Approximate solution of equations with normally resolvable operators

APPROXIMATE SOLUTION OF EQUATIONS WITH NORMALLY RESOLVABLE OPERATORS* L. S.

RAKOVSHCHIK Leningrad

(Received

THJZsimplest

scheme for

solving

20 February

the equation Ax =

consists

in replacing

this

1965)

equation

y

by the approximate

&a: =

yn,

104 equation (0.2)

in which the operator A,, and the element y,, are chosen near in some sense to the operator A and the element y respectively and are such that equation (0.2) is easily solvable. The solution of the last equation is also taken as the approximate solution of equation (0. I.). This scheme in its most general form has been investigated by L.V. Kantorovich [ll, where a number of assumptions is proved which enable us to obtain, from the information about the solvability of one of the equations (0.1) or (0.2). definite information about the solvability of the other and about the convergence of the approximate solutions to the exact one (if A,, ---)A, y,, .+ y). In the present paper the scheme described above is investigated for an equation with a normally resolvable operator A. The additional assymption, compared with Kantorovich’s theory, about the normal resolvability of the operator A enables us to formulate a very simple necessary and sufficient condition for the convergence of the approximate solutions to the exact one. Note that in the case we are considering additional difficulties arise in comparison with Kantorovich’s scheme: the equations considered are as a rule not solvable with any right-hand side, and in the case of solvability their solutions are not unique.

l

Zh.

vychisl.

Mat.

mat.

Fiz.

6,

1, 3 - 11. 1966. 1

2

L.S.

Rakovshchik

Por particular types of equations* schemes, additional to ours, have been considered in [2 - 51. In the first three equations of the Wiener H&f type, in the simplest case of a zero index, have been considered, and in the last singular equations with a Cauchy kernel on the segment L-1, 11. Projection methods of solving equations with non-zero index have been considered in [6 - 81. In the first two works specific properties of the types of equations considered in them were used and their results were not capable of being transferred to the general case. The application of the general method of [8] was difficult, because it required a knowledge of the subspaces of zeros of the operator A and its conjugate operator A*. Tine method given below requires a knowledge of the dimensionalities of their subspaces only.

1. ‘4 description

of the schCme of approximate solution

Let .ri and Y be Eilbert spaces, .4 a linear normally resolvable operator from X into Y (i.e. tire range of values AA of the operator .4 is closed). We shall denote by 7.4 and ;:A= the subspaces of zeros of the operators il and .4* respectively. In all future work it is assumed that at leat one of these subspaces is of finite dimensions. We here note some well-known ators.

properties

Of normally

resolvaole

oper-

sum ,Y’ = 7.4 @ A,=, 1. The space .i is decomposed into the orthogonal is the range of values of the operator A* (which is also normally AA*

resolvable).

2. The space

:’ is decomposeci

into

the

orthogonal

SUM Y = T,,-

@

AA..

3. If the operator .4 is considered as an operator from 4.4. into AA, ;Ire shall denote by 4-l the inverse operator then it is the inverse. (from AA into AA-). 4. Similarly, into

*

AA- then

it

if the operator is considereti as an operator 1 If (,4*)-l is the inverse iS alS0 the inverse.

This concerns equations which form (I + T).z = y, where r is

are different a completely

from AA operator

from equations of the continuous operator.

Anoroximate

(from

a.4 into

A.4-j then

5. If the sequence of solvable operator 4, for d iin -7.4. is finite, then able, dim ZA,< dim ZA, If both subspaces the indices

of equations

solution

11A-* 11=

11(A-l) * 11=

/I (A”) -l 11.

operators which at for large if dim .?A

converges in norm to the norinally releast one of the numbers dim ?A or n the operators .&, are normally resolv< Co; dim ,?A *< dim zA_, if dim ?A!A--‘sr,.

