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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXV (1985) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXV (1985)

S. St o in s k i (Poznan)

1. Let F be an T7-operation in R + = <0, oo), i.e., let F be a mapping F : R + x R+ -*■ R +, satisfying the following conditions (see [1], [2]):

(a) F (и, v) = F(v, и),

(b) F (u , F( v, w) ) = F(F(u, v),w), (c) F(u, 0) = u, F{0, v) = v,

(d) F(u, v) is non-decreasing in each variable separately, (e) F is continuous.

If F is an F-operation, then:

(a) F (m1s i^) < F (u2, v2) for 0 ^ ux ^ u2, 0 ^ vx ^ v2, (b) F(u, v) ^ 0, F ( 0, 0) = 0.

we can extend F to a function F: R+ x R+ -> R+, where R+ = <0, oo).

Let X be a real linear space and let F be an F-operation. A functional

q: X -+<0, o o ) is called an F-pseudomodular, if for every x, y e X:

( a ) e(0) = 0, (b) e ( - x ) = e(x),

(c) e(ax + py) ^ F(g(x), p(y)) for a, £ ^ 0, a + j8 = 1.

In case when q satisfies conditions (b), (c) and, in place of (a), it satisfies (a') g(x) = 0 o x = 0,

Q is called an F-modular. Let

X Q is called an F-modular space (see [1], [2]). In the following we shall assume that X e contains elements Ф 0, i.e., there exists an x e l such that x Ф 0 and g (Дх) —► 0 as Д -> 0 + .

An application of modular spaces to approximation problems, III

If we set

X Q = {x = X: q(Xx) -> 0 as Д -* 0 + } .

— Roczniki PTM — Prace Matemàtyczne XXV

(2)

Let h be a homeomorphism R + -►/?+. For any F-operation F we define the function G = h* F by

G {и, v) = ft_1(F(fi(u), /i(r)))

for u, v ^ 0, where /i_1 is the function inverse to h for и ^ 0. If there exists an h such that G = h* F, we say that G is equivalent to F (see [1], [2]).

Let к : /?+ -» /?+ be a mapping such that k{u) > 0 for и > 0 and let F be an F-operation. We say that к is F -super additive if

F(k(u), k(v)) ^ k(u + v) for u, v ^ 0.

If g is an F-pseudomodular in X, к is F-superadditive, then

|x|e>fc = inf {u > 0: q(x/u) ^ k(u)}

is an F-pseudonorm in X Q (see [1], [2]).

If q is an F-modular in X, к is F-superadditive, lim k(u) = 0, then | • |e>k u-»0 +

is an F-norm in X Q ([1], [2]).

The following theorem is true (see [1], [2]):

Theorem 1. Let q be an F-modular in X and h be a homeomorphism R + -+R+, G = h* F, к is F-superadditive. Then h xoq is a G-modular, where h( + oo) = + oo, and

for every x e X Q.

2. Let (G, X, p) denote a space with a finite and complete measure p, defined on S, a c-algebra of subsets of the set Q Ф 0 , p(Q) > 0, g„{t, x):

Q x X -*■ <0, oo) for n = 1, 2, ... and x e X — a space of functions x: Q oo, oo) which are F-measurable and almost everywhere finite, where x

= у iff x(f) = y(f) almost everywhere.

We define the following operator

F^iu^v) = (p~x((p{u) + <p{vj), u , v ^ 0,

for a ^-function (p (see [2]). F v is an F-operation in R + . In the special case for (p (u) = up, p > 0, и ^ 0, let us denote

F p(u, v) = (up + vp)1/p for u, v ^ 0.

If u, v ^ 0, then

FPl(u, v) ^ F P2(u, v)^ F ^ i u , v) = max(u, r) for 0 < pj ^ p2 < °o.

(3)

Modular spaces and approximation problems, III 163

For x, y e L v(Q, I , ц) and for an F-operation Fp, 0 < p ^ 1, the next inequality

(♦) {J П(М01, \ y ( t ) Ш 11” « F ,{[f M t T d t i ] 1» , [J \ m v d n ] 4 ° } ,

q a n

where v ^ p, holds.

J u s t i f i c a t i o n . Applying Minkowski inequality for \x\p, \y\p e L vlp(Q, I , ц) we have

\x\vd t f lv+(S

### I

y\vdnYlv,

n n n

and hence follows ( * ).

For 0 < p ^ 1, un, vn ^ 0, an F-operation Fp satisfies the condition

oo 00 00

### (î) Z

an F p ( u „ , v n) < F p ( Y , anut

QnVn

### )>

where an are non-negative constants.

J u s t i f i c a t i o n . Using the Minkowski inequality for series we have

00 00 00

[ Z an(Mn + fn)1/p] p ^ ( Z anUn)p + { Z anVn)p,

n= 1 л= 1 Л= 1

and hence follows (*).

Let us assume:

1° gn(t, x) is an F p-pseudo modular in X for all t e Q and for every n = 1, 2, ... for 0 < p < 1,

2° Q„{', x) eL (Q, Z, p) for n = 1, 2, ... and x e X ,

3° if for n = 1, 2, ..., Q„(t, x) = 0 for almost all te Q, then x = 0.

Let us denote for x e X

Qns(x) = j 6n(t, x)dp, n

QS(X) =

## z

2" l+Qmix)*1 £>«(*)

— { x e £ : gs(/lx) -^ 0 as 1 -» 0 + } . X s is an F p-modular space, because:

(af ^s(x) = O o x = 0.

(b) gs( — x) = gs(x) for x e X .

(c) For x, yeJE, a, /? ^ 0, a -F >5 = 1, we have for an F-operation F p, 0 < p < 1, using ( *),

Qs(<xx + Py

### К Z 4

n= 1 Z

Fp(ûns(x), 6ns(y))

£«(>>))’

(4)

Because an F-operation Fp, 0 < p ^ 1, satisfies condition (*), so for un, v„ ^ 0 we have

00 1 / и V \ / 00 t и 00 1 и

(I)

For p > 0, u, v ^ 0 there holds the inequality

(2) (up + vp)1/p

1 + (up + vp)1/p Using (1) and (2), we have

< 1 + и1 + v

lip

00 j Qs(zx + [ jy ) ^ X — Fp

n= 1 £

g n s ( * ) g M ( ^ ) \

1+ £ « (* )’ 1 +Qns(y)J

< Fp(qs(x), Qs(y))-

In the following we shall suppose that besides conditions l°-3° the following condition is satisfied:

4° if x, ye.T, |x(r)j < |y(f)| ^-almost everywhere in Q, then for n — 1, 2, g„(t, x) ^ gn(t, y) ju-almost everywhere in Q.

We say that a sequence (£„) preservs constants, if g„{t, с) = c for every ' t e Q and for every c ^ 0, for n = 1 , 2 , . . .

Theorem 2. I f the sequence (g„) preservs constants, x e X , x ^ 0, then for every Я > 0, a, /1 > 0, a + j8 = 1, we have

Qns U [g M(-. * )-* (• )]}

^ Р А д „ , [ 2 Х - x(-) 1, e J 2Я max rr , *(•) H n I --- , Qt

H P \ gm(-, x), g„

x - x ( - ) P x - x ( - )

p for m, n = 1 , 2 , . . . , where H p(u, v) = Fp(u, v) — u, 0 < p ^ 1.

P ro o f. Let xe X, x ^ 0, a, /? > 0, a + /? = 1, Я > 0. Because the sequence (g„) preservs constants, so for t e A = {teQ: x(t)-finite}

Qm(t, x{t)) = x{t).

Therefore for t e A r \ B , where В = [feO : gm{t, 2x//l)-finite}, we have Qm(t, x) = gm , , a ^ + p

a

x —x(t)

~ ~ P ~ t, xX ( f )

~ T ~ and

gm{t, x ) - x ( r ) ^ Fp x{t) ---, Qm

0L

x — x(t) P

x(t) 0L

+ - x{t).В

(5)

Modular spaces and approximation problems

### ,

III 165

For t e A n B n C , where C = { teQ\ Qm(t, x)-fmite}, we have ax(t) — <xx

<*x(t) = Qm(t, ocx(t)) = ax + p P

^ F A Qm(t, X), t, x — x(r) and

x ( t ) - Q m(t, x) = ax(t) + P x( t )- Qm(t, x)

^ Fp \ Qm(F -*-)? Qm( t•> p QmiF “b ^ x{t).

Let us denote

y m{t) = \Qm(t, x ) - x ( t ) \ for t e A n C . Then for t e A n В n C

Ут (t) < max 'x(t) ( x - x ( t ) H A ---, gm t,

H A Qm(t, X), 0W t, x — x(t)

+ - x(r).

a Using condition 4°, we obtain for almost all t

Qn(t, k y m) < Qn L Л max H T x( - )

Qn x — x( •) p

Hp ( Qm ( ? -X)5 Qn _ _ _ _ _x — x ( ) + Я — x ( ■ ) (X and

Qn(t, k y m)< Fp<q r, 2 1 -P * ( • ) ,

£?„ t, 2Л max H A --- , Qp \ > \ 9 ' a

x - x ( - ) \ ) „ ( ( л ( X -x (* ) b H p\ Qm( -, X), Q

P P

almost everywhere in Ü. Hence, using (*), we have the thesis of Theorem 2.

For an F-operation Fp, 0 < p ^ 1, the function k(u) = ul,p, u ^ 0, is Fp-superadditive. The homeomorphism h is of the form h(u) = u1/p. Therefore for the sequence (g„) of F p-pseudomodulars, where 0 < p ^ 1, such that q„, n

= 1, 2, satisfy conditions l°-4°, we have

(6)

For x e X ^ , x ne X eS, n = 1, 2,

1вп(-> х)-х(-)1^р ->0*> V (gy{A[g„('> * ) - * ( • ) ] } ->o w ' ’ A>0

(see [2]). We shall give conditions, under which

(qY U Cg, (% * ) - * ( •) ] } - * 0 with n -+ 00 for every A > 0.

From Theorem 2 it follows

1 Co r o l l a r y. I f the sequence (q„) of Fp-pseudomodulars preservs constants,

then for every A > 0, e > 0, x e X^s, x ^ 0, there exists f > 0 such that for every m = 1 , 2 , . . .

(gs {A lQm ( -, x) - x ( • )]} Y < ( V 1 2A max H r *(•) 1 -/?» 0л

x (-)

Qm ( "> •*-)> Qm

where H p(u, v) — Fp(u, v) — u, 0 < p ^ 1.

P ro o f. Let us denote for a, j6 > 0, ct + f} — 1,

/ x - x ( ' )

### I

p+ e,

Л(х, p, m, a, j3) max Hr x (-)

Qm

X — x ( - )

______ ? ^ p l Qmi ’

x — x ( )

From Theorem 2 it follows e « U [g m (s * )- * ( •) ] } 1 + g w { ^[g « ('. * )-* (* )]}

< Fp ^&ns [2/W a ) * ( ’)]> [2ЛЛ (x, p, m, a, /?)]}

^ 1 + Fp {Qnsl 2 A { P M x ( f ] , Q^l lAAi x, p, m, а, Д)]} ' Using inequality (2) and inequality (J ), we have

* ) - * ( * ) ] } < FpiQ 2A — x( •) qs[2AA(x, p, m, a, /?)]

and

(e"U [&»(*» * ) - * ( - ) ] } ) p ^ 2A — x ( ) a

+ (es[2U4(x, Pi a> 0)])p.

(7)

Modular spaces and approximation problems, III 167

Because x e X s, so gs [2Я(/?/ос)х( •)] -^ 0 for /? ->0, e.a. for e > 0 there exists P = P(e) > 0 such that qs[ 2 A ( P /ol)x(•)] < e1/p. From this follows the thesis.

The sequence (g„) of F p-pseudomodulars, 0 < p ^ 1, is called singular at the point xeX^s, x ^ 0, if for every a! > 0, b' > 1 and for n = 1 , 2 , . . .

### ■/«W = J a' max

n

H p ( y i 7 l &»('’ b ' ( x - x ( - ) ) ) J ,

H @m( 9 Ь (x x( )))) dfl~+ 0

with т —>со.

Theorem 3. I f the sequence {q„) of Fp-pseudomodulars, 0 < p ^ 1, preservs constants and is singular at the point x e X ^ s, x ^ 0, then for every Я > 0

É?s U[@m(‘9 *) — * ( ’)]} -*0 w i t h m - * c c .

P ro o f. By Corollary of Theorem 2 it follows that for every Я > 0, s > 0, x e X ^ s, x ^ 0, there exists f > 0 such that for every m = 1 , 2 , . . .

(es * )-* (* )]} )p < (qs<2A max Hr *(•) l - P ’ Qn

x — x( •) p

hi n I Qmi. 9 7C), Qn *(•) P + e,

where H p(u, v) — F p(u, v) — u, 0 < p ^ 1. Because (q„) is singular at the point

x eI jS, so, putting a' = 2Я, b' = 1/f, we obtain the thesis.

3. In the sequel we are going to consider the following special case. Let Q = <a, b >, F-d-algebra of Lebesgue measurable sets in (a, b>, ^ — the Lebesgue measure. Let X denote the set of Г -measuraoie and almost every­

where finite functions in <a, b), extended periodically, with period b — a, outside {a, b). Let K n, n = 1, 2, ..., be functions Г -measurable and positive almost everywhere in (a, b) such that

b

j K n(u)du = 1 for n = 1, 2, ...

(и = 1, 2, ...) is a concave (^-function and (p~ 1 is the function inverse to (p„ for t O 0.

We define the following sequence of functionals b

gn(t, x) = (p~x {J K H(u)<p„(\x{u + t)\)du}

(8)

with n = 1 , 2 , . . . , t e {a, b), x e X . For n = 1 , 2 , . . . and a, /? ^ 0, a + /? = 1, we have

6 b

Qn(t, ocx + py) ^ <p„-1 {J X„(w)^„(a|x(w + OI)dM+ j K„(u)<pn(£|y(t + u)|)du}

a a

= F<p„(Qn(t, ocx), g„(t, py)) < F Vn(Qn(t, x), Qn(t, y ) ) .

Therefore q„ is an F^-pseudom odular.

We shall now investigate the next sequence

Qn(t, x) = {j K n(u)\x(u + t)\pdu}1/p,

n = 1 ,2 , with fe < a , ft), x e L « e , ft), Z, p), 0 < p ^ 1. For n — 1, 2, p„(f, x) satisfy conditions l°-4° and the sequence (q„) preservs constants.

We say that (K „) is a singular kernel, if

b - 0

lim j K„ (w) du = 0

n-»oo a + 0

for every <5e(0, j ( b — aj).

Th e o r e m 4. I f is the F p-modular space generated by the sequence (gm)

of the form

b

Qmit, x ) = {{ K m(u)[x(u + t )Y du }1/p, m = 1, 2,

where (K m) is f/ie singular kernel, 0 < p < 1, x e L9((a, ft), Z, p), x ^ 0, where q = 2/Д/ + l) + e, / is the natural number such that 1/(1 + 1) ^ p < 1//, ee(0, 2 Д /+ 1)), fften

le « (’. x ) - x ( - ) le*fk -*0 with m ^ go, where k(u) = u 1Jp for и ^ 0.

P ro o f. