ď²y - +q(x)y

(1)

Volume 1, Number1 (2010), 137 146

EQUICONVERGENCE THEOREMS

FOR STURMLIOVILLE OPERATORS

WITH SINGULAR POTENTIALS

(RATE OF EQUICONVERGENCE IN

W ₂ ^θ

^NORM)

I.V. Sadovnihaya

Communiated by M.Otelbaev

Keywords and phrases: equionvergene, SturmLioville operators, singular

potential.

Mathematis Subjet Classiation: 34L10, 34L20,47E05.

Abstrat.Westudy the SturmLiouville operator

Ly = l(y) = − d ² y

dx ² + q(x)y

^with

Dirihlet boundary onditions

y(0) = y(π) = 0

ⁱⁿ ^the ^spae

L ₂ [0, π]

^. ^W^e ^assume

that the potential has the form

q(x) = u ^′ (x)

^, ^where

u ∈ W ₂ ^θ [0, π]

^with

0 < θ <

1/2

^. ^Here

W ₂ ^θ [0, π] = [L 2 , W ₂ ¹ ] θ

îs ^the ^Sobolev ^spae. ^Wê ônsider ^the ^problem ôf

equionvergene in

W ₂ ^θ [0, π]

^norm ôf ^two êxpansions ôf â ^funtion

f ∈ L ₂ [0, π]

^.

The rst one is onstruted using the system of the eigenfuntions and assoiated

funtions of the operator

L

^. ^The ^seond ône îs ^the ^Fôurier êxpansion ⁱⁿ ^the ^series

of sines. We show that the equionvergene holds for any funtion

f

ⁱⁿ ^the ^spae

L 2 [0, π]

^.

1 Introdution

In this paperwe deal with the SturmLiouville operator

Ly = l(y) = − d ² y

dx ² + q(x)y, (1)

with Dirihlet boundary onditions

y(0) = y(π) = 0

ⁱⁿ ^the ^spae

L 2 [0, π]

^. ^We

assume that the potential

q

^is omplex-valued and has the form

q(x) = u ^′ (x)

^,

where

u ∈ W ₂ ^θ [0, π]

^with

0 < θ < 1/2

^. ^Here ^the ^derivative ^is ^treated ⁱⁿ ^the

distributional sense, and

W ₂ ^θ [0, π] = [L 2 , W ₂ ¹ ] θ

^is ^the ^Sobolev ^spae ^with ^frational

orderof smoothnessdened by interpolation.This lass of operatorswas dened in

the papers of A.M. Savhuk and A.A. Shkalikov [7℄[9℄. In partiular, itwas shown

there that

L

^is ^bounded ^from^below^and ^has ^purely ^disrete^spetrum.

Weonsider the problemof equionvergene in

W ₂ ^θ [0, π]

^norm^of ^two ^expansion

of a funtion

f ∈ L 2 [0, π]

^. ^The ^rst ône îs ônstruted ûsing ^the ^system ôf ^the

eigenfuntions and assoiated funtions of the operator

L

^, ^while ^the ^seond ^one ^is

(2)

Theproblemofuniformequionvergene (i.e.inthenormofthespae

C[0, π]

⁾^is

wellstudiedinthelassialtheoryofSturmLiouvilleoperatorsforregularpotentials

q ∈ L 1 (0, π)

^. În^the ^monograph ôf ^V.A.^Marhenko ^[4℄^the ûniform equionvergene wasproved foranyfuntion

f ∈ L ₂ [0, π]

^.^In ¹⁹⁹¹^V.A. ^Il'in^[2℄^proved ^this^result ^for

any

f ∈ L 1 [0, π]

^. ^V.A. ^Vinokurov ând ^V.A. ^Sadovnihii ^[10℄ ^proved â ^theorem ôn

the equionvergene for SturmLiouvilleoperatorswhose potentialisthe derivative

of a funtionof bounded variation (and alsofor any

f ∈ L 1 [0, π]

^).

The problem of the rate of equionvergene (for regularpotentials) was studied

in the paper of A.M. Gomilko and G.V. Radzievskii [1℄. In the author's paper [5℄

results were obtained in the ase in whih

q = u ^′

^,

u ∈ W ₂ ^θ [0, π]

^with

0 < θ < 1/2

^.

It wasshown that forany

f ∈ L 2 [0, π]

ône ân êstimate^the ^rateôf equionvergene uniformlyover the ball

u ∈ B θ,R = { v ∈ W ₂ ^θ [0, π] : k v k W ₂ ^θ 6 R }

^. ^This ^result ^is ^new

even inthe lassialase

q ∈ L 2 [0, π]

^.