7.4 and ?A, are finite-dimeniional,

;xA = dim ?A

- dim

the index x;i 0: the oper:tor For the approximate

?A”*of

ciien for

the onerators

large

.I, coincide

n

with

‘4.

solution

of the equation Ax =

y

(1.1)

we take the sequence of operators A,.,, which converges in norm to tne operator 4, and the sequence of elements yn E AA,, which converges in norm to the element f-‘y, where P is a projector (orthogonal) onto the subspace AA. If equation (1.1) is solvable then y E AA and Py = y. In tnis case we can take as yn elements P,,y, where F’n is a projector onto AA n’

In fact fore

since IIP,Y -

,y E y II d

AA, y = Ax for IIA,z

some x. Consequently

of tne theorem are satisfied

sible

in the case where ,y E AA.

also

4s an approximation

It will

-- y II-+ 0.

conditions

,4,x -t ,y and there-

be shown be.low that

such a choice

if

the

of yn is permis-

we take the equation A,s

=

yn.

In view of the method of construction is not uniquely so in general. 4s an approxi.mate solution which belongs to the subspace

(1.2) equation

(1.2)

is solvable,

but

we take the solution of equation (1.2) AA,*, i.e. x, = A,,-ly, (see properties

2

and 3). If dim .ZA < a, then the construction can be carried out thus: let Zn be an arbitrary solution of (1.2) and 21, 22, . . . , zJ( be a complete system of solutions of the homogeneous equation f$.,x = 0. Then

where the coefficients If

{Zi?

Ci are determined

is an orthonormal

system,

then

from the conditions

(x,,

Zi) =(I.

4

L.S.

Rakovshchik

-F(ii!,, Zi)Zi.

2, -

xn =

j

i=i does not ensure the convergence Note that condition IIA - A, II-+O of tile sequence X, to the solution of equation (1.1) (in the case of its solvability). This is obvious from the following simple example. 11e 11 = 1. For this operator x = 0, dim 7.4 = 0. Let .h = x - (x, e)e, only if (y, e) = 0. Let this The equation x - (nc, e)e = y has a solution condition be satisfied. The solution has the form x = y + ce, where c iS aI1 arbitrary constant. Now let (91, e) = 0, 11el 11 = 1 and A,x = x (x, a,e + ,3,,el)e, where a,, < 1, a, --t 1, 13, --t 0 so that the ratio &/ (I - xn) does not approach any limit. It is easy to see that the operstar :I, is invertible and PO the approximate equation assumes the form

xand is uniquely

solvable.

(x,che+fi,ei)e=y

Its

xn= If

(y,

el)

solution

yn+---Fan (y7de. 1

# 0, then the sequence

x, has no limit.

The example considered shows that to ensure the convergence of the approximate solutions to the exact one it is necessary to impose on the operators some additional restrictions.

2.

Supplementary

assumptions

Lemma 1 onto the subspaces Let dim ZA < co. Let P,,, P, Qn and Q be projectors AA, AA* and AA. respectively. If for each z E ZA (or z 62 zA.1 the bA,n of equasequence Q,,; -+ 0 (Pnz -t O), then in the case of the solvability tion (X.1) the sequence x, converges in norm to the solution x E AA* of this equation. If, however, (1.1) is not solvable for a given y then this sequence converges to the solution of the equation Ax =

Py.

(2.1)

Proof. 1. The sequence {x,) is bounded. In the opposite case we can consider that 11xn (1 - cc. Let us assume that V, = X, / II X, I(. Then

Approximate

solution

of

equations

5

II II =

Xn 1, and Anvn- 0. It can be easily shown that there exists a sequence of left-hand regulators H, of the operators A,, which converges in norm to the left-hand regulator f: of the operator A. For such a sequence R,A, = I t T, -RA = I t T and R,A,v, = v, + T,v, -+ 0, where T and T, are completely continuous operators. From these relations it is obvious that the operators T,, converge in norm to the operator T and, consequently, the set {T,v,) is compact. In such a case we can choose from the sequence {T,v,) a convergent subsequence {Tnkunk?. Since v, +

T

-+

“k”‘k

0, the sequence

By the condition lim

vnk also

converges.

of the lemma QnkvO - 0. Therefore

[Qnk(vO - unk) = Q”kvnkl = lim

impossible

since

boundedness

and in addition

An-l and (A,*)-1

where P,z -+ 0 for

(since

II VII 1’1=1.

0 = lim QnkvO =

[Qnk(vu - vnk) +

(1 v. II = I. The contradiction

of the sequence

the operators

Let vg = lim vnk,

vnkl = VO, which

obtained

proves

the

the uniform boundedness

(( .A,-1 11 = (1 (A,*)-l

any z E ZA* is investigated

is

of

(1); the case

in a similar

way.

2. In the conditions of the lemma the projectors P, and 0, converge in norm to the projectors P and (z onto the subspaces AA and AA* respectively. This assertion follows immediately from the boundedness of the norms 11A,-’ II and the easily established inequalities [Sl

ll P, -P II < max[II A,-’ II, II Ami III II A -An llQ,, which are true 3.

for

Qll < max[llA,-‘II, any normally

II,

IIA-‘II] ID - .&II,

resolvable

G.2)

operator.

lim Ax, = Py. In fact

ll~~n-PYII~II~--A~IIllznll+llAnzn-Pyll~ < by virtue

IM - &II II&-‘II

of the choice

-

PY

yn = P,,y then instead

II < II A -An II II Al-’ <

{II A -Al

(2.31

of yn.

If is (1.2) we assume that equality we obtain 11-&I

llyll + llyn - Pyll --t 0

of the preceding

in-

II II y II + II Pny - Py II <

II II Al-i II + II P, - P II } II y II + 0.

(2.4)

L.S.

6

4. If x0 is a solution AA-, then x, -+ x0. In fact II Qxn

since

$n

of equation

- &O E

- Qxo II = II A-l(AQ~n

Rakovshchik

AA.,

which lies

(2.1,.

in the subspace

then

- AQzo) II <

< II A-’ II [II A II II Q - Qn II II in II +II~c-PyII]+O in view of

the

From (2.3),

above proof.

(2.5)

It remains to be remarked that

zn -

xn =

(Qn - Q)z, +

(2. S),

(2.4)

we ootain

(QG - Qzo) :

(2.6)

the evaluation

50 II < II Qn -- Q II II An-’ II II yn II + + II A-’ II [II A II II Qn - Q II II An-’ II II yn II + II fi - An II II An-’ II II yn II + + llyn - Pdlll < IIA - &II max [II-k-ill, llA-ill] X (2.7) x [II&-’ II + II A II II &-i II II A-’ II + 11 II Yn II + II A-’ II II yn - Py II.

IIxn -

If Y" = P,y then

II &I - x0 ll < II A -A,

II max[II A-’

II, II ~4-lIIIX II+ II A-’ II + 11 IIY II.

X [II A,-i II+ II A II II A-l II II A-’

(2.8)

operNote that Paragraphs 2 - 4 are true for any normally resolvable On the assumpator only if the norms 11A,,-’ 11 are bounded in aggregate. tion that dim ZA < ~0,para. 4 can be obtained as a corollary of Theorem 3 of hl. Lemma

2

Suppose that in the pair (dim ZA, dim ZA.) at least one of the components is finite. Suppose that for large n in the pairs (dim ZA, components coincide, then the dim ZA.)and (dim &i , dim ZAz) the finite projectors

P, and /;)nnconverge to the projectors

P and Q respectively.

Proof. In the case where dim ZA < cothe lemma is established in [9] (Theorem 2). If, however, dim 2.4 = 03, but dim ~?AW< cc, then if we consider that (A”)* = A and apply the lemma to the operator A’, we establish the lemma also in this case (since Q is the projector onto AA*, and P onto A(A.,. = AA and 11A,* - A* II = II A, - A I\ -+ 0).

,4pprd.

3.

The

L;.~te

conditions

solution

of

equations

for the convergence approximations

7

of

the

Theorem

Let A be a bounded, normally resolvable operator and let at least one of the components of the pair (dim ,?A, #dimz~=, be finite. Then for the convergence of the solutions x, E a~* of equations (1. ?), for any Y, to the solution x0 E &A= of Equation (2.1) it is necessary (if = P,y) and sufficient that for large n tne finite components of the Yn pairs (dim Z*, dim z~*) and (dim z~ dim Z,A~\ shoul;l coincide. n’ y E

Proof. Suppose that for any Y E y the approximations constructed in accordance with Section 2 couverge, i.e. for any Y the sequence n, = A,-lP,,Y converges. By tne Ranach - Steinhaus theorem the norms of the are bounded in aggregate. The more so are the norms of operators ?In-li’, their contractions oounded on the subspace AA , i.e. the norms of tile

operators An-l are uniformly bounded. The ine&alities (2.2) in this case prove that for large II we have 11Pn - r7 )( < 1 and (10, - 0 11 < 1. 4ccording to [lOIfrom this there follows the equality of the dimensionalities of the subspaces 7~ and .?A (ZA. and zA* respectively). n

n

The sufficiency of the conditions for the case where dim ;:A < o follows immediately from Lemmas 1 and 2. If dim BAA’< Lo, and dim .zA = .x, then oy Lemma 2 the norms II A,-’ /I are uniformly bounded. From (2.7) it then follows that X, - x0. Votes. I. From the above it is obvious that the condition of the theorem is equivalent to the condition !I A,,-’ II 11A - A, \I < 1 for large n [l, ?I. 2. If dim 2.4 = 0 or dim z~. = 0 the conditions of the theorem are automatically satisfied. In fact if dim z~ = 3 (dim ?A- = I)) then for large n, according to [IO], 0 < dim Z,4 < dim ?A = 0 (0 52 dim Z,1* < ” n dim ?A- = 0). The situation investigated in this note bolls for characteristic singular equations with a Cauchy kernel, Wiener - HBpf equations and dual equations with difference kernels. Therefore, from the theorem proved above, a number of results obtained in [2 - 11 follow, relating to the Wiener - tldpf equations. 3. .4ny convergent sequence of solutions of equations (1.2) converges to the solution of equation (2.?). If dim 2~ ( u; then from any bounded

8

L.S.

Rakoushchik

sequence of solutions of equations (1.2) we can extract a convergent subsequence. The first of these assertions is obvious. We shall prove the second. Let x, be a given sequence. Then A,x, = yn -+ Py. Let R, be the lefthand regulators of the operators A,,, introduced into the proof of Para. 1, Lemma 1. Then 3,4,x, = x, + T,n, = R,y, -RPy. 4s has been remarked in the lemma mentioned, the operators T, converge in norm. Therefore the sequence convergent

{T,,x,)

is compact

subsequence

(the set

then it

{x,1

is obvious

is bounded). that x

“k

If T,,kn,k

is also

is its

convergent.

4. If dim 2~ < cc, then in the conditions of the theorem any solution of equation (2.1) can be put as the limit of solutions of equations (1.2). Let us consider those values of n for which dim .zA = dim ZA . For n(n) any II we shall consider the orthonormal basis z1 (n), z*(n) . . . . zcL a = dim ZA, of the subspace ZA . Using the above note a times we choose

a sequence

{zl

(rlk)

(nk’)

of such bases,

possessing

that property

for which the sequence.~itk’ converges for any i. The limits of these sequences form the orthonormal system, which is the basis of the subspace 2,. Hence the truth of the present note follows. 5. Let yn E Y be an arbitrary sequence which converges to y. If in the conditions of the theorem the equations l4nxn = yn are solvable for all n, then the equation Ax = y is also solvable. Let x, = A,-‘y,. From the preceding it follows that the norms of the operators A,-1 are bounded in the aggregate. Consequently this can also be said about the norms (1 x, II. Let z E 7,~~. Also

t!/, z) = lim (L/n, Z) = lim (A,x,, Since A,,+z -+ A+z = 0, then (y, z) = Z E ZA., it follows from this that that, if we exclude the trivial case of the approximate equation does not exact equation.

Z) =

Jim

(x,,

A,4tZ).

0. In view of the arbitrariness of y c Y 0 ZA. = AA. It is obvious where dim 2,. = 0, the solvability follow from the solvability of the

Now let A be a closed, normally resolvable operator. Let us introduce into its domain of definition DA a new scalar product, assuming that [X, yl = (x,,Y)+ (.4x, 4~). In the new metric -0~ becomes a complete Hilbert space and the operator A a bounded operator [lo]. As to the “approximate” operators, we assume that they have the same domain of definition n;l, are closed in the original metric and converge in thenew

4ppronimate

solution

equations

of

9

norm to the operator A, i.e. that 11A,x - Ax (1 9 E, [ I( x 11 + 1).4x !I I, where E, + 0. If we now consider the space 8~ instead of the space X (in the new metric) then it is easy to see that we find ourselves in the former conditions. Therefore the condition that dim zA = dim ZA n (dim 2,. = dim ZAi) guarantees the convergence of the solutions x, E AA;

n /)A of

equations

(1.2)

to the solution

of equation

the new norm is stronger than the old, the approximation to the exact solution in the old norm also.

(2.1). will

Since

converge

4. Examples Enample I. Let us consider

the equation

n(t)m(t)+$-jq$ = I(t),

(4.1)

in the space Lz(r1, on the assumption that the ordinary requirements, which ensure the validity of Neter’s theory for this equation, are satisfied [ll]. In every case we shall consider that the coefficients a(t) and b(t) cau be approximated on the contour r with any accuracy by the polynomials an(t) and b,(t). We shall

take the “approximate”

bn(t)

-4&p = an(t)cp+where vrai

max la(t)

- a,(t)

4” in the form

operator

ni

s

r

( + vrai max lb(t)

1-t -

(4.21

) b,(t)

1 + 0 as n 4~0.

For large n the indices of equations (4.1) and (4.2) coincide. In view of the well-known theorems of the theory of singular equations, the dimensionalities of the subspaces of zeros also coincide for large n (from Rouche’s theorem it follows that these conditions hold as soon as ((a - b) /(a

+ b) - (a - b,) /(a,

+ b,) 1 < \(a - b) / (a + b) I).

If the index of the equation x> 0, then AA coincides for large with the whole space L2tl-j. The approximate equation has the form

n

(4.3) Let r be a simple

closed

contour.

The solution

of equation

(4.3)

10

L.S.

takes

the form

[ill

+#bn(t)+

%(t)=GL(t)f(t)-

z-t

r

where P,+(t)

Ai

is a polrnonial

AZ(t),

(outside) (t)d

Bz(t)fAL

the curve

+

x - 1 with arbitrary

the equalities

B;(t)&

coeffi-

with zeros a,(t)

-

inside

b,, (t) =

(t). P,-I

of the polynomial

of orthogonality

9

are polynomials

BL (‘t)]

k(t)=

px_l(j;~*~

WWL-(t)

B,-(t)

of degree

(t),

x

W)bn(t)+

r, wnich satisfy

(t)64t)+

The coefficients dition

W)]&-(t)

niB-(t) dz

XSh(z)+Bn-(4fbG) ~n(~)lAn-(7) cients,

Rakovshchik

of the solution

are determined

to the subspace

from the con-

Z,n.

The functions

‘pdn)(t)= f&(t)bn(t)+ serve

as the basis

To determine

;q;;

&l(t)1

*tk,

n.

k = 0, 1, . . . , x -

1.

in ZA, [11].

the coefficients

s (Pn(t) q,(n) (t) ds = r

we have the system k = 0, 1, . . . , x -

0,

1;

ds is the element of arc. If f(t) has a complicated structure then it advisable to approximate it in norm Lz(r) by simpler functions. If x < 0 the approximate

equation

bn(t) Un(t)T(t)+ ~ tii The solutions tions

qjJn)(t)=(Ia,(t)+ The projector

has the form

cp(z)dz= s____ r

Pnf.

T--t

of the homogeneous conjugate B,-(t)tk

takes

b,(tj]A,-(t) the form

Pnf =f-

is

t’(s)

>

,

1X1-1

2 &‘pdyt). k=O

equation

k=O,

(4.5) will

be the func-

1, . . . . I$--1.

Approximate

The coefficients

dk are determined

‘%idk 1

of

solution

equations

from the system

= 1f(t)qj(n)(t)dS,

~k'n'(t)~jcn)(t)dS

11

i=o,1,...,

1x1~1.

r

k=O

The description of the ticneme can be transferred to the case of complicated and open contours and to systems of equations. In the last case the scheme remains without left-hand (right-hand) partial indices vanish. is somewhat complicated. Example 2. As the second example we shall Lz(O, 00) the Wiener - Hdpf equation

any cnanges if In the general

consider

all the case it

in the space

We shall assume that the kernel k(t) E L( --CO,a) and that its Fourier transform K(A) satisfies the condition 1 - K(A) # 0. Equation (4.6) is easily solved if the function K(h) is rational [12, 131. Using the fact that linear combinations of functions e*%%(Tt), 8 is Heaviside’ s function, are dense in L( -co. a)), we approximate the function k(t) by So that these combinations of ktn)(tj, _ K(“)(L) These conditions

#

0 D Arg [I - K@)@)]

will

be satisfied

s IW)(t)--Q) If

the index of equation

the approximate

equations

b(t)

I-” = ArgCl -K(h)]

12.

if (dt < minISi -K(h)

I.

(4.6)

can be taken in the form

A,cp = q(t)-

jii+)(t

- z)cp(z)dz = f(t).

(4.7)

0

Its

solution

assumes the form

%tt) =

Rnf +

2 i=i

C+n)qn(i) (t)

,’

(4.8)

L.S.

12

Rakovshchik

are where 3, is some right-hand resolvent of equation (4.7) and I, linearly independent solutions of the homogeneous equation A,9 = o (since K(nJ(A) is a rational function its resolvent R,, and the functions qnci) are easily constructed in explicit form). The constants

i

in . (4.8)

ci(“)

CP (qJ*(i), qln@‘) =

-

are determined

by the conditions

(RJ?

k =

I$#‘) ,

1, 2, . . . , x.

i=i If the index x is negative, then on the right-hand side of (4.5) instead of the element f we must have its projection P,,fonto the subspace of values of the operator A,,. To construct tion

the projector

An** =

g -

P, we shall

consider

%ti”‘(r - t)*(r)&,

the conjugate

t >

equa-

o.,

The Fourier transform of the kernel of this equation is also and so its solutions yrnCk)(t), k = 1, 2, . . . . 1x1 can be found.

rational

Then

P,f = f and the coefficients

citn)

2 Cp)qJ,(i), i=i

are determined

from the conditions

IX1 z

ci (qh(i),

l+,(k))

=

k =

(f, $7.(k)),

1,

2, . . . , 1x 1.

i=i by H.F.

Translated

Cleaves

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on the Akad.

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kompleksnogo

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idea

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2,

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28 - 39,

1,

IVANOV, V.V.

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583

a complex

of

seminar

cybernetics.

solution

4,

Dokl.

The

and

13,

Zh.,

11.

Univ.

funktsii

450,

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Approximate

1,

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I?

1954.

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scientific

the of

Two theorems

3,

IVANOV, V.V.

teorii

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PKHEN, KIM ZE Approximate method of systems of linear singular integral

In

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CHERSKII,

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to

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The effective leningr.

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type

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