Some estimates of the rate of convergence of regularized approximations for equations of the convolution type

SOME ESTIMATESOF THE RATE OF CONVERGENCE OF REGULARIZED APPROXIMATIONSFOR EQUATIONS OF THE CONVOLUTION TYPE* A.V.

GONCHARSKII,

A, S. LEONOV and A, G. YAGOLA Moscow

(Received 21 May 1971; revised

A NEW mathod of obtaining

estimates

version

27 July 1971)

of the rate of convergence

of regularized

approximations to the exact solution for one-dimensional Fredholm integral equations of the first kind of the convolution type is proposed. The method is based

on a study

transform

of the behaviour

of the kernel

hood of its zeros We consider

of the square

of an integral

equation

of the modulus at infinity

of the Fourier

and in the neighbour-

on the real axis. the Fredholm

integral

equation

of the first kind of the convolu-

tion type +*

K(s-@z(s)& = a(s), -cQ
44[x9g(s)1 =

(1)

which is often encountered Let the functions

occurring

K(v) &Li(&f7 Also, to some function

equation

(I), the function

information

in applications. in the equation

u (5) =

L2 (Ed

2 (s)

Usually

u (x) itself,

u6 (3~)such that

Mat. mat. Fiz., 12, 3. 762-770, 1972.

243

the requirements

W2’

(Ed).

a unique

in the treatment

llu(t) - Q(s) I/tz =C 6.

+ Zh. vychisl.

E

u (x) let there correspond z (s).

we know not the function

it, a function

9

satisfy

solution

of

of experimental

but some &approximation

of

V. Goncharskii,

A.

244

We

consider

the Tikhonov

A. S. Leonov

functional

and A. G. Yagola

[l]

In [2] it is shown that for any a > 0 there exists realizing the minimum of the functional

a unique

way that 6’ /a (6) + 0, the sequence

z6* (6) (s) converges

solution

and it follows

of (l), in the norm W,‘(E,),

[3] that the convergence

element

zsa (s)

(4), and if C-I (8) -, 0 as S + 0 in such a

of the approximate

to z (s), the exact

from the imbedding

to the exact

solution

theorem

is uniform.

Moreover, if we consider the functional Ma [z, u], we can find a unique function za (s) on which the minimum of Ma [z, u] is realized for a fixed a > 0, where za (s) -+ z (s) as a + 0 [l, 21. After using the convolution theorem, Plancherel’s equation 141 and varying the functionals Ma [a, u,], Ma [z, U] on a set of functions of Wll, we obtain that the minimum of the functional Ma [z, u,] is attained on the function E.51

(3)

i +-.r(O)E*(W)et@* do ZIP(0) = zft s L(o)+ a(d + 1) ’ -LI1

and the minimum of the functional i

(4)

z”(s)

=

-

(0) fj (a)

2% f_,Lfti)+

where K’(o) = X(-u,)), f(w) are the Fourier We also notice

+-r

Ma [z,

ul on

the function

eisr do

No2 + 4) ’

L(W) = ]ff(a) 1’ -Ill(w)I transforms of the functions

and K(O), aa(

g(w),

K(Y), r&(z), u(z), Z(t).

that (1) implies

(5) We consider the question of the rate of convergence of the regularized approximation to the exact solution of (I). As will be shown below, the estimates of the rate of convergence depend essentially on the behaviour of the square of the modulus of the Fourier transform of the kernel at infinity and in the neighbourhoods of its zeros on the real o-axis. Very interesting estimates of the rate of convergence, based on a study of the behaviour of the Fourier

transform of the kernel at infinity, and also on the behaviour of the function in the denominator of (3) in the neighbourhoods of its zeros in the complex o-plane, were obtained in [5-71. The method of obtaining the rate of convergence

of the rate of convergence

Some estimates

presented

below is simpler

15-71 and enables

than the method of obtaining

estimates

of this type to he obtained

estimates

245

explained

for more complex

in

classes

of kernels. Usually z$‘)

(s)

particular,

the interest to the exact

is in estimating solution

such estimates

the accuracy

of the approximation

.z (s) in the norm of some Banach

will be obtained

below for the spaces

space

B.

of In

C and t,.

It is easy to see that the inequality

s.slb(s) -z=ts)IIrJ + W(S) - ZP(S)Ils.

llz(s) -zaa(s)Ils

(6) holds.

It follows

from (S), (6) that-

J L(o)+ a(02 +1)’ -m ~a(~)~ef~n~~ 1*Y?rfo)[s(W)(s) - Z&U (s) =--g (8) za J L(o)+a(w2+~) * +OD(02 + l)P(o)ei”* do

(7)

za (s) = ;

z(s)-

It remains to estimate (7) and;;) as functions of a and 6. We show that for the assumptions formulated above the following estimates hold (see 18, 91) (9)

8 lfZ=(s) - za=(s) ilL*< -3 l’a

(f0)

112&(s)-

6

1

ZP(S)Ilc f --.

ya

1’2

We use Plancherel’s

theorem: 1

(11)

llP(s)-

-

z8ys)llt,2= (24’

+II

H”(w)[E@)-

J _m

L(o)+

1 +~~(o)K(~)]~(~)-~a(~)I'~

2x 5 -0D

[L(o)+

a(02 + ill”

=G

u”g(@l)]e*‘b* do a(02 + 1)

a =

II L2

A. V. Goncharskii,

A. S. Leonov

and A. G. Yagola

Here we have used the fact that

estimate

(9) follows

from (11).

Also,

Using

(12) and Plancherel’s

We notice independent

We estimate 14(w) I

that the estimates

of the exact

solution

theorem,

we obtain

obtained

depend

(7) in the metric of the space

C.

from (7) we obtain OD(CO2 + 1) ff(w) [do

Ilz(s)-- z”(s)/lc f

”3x s

0

We partition

the last integral

It is easy to obtain (15)

We now obtain

L(o) +c@+11’ into two:

a lower bound for T(a>:

I(a) 3 const a.

an upper bound:

only on 8 and a but are

z (s). Because

of the evenness

of

of the rate of convergence

Some estimates

247

where

‘la

dp(o)

since

< +ofh

12do

z(t) E W2i. We obtain

an estimate

about the Fourier

for I,.

transform

(1) Uo) = IRW I” there are no zeros

Here we will make the following

assumptions

of the kernel: has a countable

or a finite

number

number of zeros

of them the estimate

on the real w-axis obtained

remains

8(if

all

the more valid); (2) there exists of the function L(0)

where

of non-intersecting

3

fZi(CO~-W)2P,

OE

(3) outside

of the i-th c-neighbourhood

the +neighbourhoods

L(O) >,

(4) points exception,

Azc.+,

of condensation

perhaps,

k > 0,

~-neighbourho~s

Ei, Oi + Ei],

are the zeros

of L (w), i = 1, 2,. . . ;

(the point w = 0 is considered

indicated AZ > 0;

of the zeros

of L (0) do not exist

(with the

of w = + -), and if N (MM> is the number of zeros

Ml,

(the case

of the zeros

w 2 0 such that

[a-

ai > a = const > 0, p > ‘14, WI> 0

ci is the radius separately);

(0,

a system

L (0) on the real semiaxis

y = 0 corresponds

to a finite

number

of zeros).

of L (w) on

248

A. V. Goncharskii,

These

constraints

of kernels

A. S. Leonou

are not too stringent,

encountered

in applications

and A. G. Yagola

and there exist

which satisfy

fairly wide classes

the requirements

formulated,

for example, sin29 mo

L(0) =

u2r

We now estimate

m = cone_

’

12(o) Ido =a2-t6oL(o)+a(u2+1)

>

1

etc.

o’-e’(~2+1) II(o) Ido

1)

{J

J

+

e, L(o)+a(a’+l)

0

N(M)

r

I,: (d+

Ii

(17)

CI >

ol+at

L(o)+ a(02 +1) (o2

+

1)

If(&))

Ido

J

N(M)-’

“i+‘-‘i+‘(02

+

CJ t=I Cli-ei

Z

i=i

Oifei

+

1)

+

If(o)

I&l

L(o)+a.(d+1)

+

M

to2 + 1) I:(o) Ido

I:” + I:” + I:“’ + I:“’ + I:“‘.

G

J L(o)+ ~(N)P(N)M For I(:)

a(o2 + 1) >

we obtain ai+i-et+r

“l+,-cl+r

(02 + 1) 12(o) Ido

(18)

a

GJ oi+L1 L(o)+ a(a2 + 1) 2%

+

J

(02 + i) I?(O) I&l -< A20-k

@l+=l

Ido = J ok+2lz’(o)

al+b-ei+l

2

3

1

I+ min(wi + ( i ui+l-Fitl UA3

-J3t

ok+*li(o)

6)’

)

Of+81

AS = convt > 0,

Ido,

mitEi

since

(1%

69 + 1 < (I+

min(oi +

Ei)-2)W2

i

(here we have used the property It follows

from (18) that

(3)).

=

P02,

!2 =

const > 0

Some estimates

from which,

Similarly,

using

the Cauchy

249

of the rate of convergence

inequality,

we obtain

for 1i2) (or {,‘I’ + f:‘)n if L (0) f 0) we obtain

where

It is also obvious fi3).

that the estimate

For convenience

we write n = l/a*

and (19), we obtain

We use the Cauchy

(20) is also valid for 1:).

inequality

Then taking

We will estimate

into account

property

(2)

250

A.

From (21)-(23)

Similarly,

it follows

A. S. Leonov and A. G. Yagola

that

if L (0) = 0,

I:” g Finally,

V. Goncharskii,

A,allQ

A, = const > 0.

,

for I, we obtain

(24)

Hence from (14), (16) and (24) we have

$ +A,abW%

l(a)< As =

Using

const > 0.

(4) and introducing

-!k ‘)‘M + (A

‘(u)<

Minimizing smallness, (25)

+ Da ll*hltN (M) + Bia + AIaLlhp,

4 +

with respect

r = max{k + *lzlv),

Bai14P)Mr+i + AIa’/‘P + Bia,

to M and discarding

we arrive at an estimate lIza(s) -

where 0 depends

2

(s) IIc &

I (a) =

A,B=const>O.

terms of a higher order of

of the rate of convergence 0{a114~~~2r+3)),

on z, K, CU,~1, 01, a, r, k, P.

It is thereby proved that when conditions estimate (25) holds for (7). We estimate integral

we obtain

(7) in the metric of the space

I(a) =llz(s)-

za(s)llLa2

=

S

(l)-(4)

L,.

are satisfied

the uniform

We now denote

by I (a) the

(02 + i)Zjf(o)

J 0

[L(o)+

ato’+

I’&0 1)12’

*

251

of convergence

Some estimates of the fate

As for (15) we find (Z(i)

f(a) 3 con& u2. For an upper bound of I(a) we will suppose

(4) are satisfied.

We also require

We partition

(l)-

that

F=const>O.

Iz”(o)/ <:,

(27)

as before that conditions

the integral

Z(a) = Ii + I2 = %

f(a) into two:

y J o

I” do

(d + q’p(0)

[L(o)+

aw +

=

U12 +$J

~

(02 + i)*fz(o)]a

do

IQo)+a(~‘+~)P

.

It is easy to see that

f2
CI =

In order to estimate same ranges

the integral

of integration

The integrals (2)

I, we partition

it into 5 integrals

as in (17):

I:” + p+ I:“‘+

I, =

Z:“, II

const > 0.

zp+I:“‘.

are estimated

I&Q, I,(5)

in the same way as (20):

g B3a2M2k+2 9

where BJ = Bs(k, 51, z, Kl. For I,(z) (or I&Q + Z$*l, if L (0) 4 0) we obtain Z1@)=g BIa2, We estimate

f 1t3). Introducing NW

(281

B, = &(eo,

a(+‘1

CJ id

Also,

taking

n = l/a,

(o”+f)‘lz(o) [nL(63)+

q-q

into account

et, w,

2,

K).

we have

N(w

1” do 02 + 112

“’

c cm. i-1

(2), (25) and (41), we obtain

over the

A. V. Goncharskii,

252

Using

A. S. Leonov

and A. G. Yagola

of + et < M, we obtain

w

P=COIlSt>O*

[(Oi+&)2+i]z
Hence

from (28)-(30)

I:” <

taking

at/aPD @)

(23) it follows

that

FpM’N(M). 3-P

I,“’ similarly

We estimate

into account

(if L (0) = 0):

Adding the estimates for I, and I,, taking the minimum with respect neglecting terms of a higher order of smallness in a, we obtain

II2(8) -z=(s)

(31)

to M and

11~~ = l([Z(a)] = 0(a1/2P(~+r)},

where q = max 1~ + 2, 2k].

Here 0 depends

on the same quantities

as in (25),

and also on F. It is thereby

proved that if conditions

(l)-(4)

are satisfied

for (7) in the

metric L,, then (31) is satisfied. We have obtained estimate

estimates

for the integrals

We show that if conditions (l)-(4) are satisfied holds for the rate of convergence of the regularized solution

(7), (8).

We now obtain

an

for (6).

of equation

(1):

(32)

l/z(s)-

Q??(S) llC = 0

(3%

b(r)

2’(s)

Indeed,

-

II 4

x

0

{~M~+~P(~+~)l},

{~m+P(~+4)1}.

(6), (13) and (25) imply that

the following approximations

estimate to the exact

Some estimates

I/z(s)-

z=“(s)

Minimizing

manner,

;

ai/bP(37+3)

’ ,

to a, that is, selecting we obtain

from the estimates

approximations

=

defining

in (25), (32) determines

essentially

parameter

the rate of convergence on the behaviour

Thus,

the maximum order of the zeros

of its zeros

the regularization

max {2k, y + 2).

and at infinity.

increases,

const.

=

(33), if

obtained,

depends

of its zeros

D,

253

(32) if

Q

r occurring behaviour

+

0{fy-2/r~+P(P+4,1},

As is obvious neighbourhoods

Di

from (6), (9) and (31) we obtain

#-J(6) =

regularized

<

with respect

in some “optimal”

Similarly,

IIc

of the rate of convergence

the estimates of L (w).

the rate of decrease

as o + DO. The slower

of the

of L (0) in the worsen

Also,

as p

the quantity

of L (w) and also the

L (0) decreases

at infinity

and the

more its zeros cluster at the point at infinity, the worse, in general, is the estimate of the rate of convergence of the regularized approximations to the exact solution. The estimates (15), (26) give constraints on the rate of convergence

as a + 0 downwards.

We notice

that by using

a similar

type (25), (32) for other classes is changed

method we can obtain

of kernels.

For example,

to the following: L (0) > A2e-mm;

m > 0,

Az = const > 0,

the estimates

b(s)-

z”(s)IIc = 0

1

1

ln(W

>

hold, and as for (32),

llw-z;(a)(s)I,c =

0

. 1,.___!_l W/Q

estimates

of the

if the requirement

(3)

A. V. Goncharskii,

254

In conclusion

the authors

and also V. Ya. Arsenin

A. 5”. Leonov

and A. G. Yagola

thank A. N. Tikhonov

and V. B. Glasko

for supervising

for many discussions

the work,

and for reading

the manuscript. Translated

by J. Berry

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of order p.