In thispaperweproveequionvergene and obtainasimilarestimateof the rate

of equionvergene inthe norm of the spae

W ₂ ^θ [0, π]

^with

0 < θ < 1/2

^.

2 Preliminary results

Webegin with the following results about operator (1).

Letus reallthat the operator

L

^is ^bounded ^from^below ^and^has ^purely ^disrete

spetrum.Let

{ λ n } ^∞ 1

^be^the ^sequene ^of ^alleigenvaluesof the operator

L

^.^We^shall

enumerate theeigenvalues insuh away that

| λ ₁ | 6 | λ ₂ | 6 . . .

^, ^and^we^assume ^that

eah eigenvalue is repeated as many times as its algebrai multipliity. By

{ y n } ^∞ 1

we denote the system of the eigenfuntions and assoiated funtions. We assume

that the funtion

y _n

^orresponds ^to ^the ^eigenvalue

λ _n

^and

k y _n k L 2 = 1

^if

y _n

^is ^an

eigenfuntion.

Statement 1 (A.M. Savhuk, A.A. Shkalikov). Let

u ∈ L 2 [0, π]

^. ^Then ^the ^system

{ y n } ^∞ 1

^of ^theeigenfuntionsandassoiatedfuntionsof theoperator

L

^forms ^a^Riesz

basis in the spae

L 2 [0, π]

^.

By the above there exists the biorthogonal system

{ w n } ^∞ 1

^, ^i.e.

(y n , w m ) = δ nm

where

(f, g) = R π

0 f (x)g(x)dx

^. ^F^or ^more ^details^see ^[5℄.

Let

l ^θ ₂ =

 

 x = { x _n } ^∞ n=1 :

X ∞ n=1

| x _n | ² n ^2θ

! 1/2

= k{ x _n }k l ₂ ^θ < ∞

 

 .

Wereall that for any

u ∈ W ₂ ^θ [0, π]

^with

0 < θ < 1/2 C 1 k{ u n }k l ₂ ^θ 6 k u k W ₂ ^θ 6 C 2 k{ u n }k l ^θ ₂

where

u _n = r 2

π (u(x), sin nx)

^and

C ₁ , C ₂ > 0

^are independent of

u

^.

(3)

Statement 2 (A.M. Savhuk, A.A. Shkalikov). Let

R > 0

^,

0 < θ < 1/2

^, ^and

u ∈ B θ,R

^. ^Then ^there ^exists ^a ^natural ^number ²

N = N θ,R

^suh ^that ^for ^all

n > N

the eigenvalues

λ n

^of ^the ^operator

L

^are ^simple,

y n (x) = q

2 π sin nx + ϕ n (x), w n (x) = q

2 π sin nx + ψ n (x), y _n ^′ (x) = n q

2 π cos nx + η n (x)

+ u(x) q

2 π sin nx + ϕ n (x)

, (2)

where the funtions

ϕ _n

^,

ψ _n

^and

η _n

^are ^suh ^that ^the ^sequen^e

{ γ _n } ^∞ n=N = {k ϕ n (x) k ^C + k ψ n (x) k ^C + k η n (x) k ^C } ^∞ n=N ∈ l ₂ ^θ

ând îts ^norm ⁱⁿ ^this ^spae îs ^bounded

by a quantity depending only on

θ

^and

R

^. ^Moreover ^for

n > N ψ n (x) = ψ n,0 (x) + ψ n,1 (x),

where

ψ n,0 (x) = α n sin nx + β n x cos nx − r 2

π Z x

0 u(t) sin n(x − 2t)dt, (3)

andthenumbers

α n

^,

β n

^and^the^funtions

ψ n,1

^are^suh^that^the^norm^of^the^sequene

{| α n | + | β n |} ^∞ n=N

ⁱⁿ ^the ^spae

l ₂ ^θ

îs ^bounded ^by â ^quantity ^depending ônly ôn

θ

^and

R

^, ^and ^the ^sequene

{k ψ n,1 (x) k ^C } ^∞ n=N

^belongs ^to ^the ^spae

l ₁ ^τ

^for ^any

τ < 2θ

^.

Statement 3. Let

R > 0

^,

0 < θ < 1/2

^and

u ∈ B θ,R

^. ^Then ^there ^exists ^a ^natural

number

N = N θ,R

^suh ^that ^for ^all

n > N

^the ^operator

P n (u) : L 2 [0, π] → W ₂ ¹ [0, π], P n (u) : f 7→

X n k=1

(f, w k )y k ,

is ontinuous on

B θ,R

^, ^i.e. ^for ^any

u 0 ∈ B θ,R

k P n (u) − P n (u 0 ) k L 2 → W ₂ ¹ → 0

^as

k u − u 0 k W ₂ ^θ → 0, u ∈ B θ,R .

This statement follows from the results of the paper [8℄ (Theorem 1.9) and

lassial theorems (see [3, Theorems IV.2.23 and IV.3.16℄). For detailed proof see

[6℄.

3 Main theorem

Theorem. Let

R > 0

^,

0 < θ < 1/2

^.^Consider^operator⁽¹⁾^atingⁱⁿ^the^spae

L 2 [0, π]

with the homogeneous Dirihlet boundary onditions. Suppose that the omplex-

valued potential

q(x) = u ^′ (x)

^, ^where

u(x) ∈ B θ,R

^.

Let

{ y n (x) } ^∞ n=1

^be ^the ^system ^of ^the eigenfuntions and assoiated funtions of the operator

L

^and

{ w n (x) } ^∞ n=1

^be ^the biorthogonal system.

2

Here andin thesequelwhen we write

N θ,R

^,

C θ,R

^et^we^mean ^that ^these ^quantities ^depend

onlyon

θ

^and

R

^.

(4)

For an arbitrary funtion

f ∈ L ₂ [0, π]

^denote

c n := (f(x), w n (x)), c n,0 := p

2/π(f (x), sin nx).

Then there exist a natural number

M = M _θ,R

^and ^a ^positive ^number

C = C _θ,R

^suh

that for all

m > M

^and ^for ^all

f ∈ L 2 [0, π]

X m n=1

c n y n (x) − X m n=1

r 2

π c n,0 sin nx W ₂ ^θ

6 C



 s X

n>m ^1−θ

| c n,0 | ² + k f k L 2

m ^θ(1 ⁻ ^θ)



 . (4)

Proof.

Step 1 (operators

B m,N

^and

B m

^).

For given integers

m

^,

m > N

^, ^we ^introdue ^the ^operator

B m,N : L 2 [0, π] → W ₂ ^θ [0, π]

^dened ^by

B m,N f (x) :=

X m n=N

c n y n (x) − X m n=N

r 2

π c n,0 sin nx

and denote

B _m,1 =: B _m

^. ^Then ^for ^any ^funtion

f ∈ L ₂ [0, π]

^the ^following ^equality

holds:

B m,N f (x) = X m n=N

r 2

π (f (t), ψ n (t)) sin nx+

+ X m n=N

r 2

π (f (t), sin nt)ϕ n (x) + X m n=N

(f(t), ψ n (t))ϕ n (x). (5)

Let

N = N θ,R

^be ^the ^maximal ^of ^the ^positive ^integers ^dened ⁱⁿ ^Statements ²

and 3.

Step 2 (estimation of the norm of the operator

B m,N

^).

There existsapositive number

C θ,R

^suh^that^for^all^natural

m

^satisfying

m > N k B m,N (u) k L 2 → W ₂ ^θ 6 C θ,R . (6)

Letusestimateeahtermoftheright-handsidein(5)separately.Byasymptoti

formulae (2)we have:

X m n=N

(f (t), ψ _n (t)) sin nx W ₂ ^θ

6 C ₁ X m n=N

| (f (t), ψ _n (t)) | ² n ^2θ

! 1/2

6 6 C 1

X m n=N

k f k ² L 2 k ψ n k ² L 2 n ^2θ

! 1/2

6 C 2 k f k L 2 ,

where

C 1 , C 2 > 0

^are independent of

f

^. ^Consider ^the ^seond ^and ^the ^third ^terms.

We willshow that

{k ϕ n k W ₂ ^θ } ^∞ n=N ∈ l 2

^provided ^that

0 < θ < 1/2

^. ^Asymptotis ⁽²⁾

implythe estimate

k ϕ n k W ₂ ^θ 6 C 3 k ϕ n k ¹ L ⁻ 2 ^θ k ϕ n k ^θ W ₂ ¹ 6 C 4 k ϕ n k ¹ L ⁻ 2 ^θ k ϕ ^′ _n k ^θ L 2 6

(5)

6 C 4 k ϕ n k ¹ L ⁻ 2 ^θ



 k η n k ^θ L 2 n ^θ + r 2

π

! θ

k u k ^θ L 2 + k uϕ n k ^θ L 2



 6 C 5 k ϕ n k ¹ L ⁻ 2 ^θ k η n k ^θ L 2 n ^θ + 1

=

= C 5 k ϕ n k L 2 n ^θ 1 − θ

( k η n k L 2 n ^θ ) ^θ + (n ^θ ⁻ ¹ ) ^θ

, (7)

where

C ₃ , C ₄ , C ₅ > 0

^are independent of

n

^. ^Here ^we ^used ^the ^inequality

k f k W ₂ ¹ 6

√ π ² + 1 k f ^′ k L 2

^fullled^for^all

f ∈ W ₂ ¹ [0, π]

^satisfying

f(0) = 0

^.^Next^we^use^H^older's

inequality

X m n=N

(a ¹ _n ⁻ ^θ b ^θ _n ) ² = X m n=N

(a ² _n ) ¹ ⁻ ^θ (b ² _n ) ^θ 6 X m n=N

a ² _n

! 1 − θ X m n=N

b ² _n

! θ

andnote thatthe sequenes

{k ϕ n k L 2 n ^θ } ^∞ n=N

^,

{k η n k L 2 n ^θ } ^∞ n=N

^,^and

{ n ^θ ⁻ ¹ } ^∞ n=1

^belong

to

l 2

^spae ^for ^any

0 < θ < 1/2

^. ^Therefore ⁽⁷⁾ ^implies^the ^inequality

X m

n=N

k ϕ n k ² W ₂ ^θ 6 C 6 , (8)

where

C 6 > 0

^is independent of

m

^and

N

^. ^Then

X m n=N

(f (t), sin nt)ϕ n (x) W ₂ ^θ

6 X m n=N

| f n |k ϕ n k W ₂ ^θ 6 C 7 k f k ^L 2 , (9)

where

f n = (f (t), sin nt)

^and

C 7 = √ C 6

^.

For the lastterm of the right-handside in(5)wehave:

X m n=N

(f (t), ψ n (t))ϕ n (x) W ₂ ^θ

6 X m n=N

k f k ^L 2 k ψ n k ^L 2 k ϕ n k W ₂ ^θ 6

6 k f k ^L 2

X m n=N

k ψ n k ² L 2

! 1/2 X m n=N

k ϕ n k ² L 2

! 1/2

6 C 8 k f k ^L 2 , (10)

where

C 8 > 0

^is independent of

m

^and

N

^, ^beause ^the ônditions ôf ôur ^theorem

implythat

{k ψ n k L 2 } ^∞ n=N ∈ l 2

^.

Weproved inequality (6). Step 2is ompleted.

Step 3 (estimation of the norm of the operator

B m

^).

There existsapositive number

C θ,R

^suh^that^for ^all^natural

m

^satisfying

m > N k B m (u) k L 2 → W ₂ ^θ 6 C θ,R . (11)

Obviously the operator

B m (u)

^an ^berepresented asthe sum:

B m (u) = B N − 1 (u) + B m,N (u) = P N − 1 (u) − P N − 1 (0) + B m,N (u),

(6)

where

P _n (u)

^is ^the ^operator ^dened ⁱⁿ ^statement ^3. ^Hene

k B m (u) k L 2 → W ₂ ^θ 6 2 sup

v ∈ B θ,R

k P N − 1 (v) k L 2 → W ₂ ^θ + k B m,N (u) k L 2 → W ₂ ^θ 6 6 2 sup

v ∈ B θ,R

k P N − 1 (v) k L 2 → W ₂ ¹ + k B m,N (u) k L 2 → W ₂ ^θ .

Let

0 < θ 1 < θ

^, ^say

θ 1 = ^θ ₂

^. ^By^statement³ ^the^funtion

k P N − 1 (v ) k L 2 → W ₂ ¹ : B θ 1 ,R → R

is ontinuous on

B θ 1 ,R

^. ^It ^is ^known ^that ^the ^ball

B θ,R

îs â ômpat ^subset ôf

B θ 1 ,R

^. ^Therefore ^the ^funtion

k P N − 1 (v ) k L 2 → W ₂ ¹

^is ^bounded ^on

B θ,R

^by ^a ^onstant

depending only on

θ

^and

R

^. ^This, ^together ^with ^inequality ^(6), ^imply ^(11). ^Step ³

is ompleted.

Step 4 (proof of the equionvergene).

Let

f ∈ L ₂ [0, π]

^. ^Then

m lim →∞ k B m f k W ₂ ^θ = 0. (12)

First let us hek that the system

{ y _k } ^∞ 1

^of ^the eigenfuntions and assoiated funtions is minimal in the spae

W ₂ ^θ [0, π]

^. ^Assume ^the ^onverse. ^Then

y n ∈ span { y k } k 6 =n

^(here ^the ^losure ^is ^taken ^subjet ^to

W ₂ ^θ [0, π]

^norm) ^for ^some ^natural

number

n

^. ^Therefore

y _n ∈ span { y _k } k 6 =n

^, ^where ^the ^losure ^is ^taken ^subjet ^to

L 2 [0, π]

^norm. ^It ^means ^that ^the ^system

{ y k } ^∞ 1

^is ^not ^minimal ⁱⁿ^the ^spae

L 2 [0, π]

ontradition with statement 1. Next, we proved (see (8)) that the system

{ y _k } ^∞ 1

^is ^lose ^to ^the ^orthogonal ^basis

{ p

2/π sin kx } ^∞ 1

ⁱⁿ ^the ^spae

W ₂ ^θ [0, π]

^. ^Let

us now onsider the orthonormal basis

nq 2

π (1 + k ² ) ⁻ ^θ/2 sin kx o ∞ 1

and the system

{ (1 + k ² ) ⁻ ^θ/2 y k } ^∞ 1

ⁱⁿ

W ₂ ^θ [0, π]

^. ^Wê ^see ^that ^these ^two ^systems âre âlso ^lose ⁱⁿ ^the

spae

W ₂ ^θ [0, π]

^.Îtân^beêasily^proved ^that^the ^system

{ (1 + k ² ) ⁻ ^θ/2 y k } ^∞ 1

^is^minimal

inthe spae

W ₂ ^θ [0, π]

^.^Hene, ^this ^system ^forms^the ^Bari^basis ⁱⁿ^the ^spae

W ₂ ^θ [0, π]

and onsequently the system

{ y k } ^∞ 1

^is ^totalⁱⁿ ^this ^spae.

Let usonsider the image ofthe funtion

y k

^under^the ^map

B m

^:

(B m y k )(x) = X m n=1

(y k (x), w n (x))y n (x) − 2 π

X m n=1

(y k (x), sin nx) sin nx.

Therst term ofthe right-handsideinthe lastequalityisequalto0for

m < k

^and

is equalto

y k (x)

^for

m > k

^.^The ^seondône îs ^the ^partial^sum ôf ^the ^Fôurier ^series

for the funtion

y _k

^. ^Reall^that ^the ^funtion

y _k ∈ W ₂ ¹

^. ^This împlies ^that îts ^Fôurier

series onverges to

y k

ⁱⁿ

W ₂ ¹

^norm. Consequently this series onverges to

y k

ⁱⁿ ^the

spae

W ₂ ^θ

^. ^This ^yields ^that ⁽¹²⁾ ^holds ^for ^any ^funtion

f ∈ span { y k }

^. ^T^o ^onlude

the proof of the Step 4, it remains to apply ompleteness of the system

{ y _k }

^and

inequality(11).

Nowletusproveinequality(4).Let

g k (x) = P k n=1

c n y n (x)

^,^where

k > N

^.^It^is^lear

that for any funtion

f ∈ L 2 [0, π]

^and ^any ^natural ^number

m

k B m f k W ₂ ^θ 6 k B m (f − g k ) k W ₂ ^θ + k B m g k k W ₂ ^θ . (13)

(7)

Step 5 (estimation of the norm of

B _m (f − g _k )

ⁱⁿ ^the ^spae

W ₂ ^θ [0, π]

^).

There exists a positive number

C θ,R

^suh ^that ^for ^all ^natural

k

^,

m

^satisfying

k, m > N

^and ^for ^all

f ∈ L 2 [0, π]

k B _m (f − g _k ) k W ₂ ^θ 6 C _θ,R



 X ∞ n=k+1

| c _n,0 | ²

! 1/2

+ k f k ^L 2

k ^θ



 . (14)

By asymptotiformula(2) wehave:

k f(x) − g k (x) k ^L 2 =

X ∞ n=k+1

c n y n (x) 6

6 X ∞ n=k+1

2 π (f (x), sin nx) sin nx L 2

+

X ∞ n=k+1

r 2

π (f (x), ψ _n (x)) sin nx L 2

+

X ∞ n=k+1

r 2

π (f (x), sin nx)ϕ _n (x) L 2

+

X ∞ n=k+1

(f (x), ψ _n (x))ϕ _n (x) L 2

6

6 

 X ^∞

n=k+1

| c n,0 | ²

! 1/2

+

X ∞ n=k+1

| (f(x), ψ n (x)) | ²

! 1/2 

 ×

×



1 + X ^∞

n=k+1

k ϕ n k ² L 2

! 1/2 

 6

6 

 X ^∞

n=k+1

| c n,0 | ²

! 1/2

+ k f k L 2

k ^θ





1 + C θ,R

k ^θ

. (15)

Now inequality (14) follows from (15) and (11). Step 5is ompleted.

Next we estimate the seond term in (13). For a given natural

m

^,

m > k

^, ^we

introdue the operator

S _m : W ₂ ¹ [0, π] → W ₂ ^θ [0, π]

^dened ^by

S m h(x) = 2/π

X ∞ n=m+1

(h(t), sin nt) sin nx .

Notethat

B m g k (x) = g k (x) − 2 π

X m n=1

(g k (t), sin nt) sin nx = S m g k (x) .

(The expression

S m g k (x)

^is ^well ^dened, ^sine ^all eigenfuntions and assoiated funtionsof the operator

L

^belong ^to ^the ^spae

W ₂ ¹ [0, π]

^.) ^Therefore

k B m g k k W ₂ ^θ 6 k S m (g k − g N ) k W ₂ ^θ + k S m g N k W ₂ ^θ . (16)

(8)

S _m (g _k − g _N )

ⁱⁿ ^the ^spae

W ₂ ^θ [0, π]

^).

There existsa positive number

C θ,R

^suh ^that^for^all^natural

k, m

^satisfying

m >

k > N

^and ^for ^all

f ∈ L 2 [0, π]

k S m (g k − g N ) k W ₂ ^θ 6 C θ,R k f k L 2 m ^θ ⁻ ¹ k ¹ ⁻ ^θ . (17)

First let usreall that

y n (x) = p

2/π sin nx + ϕ n (x)

^(see ⁽²⁾⁾ ^and ^represent ^the

left-hand side of inequality (17) as follows:

k S m (g k (x) − g N (x)) k W ₂ ^θ =

X k n=N +1

c n S m y n (x) W ₂ ^θ

=

X k n=N +1

c n S m ϕ n (x) W ₂ ^θ

.

Notethat 3

X k n=N+1

| c n | ² 6 X ∞ n=N+1

| c n | ² 6

6 2 2 π

X ∞ n=N+1

| (f (x), sin nx) | ² + X ∞ n=N+1

| (f (x), ψ n (x)) | ²

!

6 C _θ,R ⁽¹⁾ k f k ² L 2 ,

Therefore

X k n=N +1

c n S m ϕ n (x) W ₂ ^θ

6 X k n=N+1

| c n | ²

! ^1/2 _k X

n=N +1

k S m ϕ n (x) k ² _W ₂ ^θ

! ^1/2 6

6 q

C _θ,R ⁽¹⁾ k f k L 2

X k n=N +1

k S m ϕ n (x) k ² _W ₂ ^θ

! 1/2

.

Furthermore

k S _m ϕ _n (x) k W ₂ ^θ 6 C _θ,R ⁽²⁾

X ∞ j=m+1

(j ^2θ + 1) | (ϕ _n (x), sin jx) | ²

! 1/2

=

= C _θ,R ⁽²⁾

X ∞ j=m+1

j ^2θ + 1

j ² | (ϕ ^′ _n (x), cos jx) | ²

! 1/2

6 C _θ,R ⁽²⁾ m ^θ ⁻ ¹ k ϕ n (x) k W ₂ ¹ . (18)

Here we applied integration by parts and boundary onditions

ϕ n (0) = ϕ n (π) = 0

^. ^By ^asymptoti ^formulae ⁽²⁾ ^we ^obtain

ϕ ^′ _n (x) = nη n (x) + u(x)y n (x)

^, ^where

{k η n (x) k L 2 + n ^θ ⁻ ¹ } ∈ l ^θ ₂

^. Consequently

k ϕ ^′ _n (x) k ^L 2 6 (n k η n (x) k ^L 2 + k u(x)y n (x) k ^L 2 ) = n ¹ ⁻ ^θ τ n ,

3

Here

C _θ,R ⁽¹⁾

^andⁱⁿ ^the^sequel

C _θ,R ^(j)

^with

j = 2, 3...

^are^some^positive^quantities^depending^only

on

θ

^and

R

^.

(9)

where

k{ τ n }k l 2 6 C _θ,R ⁽³⁾

^.^Therefore,

k S _m ϕ _n (x) k W ₂ ^θ 6 C _θ,R ⁽⁴⁾ m ^θ ⁻ ¹ n ¹ ⁻ ^θ τ _n .

This impliesthat the rst term in(16) satises the followinginequality:

k S m (g k − g N ) k W ₂ ^θ 6 C _θ,R ⁽⁵⁾ k f k L 2

X k n=1

m ^2θ ⁻ ² n ² ⁻ ^2θ τ _n ²

! ^1/2

6 C _θ,R ⁽⁶⁾ k f k L 2 m ^θ ⁻ ¹ k ¹ ⁻ ^θ .

Step 6 isompleted.

S m g N

ⁱⁿ ^the ^spae

W ₂ ^θ [0, π]

^).

Finallyweonsidertheseondtermin(16).Weshallestimatethenorm

k S m g N k ^C

inthe same way as in(18):

k S m g N (x) k C 6 C _θ,R ⁽⁷⁾ m ^θ ⁻ ¹ k g _N ^′ (x) k L 2 6 C _θ,R ⁽⁷⁾ m ^θ ⁻ ¹ k g N k W ₂ ¹ .

Sine

k P _N (u) k L 2 → W ₂ ¹ 6 C _θ,R

^(see ^Step ³⁾ ^and

g _N = P _N f

^we ^have

k g _N k W ₂ ¹ 6 C θ,R k f k ^L 2

^.^Therefore,

k S m g N k C 6 C _θ,R ⁽⁸⁾ k f k L 2 m ^θ ⁻ ¹ . (19)

Hene inequalities(16), (17) and (19) imply that

k B m g k k ^C 6 C _θ,R ⁽⁹⁾ k f k ^L 2 m ^θ ⁻ ¹ k ¹ ⁻ ^θ . (20)

Let us put

k = [m ¹ ⁻ ^θ ] + 1

^for ^any ^natural^number

m > N ²

^. Inequalities (13), (14)

and (20) implyinequality(4).

(10)

Referenes

[1℄ A.M. Gomilko,G.V.Radzievskii, The equionvergene ofthe series in the eigenfuntions of

ordinaryfuntional-dierential operators. DokladyRoss.Akad. Nauk.Matematika,316,no.

2(1991),265269(in Russian).

[2℄ V.A.Il'in,Equionvergeneofthetrigonometriseriesandtheexpansionineigenfuntionsof

the one-dimensional Shrodinger operator with the omplex-valued

L 1

-potential.Di. Uravn., 27,no.4(1991),577597(inRussian).

[3℄ T. Kato, Perturbation theory for linear operators. Springer Verlag, Berlin-Heidelberg-New

York,1966.

[4℄ V.A. Marhenko, SturmLiouville operators and their appliations. Naukova dumka, Kiev,

1977(inRussian).

[5℄ I.V.Sadovnihaya,Onthe rate of equionvergeneof expansionsin the trigonometri series

andin the eigenfuntions of SturmLiouvilleoperator with the distributional potential.Di.

Uravn., 44,no.5(2008),656664(inRussian).

[6℄ A.M.Savhuk,Uniformasymptotiformulaefor eigenfuntionsofSturmLiouvilleoperators

withsingular potentials. arXiv0801.1950,2008.

[7℄ A.M. Savhuk, A.A. Shkalikov, SturmLiouville operators with singular potentials. Mat.

zametki,66,no.6(1999),897912(in Russian).

[8℄ A.M.Savhuk,A.A.Shkalikov,SturmLiouvilleoperatorswithdistributionpotentials.Trudy

MosowMathematialSoiety,64(2003),159219(inRussian).

[9℄ A.M. Savhuk, A.A.Shkalikov,On eigenvalues of SturmLiouville operators with potentials

fromSobolev spaes.Mat.zametki, 80,no.6(2006),864884(inRussian).

[10℄ V.A.Vinokurov,V.A.Sadovnihy,The uniformequionvergene ofthe Fourierseriesinthe

eigenfuntions of the rst boundary value problem and of the trigonometri Fourier series.

DokladyRoss.Akad.Nauk.Matematika,380,no.6(2001),731735(inRussian).

IrinaSadovnihaya

FaultyofComputationalMathematisand Cybernetis

M.V.LomonosovMosowStateUniversity

Leninskiegory

119991Mosow,Russia

E-mail:ivsadyandex.ru

Reeived:14.10.2009

ď²y - +q(x)y

W 2 θ

Ly = l(y) = − d 2 y

dx 2 + q(x)y

y(0) = y(π) = 0

L 2 [0, π]

q(x) = u ′ (x)

u ∈ W 2 θ [0, π]

0 < θ <

1/2

W 2 θ [0, π] = [L 2 , W 2 1 ] θ

W 2 θ [0, π]

f ∈ L 2 [0, π]

L

f

L 2 [0, π]

Ly = l(y) = − d 2 y

dx 2 + q(x)y, (1)

y(0) = y(π) = 0

L 2 [0, π]

q

q(x) = u ′ (x)

u ∈ W 2 θ [0, π]

0 < θ < 1/2

W 2 θ [0, π] = [L 2 , W 2 1 ] θ

L

W 2 θ [0, π]

f ∈ L 2 [0, π]

L

C[0, π]

q ∈ L 1 (0, π)

f ∈ L 2 [0, π]

f ∈ L 1 [0, π]

f ∈ L 1 [0, π]

q = u ′

u ∈ W 2 θ [0, π]

0 < θ < 1/2

f ∈ L 2 [0, π]

u ∈ B θ,R = { v ∈ W 2 θ [0, π] : k v k W 2 θ 6 R }

q ∈ L 2 [0, π]

W 2 θ [0, π]

0 < θ < 1/2

L

{ λ n } ∞ 1

L

| λ 1 | 6 | λ 2 | 6 . . .

{ y n } ∞ 1

y n

λ n

k y n k L 2 = 1

y n

u ∈ L 2 [0, π]

{ y n } ∞ 1

L

L 2 [0, π]

{ w n } ∞ 1

(y n , w m ) = δ nm

(f, g) = R π

0

f (x)g(x)dx

l θ 2 =

 

 x = { x n } ∞ n=1 :

X ∞ n=1

| x n | 2 n 2θ

! 1/2

= k{ x n }k l 2 θ < ∞

 

 .

u ∈ W 2 θ [0, π]

0 < θ < 1/2 C 1 k{ u n }k l 2 θ 6 k u k W 2 θ 6 C 2 k{ u n }k l θ 2

u n = r 2

π (u(x), sin nx)

C 1 , C 2 > 0

u

R > 0

0 < θ < 1/2

u ∈ B θ,R

N = N θ,R

n > N

W ₂ ^θ

Ly = l(y) = − d ² y

dx ² + q(x)y

L ₂ [0, π]

q(x) = u ^′ (x)

u ∈ W ₂ ^θ [0, π]

W ₂ ^θ [0, π] = [L 2 , W ₂ ¹ ] θ

W ₂ ^θ [0, π]

f ∈ L ₂ [0, π]

Ly = l(y) = − d ² y

dx ² + q(x)y, (1)

q(x) = u ^′ (x)

u ∈ W ₂ ^θ [0, π]

W ₂ ^θ [0, π] = [L 2 , W ₂ ¹ ] θ

W ₂ ^θ [0, π]

f ∈ L ₂ [0, π]

q = u ^′

u ∈ W ₂ ^θ [0, π]

u ∈ B θ,R = { v ∈ W ₂ ^θ [0, π] : k v k W ₂ ^θ 6 R }

W ₂ ^θ [0, π]

{ λ n } ^∞ 1

| λ ₁ | 6 | λ ₂ | 6 . . .

{ y n } ^∞ 1

y _n

λ _n

k y _n k L 2 = 1

y _n

{ y n } ^∞ 1

{ w n } ^∞ 1

l ^θ ₂ =

 x = { x _n } ^∞ n=1 :

| x _n | ² n ^2θ

= k{ x _n }k l ₂ ^θ < ∞

u ∈ W ₂ ^θ [0, π]

0 < θ < 1/2 C 1 k{ u n }k l ₂ ^θ 6 k u k W ₂ ^θ 6 C 2 k{ u n }k l ^θ ₂

u _n = r 2

C ₁ , C ₂ > 0

π sin nx + ψ n (x), y _n ^′ (x) = n q

ϕ _n

ψ _n

η _n

{ γ _n } ^∞ n=N = {k ϕ n (x) k ^C + k ψ n (x) k ^C + k η n (x) k ^C } ^∞ n=N ∈ l ₂ ^θ

{| α n | + | β n |} ^∞ n=N

l ₂ ^θ

{k ψ n,1 (x) k ^C } ^∞ n=N

l ₁ ^τ

P n (u) : L 2 [0, π] → W ₂ ¹ [0, π], P n (u) : f 7→

k P n (u) − P n (u 0 ) k L 2 → W ₂ ¹ → 0

k u − u 0 k W ₂ ^θ → 0, u ∈ B θ,R .

q(x) = u ^′ (x)

{ y n (x) } ^∞ n=1

{ w n (x) } ^∞ n=1

f ∈ L ₂ [0, π]

M = M _θ,R

C = C _θ,R

π c n,0 sin nx W ₂ ^θ