, b]

( ) I = ( a, b) f

f ∈ C (T ) V 2π0 f < ∞ ( f ∈ (T ))

sn (f ) → f

R

n → ∞ . [a, b] = [− π, 3π] J = [0, 2π] ( )

sn (f ) → f R .

f (φ) =

0 φ = 01log( 1

| x | ) |x| ≤ 12 ,

f [0, 12 ]

f x = 0 0 < δ < 1

δ

0

|f (0 + t) + f (0 − t) − 2(0) |t

dt = δ

0

2f (t)t

dt = 2 δ

0

1t log 1

t

dt,

u = 1log( 1

t ) du = 1

t u2dt

1t log 1t

dt = uu2 du = 1u du = log (u) .

2 δ

0

1t log 1

t

dt = lımn →∞

2 δ

1n

1t log 1

t

dt = − 2 lımn →∞

log log1t

δ1n

→ ∞

n → ∞ .



f (φ) = 0 φ = 0

φ sen 1φ |x| ≤ π,

x = 0

δ

0

|f (0 + φ) + f (0 − φ) − 2f (0) |φ

dφ = δ

0

|φ sen 1φ − φ sen 1

− φ |

φ dφ ≤ δ

0

2φφ

dφ < ∞ .

P = x0 = 0, x1 = 2

π (2n + 1), x2 =

1nπ

,...,x n − 1 = 2π

, xn = π ,

V π0 (f, P ) =n

j =1

x j sen 1x j

− x j − 1 sen 1x j − 1

= 4π

k

j =1

12 j + 1

→ ∞

k → ∞

f ∈ L1(T ), X ⊆R , 0 < δ < π

φ(t) = φ(x, t ) = f (x + t) + f (x − t) − 2f (x).

(1). lımh ↓0

1h h

0φ(t)dt = 0 x ∈ X.

(2). lımh ↓0

δ

ht− 1|φ(h + h) − φ(t)|dt = 0 x ∈ X.

sn (f, x ) → f (x) n → ∞ .

f > 0

Φ(u) = u

0φ(t)dt.



0 < δ < δ2

x ∈ X

|u− 1Φ(u)| < 0 < u ≤ δ

δ

ht− 1|φ(t + h) − φ(t)|dt < 0 < h < δ.

g(s) = s

0|f (u)|du,

0 < M < ∞ |f (x)| < M x ∈ X h > 0 x ∈ X.

δ+ h

δt− 1|φ(t)|dt ≤ δ − 1

δ+ h

δ[|f (x + t)| + |f (x − t)| + 2 |f (x)|]dt

≤ δ − 1 (2wg(h) + 2 Mh) ,

wg(h) = sup {|g(u) − g(v)| : |u − v| ≤ h} .

g > 0 s ∈R

|g(s + h) − g(s)| =

s+ h

0

|f (u)|du −

s

0

|f (s)| =

s+ h

s

|f (u)|du;

δ > 0

s+ h

s|f (u)|du < |h| < δ.

g 2π g

R , wg(h) → 0 h ↓ 0. N ∈N N > 2π

δ

x ∈ X,

δ+ h

δt− 1|φ(t)|dt < h = πn

n ≥ N.

n > N h < δ2 δ + h < δ ≤ π.

F (t) = t

0φ(u)du G(t) =

sen(nt )t

,



b

a

φ(t)sen(nt )

t dt = t− 1Φ(t)sen(nt ) b

a −

b

a

t− 1Φ(t) n cos(nt ) − t− 1 sen(nt ) dt;

|n cos(nt ) − t− 1 sen(nt )| ≤ 2n,

t ∈R x ∈ X

b

aφ(t)

sen(nt )t

dt < 2 + 2n (b − a) ≤ (4π + 2)

0 ≤ a < b ≤ 2h.

x ∈ X n > N

δ

0

φ(t)t

sen(nt )dt < 29 .

( ), sn (f, x ) → f (x) X.

h

0

φ(t)t

sen(nt )dt < 15 .

I = δ

h

φ(t)t

sen(nt )dt = δ+ h

2h

φ(t)t

sen(nt )dt + α,

α = δ

h

φ(t)t

sen(nt )dt− δ+ h

2h

φ(t)t

sen(nt )dt = 2h

h

φ(t)t

sen(nt )dt− δ+ h

δ

φ(t)t

sen(nt )dt.

|α | < (4π + 2) + < 16 .

sen(n(u + h)) = − sen(nu )



2I − α = δ

h

φ(u)u

− φ(u + h)

u + hsen(nu )du

= δ

h

φ(u) − φ(u + h)u + h sen(nu )du

+ h 2h

h

φ(u)sen(nu )u(u + h)

du + h δ

2h

φ(u)sen(nu )u(u + h)

du.

= I 1 + hI 2 + hI 3.

|I 1| < .

(u + h)− 1

[0, 2h]

|hI 2| =12 ξ

h

φ(u)sen(nu )u

du < 8 .

hI 3 = − h δ− h

h

φ(t + h) sen(nt )(t + h)(t + 2h)

dt = − h δ

h

φ(t + h)sen(nt )(t + h)(t + 2h)

dt + β ;

|β | = h δ+ h

δ

φ(u)sen(nu )u(u + h) du ≤

hδ + h

δ+ h

δu− 1|φ(u)|du < .

hI 2 + 2 hI 3 − β = h δ

h

φ(t)t(t + h)

− φ(t + h)

(t + h)(t + 2h)sen(nt )dt = A + B,

A = h δ

h

φ(t) − φ(t + h)(t + h)(t + 2h)

sen(nt )dt B = 2h2 δ

h

φ(t)sen(nt )t(t + h)(t + 2h)

dt.

|A| ≤ h δ

h

|φ(t) − φ(t + h)|t3h

dt < 3

.



B = 2h2Φ(δ ) sen(nδ )δ (δ + h)(δ + 2 h)

− 2h2 δ

hΦ(t)

ddt

sen(nt )t(t + h)(t + 2h)

dt.

ddt

sen(nt )t(t + h)(t + 2h)

=n cos(nt )

t(t + h)(t + 2h) −

sen(nt )(3t2 + 6 th + 2h2)t2(t + h)2(t + 2h)2

≤ nt − 3 + (3t2 + 6 nt + 2h2)t2(t + h)2(t + 2h)2

≤ nt − 3 + (3t2 + 6 nt + 3h2)t2(t + h)2(t + 2h)2

= nt − 3 + 3 (t2 + 2 nt + h2)t2(t + h)2(t + 2h)2

≤ nt − 3 + 3 t− 4.

|B | < + 2h2 δ

h(nt − 2 + 3 t− 3)dt < + 2π + 3 < 11 .

4I = 2α + 2 I 1 + hI 2 + β + A + B,

|4I | < 32 + 2 + 8 + + + 11 < 56 .

f ∈ L1(T ) I ⊆R

lımt →0

supx∈I

|f (x + t) − f (x)| log 1|t |

= 0 .

sn (f ) → f J ⊂ I .



J ⊂ I 0 < δ < 1 x ± t ∈ I

x ∈ J 0 < t ≤ δ . ( ) X = J.

lımt →0

supx∈I

|f (x + t) − f (x)| log 1|t |

= 0 ,

> 0 0 < δ < e − 1

|f (x + t) − f (x)| log 1|t |

< 0 < |t | < δ.

e < 1δ < 1

| t | ,

|f (x + t) − f (x)| < 0 < |t | < δ.

f I J ⊂ I .

> 0, 0 < δ < mın {δ , e− 1}

|f (u + v) − f (u)| < log 1|v|

− 1

2 u ∈ I 0 < |v| < δ.

0 < t < δ x ∈ X

|φ(t)| ≤ | f (x + t) − f (x)| + |f (x − t) − f (x)| < log1

t

− 1

< .

1h h

0φ(t)dt < x ∈ X 0 < h < δ,

( ).

x ∈ X 0 < t < δ , 0 < h < δ

|φ(t + h) − φ(t)| ≤ | f (x + t + h) − f (x + t)| + |f (x − t − h) − f (x − t)| < log

1

h

− 1

x ± t ∈ I . log(δ ) < 0 − log(h) = log( h− 1),

δ

ht− 1|φ(x + t) − φ(t)|dt < log(h− 1) − 1 δ

ht− 1dt < ,

x ∈ X 0 < h < δ. ( ).



p(T )

p(T ) L2(T )

L p(T )

1 ≤ p

p = 2

L1(T )

f, g ∈ L1(T ). t f (t − τ )g(τ )

τ T

(1). h(t) = 12π 2π

0 f (t − τ )g(τ )dτ, h ∈ L1(T ).

(2). ||h|| L 1 (T ) ≤ || f || L 1 (T ) || g|| L 1 (T ) .

(3). h(n) = f (n)g(n) n

F (t, τ ) = f (t − τ )g(τ ) τ F (t, τ )

f (t − τ )

12π 2π

0

12π 2π

0|F (t, τ )|dt dτ =

12π 2π

0|g(τ )||| f || L 1 dτ = || f || L 1 || g|| L 1 .

12π 2π

0|h(t)|dt =

12π 2π

0

12π 2π

0F (t, τ )dτ dt ≤

14π2 2π

0 2π

0|F (t, τ )|dtdτ

= || f || L 1 || g|| L 1 .

h(n) = 12π 2π

0h(t)e− int dt =

14π2 2π

0 2π

0f (t − τ )e− in ( t − τ )g(τ )e− inτ dtdτ

= 1

2π 2π

0

f (t)e− int dt 1

2π 2π

0

g(τ )e− inτ dτ = f (n)g(n).

f, g ∈ L1(T ) f g h ( ) f ∗g

L p(T ) sup || sn || p < ∞

K ||sn (f )|| L p ≤ K || f || L p f

n



sn (f ) f f ∈ L p(T )

sn (f ) f ∈ L p(T )

|| sn || L p

f ∈ L p(T ), > 0 ( )

P ||f − P || L p < 2K .

n P sn (P ) = P

|| sn (f ) − f || L p = || sn (f ) − sn (P ) + P − f || L p

≤ || sn (f − P )|| L p + || P − f || L p

≤ K 2K +

2K < .

K n (t) =n

j = − n

1 − | j |n + 1

eijt Dn (t) =

n

j = − n

eijt .

σn (f, t ) =n

j = − n

1 − | j |

n + 1

f ( j )eijt = ( K n ∗f ) ( t)

sn (f, t ) =n

j = − n

f ( j )eijt = ( Dn ∗f ) ( t).

( ) ||sn (f, t )|| L 1 ≤ || Dn (t)|| L 1 || f || L 1

|| sn || L 1 ≤ || Dn || L 1 .

|| σN (Dn )|| L 1 = || sn (K N )|| L 1 ≤ || sn || L 1 || K N || L 1 = || sn || L 1 .

Dn (t) ∈ C (T ), ( ), σN (Dn ) → Dn

N → ∞

|| sn || L 1 ≥ || Dn || L 1 .

|| sn || L 1 = ||Dn || L 1 .

( ) ||Dn || L 1 → ∞ n → ∞ . ( )

L1(T )



p(T )

1 < p < ∞

H : L p(T ) → L p(T ), 1 < p < ∞ ,

T ,

Hf (n) = − i sgn (n) f (n).

f ∈ L p(T )

Hf ∼∞

n = −∞

− i sgn (n) f (n)eint .

P : L p(T ) → L p(T )

P (f ) = 12f (0) + 12 (f − iHf ) ,

P 1 < p < ∞ |f (0) | ≤ || f || L p ,

P (f ) ∼n ≥ 0

f (n)eint ,

i(− i sgn (n)) = sgn(n). m ∈N

e− imt P eimt f − ei (m +1) t P e− i (m +1) t f = sm (f ).

eimt f ∼∞

n = −∞

f (n)ei(n + m ) t



P eimt f ∼∞

n ≥− m

f (n)ei(n + m ) t

e− imt P eimt f ∼∞

n ≥− m

f (n)eint

ei (m +1) t P e− i(m +1) t f ∼∞

n ≥ m +1

f (n)eint .

e− imt P eimt f − ei (m +1) t P e− i (m +1) t f = sm (f ).

sup || sn || L p ≤ 2|| P || L p < ∞ ,

1 < p < ∞

1 < p < ∞ f ∈ L p(T ),

lımn →∞ || sn (f ) − f || L p = 0.

sup || sn || L p < ∞ ( )

lımn →∞

|| sn (f ) − f || L p = 0.



E ⊂ [− π, π ]

λ(E ) = 0 λ f ∈ C (T ) sn (f, x ) x ∈ E.

V ⊆ [− π, π ]

δ = λ(V ) > 0.

(1). ||Q|| u = sup {|Q(t)| : t ∈R }

(2). S ∗(Q, t ) = sup {|S n (Q, t )| : n ≥ 0} > 1π log 1

3δ t ∈ V.

0 < < 1

{t1, t2, t3,...,t p} ⊂ V p < δ [t j − , t j + ] 1 ≤ j ≤ p V.

δ 1 ≤ δ 2 ≤ · · · ≤ δ q q

V, 0 < < 1 < δ 1 1 ≤ k ≤ q

Ak = {n ∈N : n ≥ δ k } ;

pk = mın {Ak } − 1 pk < δ k ≤ 2 pk .



k− pk 2 .

p = p1 + p2 + · · · + pq

p = p1 + p2 + · · · + pq < δ 1 + δ 2 + · · · + δ q ≤ δ,

pk 1 ≤ k ≤ q pk

k− {tk1, t k2,...,t kp k } [tkj − , tkj + ]

1 ≤ j ≤ q k 2

{t1, t2, t3,...,t p} ⊂ V.

z ∈C z = 1 φ(z ) = (1 − z )− 1

Re φ(re it ) = 1

2φ(re it ) + φ(re it ) =

1

2

1 − re − it

(1 − re it )(1 − re it ) +

1 − re it

(1 − re − it )(1 − re it )

= 12

1 + 1 − r 2

1 + r2 − 2r cos(t)

0 ≤ r < 1 t ∈R . φ(0) = 1 , − 1 ≤ cos(t) − 2r cos(t) ≤ 2r

Re φ(re it ) = 12

1 + 1 − r 2

1 + r2 − 2r cos(t)≥

12

1 + 1 − r 2

1 + r2 + 2 r>

12

|z | < 1. cos( ) > 1 − 2

0 < < 1

Re φ( 1

1 + eit ) =

12

1 +1 − ( 1

1+ )2

1 + ( 11+ )2 − 2

1+ cos(t)

= 12

2 − 21+ cos(t)

1 + ( 11+ )2 − 2

1+ cos(t)

> (1 + )2

(1 + ) (1 + ) − cos(t)

1 + (2 − 1)(1 + )2

≥ (1 + ) (1 + ) − 12 3 + 3 2

= + 2

2 3 + 3 2 > + 2

3 3 + 3 2 = 13

t ∈ [− , ]

f

f (z ) = 1 p

p

j =1

φ(ze− it j ).



f (0) = 1 Re [f (z )] > 12 |z | < 1.

Re f 1

1 + eit = Re1 p

p

j =1φ

11 + ei ( t − t j ) >

13 p >

13δ

t ∈ V

j |t − t j | ≤ .

F (z ) = log( f (z )) = log( |f (z )|) + iArg (f (z ))

F (0) = 0

|Im [F (z )]| = |Arg (f (z )) | < π2

,

|F (z )| ≥ Re [F (z )] = log( |f (z )|) ≥ log(Re [f (z )])

|z | < 1.

F

F (z ) =∞

n =1

An z n ,

|z | = 11+ .

|z | < R = 1 − (2a)− 1, a = 1 + − 1

11+ = 1 − a− 1 < R

|φ(z ) − a | = az − (1 − a− 1)

z − 1< a |z | < R,

α = 1 − a− 1 < 1 |z | < (1+ α )2 = R 2Re(z ) < 1 + α,

|z − 1|2 − | z − α |2 = (1 − α)(1 + α − 2Re(z )) > 0

|z − α | < |z − 1|.

|f (z ) − a | =1 p

p

j =1

φ ze− it j − p a p

≤ 1 p

p

j =1

φ ze− it j − a < a,

|z | < R. z ∈C |z | < 1



log(1 + z ) =∞

n =1

(− 1)n − 1

n z n .

z = w− aa ,

log(w) = log( a) +∞

k=1

(− 1)k− 1

kw − a

a

k

,

|w − a | < a.

f k (z ) = (− 1)k− 1

kf (z ) − a

a

k

,

w = f (z )

F (z ) = log( a) +∞

k=1

f k (z ),

|z | < R |z | ≤ r

0 < r < R.

φ |z | < 1

f

k

f k

|z | < R.

{An }∞n =1 ⊆ C

(A0 = 0 F (0) = 0) t ∈R

g(t) = 2π

F 11 +

eit =∞

n =1

an eint ,

an = 2π

11+

n An . ||Im (g)|| u < 1

β = sup |g(t)| : t ∈ V

β ≥ | g(t)| = 2π

F 1

1 + eit >

2π

log 13δ

.

gN (t) = N n =1 an eint

|| Im (gN )|| u → ||Im (g)|| u N → ∞ .



t ∈ V

|gN (t)| ≥ β − || g − gN || u → β N → ∞ . N ∈N

|| Im (gN )|| u < 1

|gN (t)| > 2π

log 13δ

t ∈ V.

Q

Q(t) = e− iNt Im (gN (t)) = e− iNt

2i gN (t) − gN (t)

= 12i

0

k = − N +1

ak+ N eikt − 12i

− N − 1

k= − 2N

a− k− N eikt .

S ∗(Q, t ) ≥ | S N (Q, t )| =e− iNt

2i gN (t) =

12

|gN (t)|,

E

E.

E ⊆ [− π, π ] λ(E ) = 0

f ∈ C (T ) s(f )

f (n) = 0 n < 0

2j > 0 j ∈N {I j,n }∞n =1 ⊆ [− π, π ]

E ⊆∞

n =1

I j,n

∞

n =1

|I j,n | < 2 j .

{I n }∞n =1 = I ( j,n ) : ( j, n ) ∈N × N ,

{I n }∞n =1

[− π, π ], ∞n =1 |I n | < t ∈ E

{n ∈N : t ∈ I n }



1 ≤ n1 < n 2 < ...

3

∞

n = n k|I n | < e

− πk 3

.

V k = n k +1n = n k

I n Qk ( ) V k

3δ k = 3λ(V k ) < e− πk 3 ,

13δk

> e πk 3 ,

log( 13δk

) > log(eπk 3 ) = πk3.

S ∗(Qk , t ) > k 3 t ∈ V k .

I n,

{k ∈N : t ∈ V k } t ∈ E.

N k Qk

N k = m ax |n | : n ∈Z , Qk (n) = 0 .

( pk )∞k=1 p1 = N 1

pk − N k > pk− 1 + N k− 1 k > 1.

f R

f (t) =∞

k=1

k− 2eip k t Qk (t).

|| Qk (t)|| u < 1

|k− 2eip k t Qk (t)| = |k− 2Qk (t)| = k− 2|| Qk (t)|| u ≤ k− 2.

M −

R

f ∈ C (T ).

f (n) =∞

k=1

k− 2

2π π

− πQk (t)e− i (n − pk ) t dt =

∞

k=1

k− 2 Qk (n − pk ),

Qk (n − pk ) = 0

− N k + pk ≤ n ≤ N k + pk .



f (n) = k− 2 Qk (n − pk ) |n − pk | ≤ N k f (n) = 0 f (n) = 0 n < 0.

n < 0 = − N 1 + p1 ≤ − N k + pk n − pk < − N k k

k ∈N 0 ≤ N ≤ N k t ∈R

S pk + N (f, t ) − S pk − N − 1(f, t ) = pk + N

n = pk − N

f (n)eint =N

j = − N

k− 2 Qk ( j )ei ( pk + j ) t

= k− 2eip k t S N (Qk , t ).

t ∈ E, j ∈N

.

k = k(t, j )

k > j pk − N k − 1 > j t ∈ V k

0 ≤ N ≤ N k

|S N (Qk , t )| > k 3.

m j = pk − N − 1 n j = pk + N

|S n j (f, t ) − S m j (f, t )| = k− 2|S N (Qk , t )| > k > j,

(S n (f, t ))∞n =1



2π R

f (x) = − 1 − π < x < 01 0 < x < π.

f

f ∼ 4π

∞

j =0

sen[(2 j + 1) t](2 j + 1)

,

sN (f, t ) = 4π

N

k=0

sen[(2k + 1) t](2k + 1)

= 4

π

N

k=0 t

0cos[(2k + 1) x]dx

= 4π t

0

N

k=0

cos[(2k + 1) x] dx.

sen(a + b) − sen(a − b) = 2 cos(a)sen(b),

N

k=0

cos[(2k + 1) x] = sen[2(N + 1) x]

2sen(x) .



sN (f, t ) = 2π

t

0

sen[2(N + 1) x]sen(x)

dx,

sN (f, t ) = 2π

sen[2(N + 1) t]sen(t)

,

(0, π ] t = π

2(N + 1).

sN f, π

2(N + 1)=

2π π

2( N +1)

0

sen[2(N + 1) x]sen(x)

dx,

u = 2( N + 1) x

sN (f, π

2(N + 1)) =

2π π

0

sen(u)u

u/ 2(N + 1)sen(u/ 2(N + 1))

du.

lımN →∞

supv∈(0 ,π )

1 − v/ 2(N + 1)

sen(v/ 2(N + 1))= 0 .

lımN →∞

2π π

0

sen(u)u

du − sN f, π

2(N + 1)

= lımN →∞

2π π

0

sen(u)u

du − 2π π

0

sen(u)u

u/ 2(N + 1)sen(u/ 2(N + 1))

du

= lımN →∞

2π π

0

sen(u)u

1 − u/ 2(N + 1)

sen(u/ 2(N + 1))du = 0.

lımN →∞

sN f, π2(N + 1)

= 2π

π

0sen(u)

u du > 1.

x = 0



2π

g(x) = 0 x = 0

π − t2 0 < x < 2π.

g ∼∞

k=1

sen(kt )k

,

sN (g, t) =∞

k=1

sen(kt )

t =

t

0

N

k=1

cos(kx) dx =

t

0

sen[(N + 12 )x]

2sen(x2 )

dx − t

2

= t

0

sen[(N + 12 )x]

x dx+ t

0

12sen( x

2 ) −

1x

sen[(N +12

)x]dx−t2

.

g(x) = 1

2sen( x2 )

− 1x

(0, π ] g ∈ C 1.

lımx →0 g(x) = lımx →0

x − 2 sen( x2 )

2x sen( x2 )

= lımx →0

1 − cos( x2 )

x cos( x2 ) + 2 sen( x

2 )

= lımx →0

12 sen( x

2 )− x2 sen( x

2 ) + 2 cos( x2 )

= 0.

g(0) = 0 g ∈ C 1 [0, π]

g ∈ C 1

t

0 12 sen( x

2 ) − 1x sen[(N + 12)x]dx → 0

0 ≤ t ≤ π N → ∞ . (hn )∞

n =1 hn → 0 n → ∞

nh n → π n → ∞ .



lımN →∞ sN (g, hN ) = lımN →∞ hN

0

sen[(N + 12 )x]

x dx+

lımN →∞ hN

0

12sen( x

2 ) −

1x

sen[(N + 12

)x]dx − lımN →∞

hN

2 .

N → ∞ .

u = ( N + 12 )x

lımN →∞ hN

0

sen[(N + 12 )x]

x dx

lımN →∞

(N

+12 )

hN

0sen(u)

u dx →

π

0sen(u)

u du > π

2.

x = 0



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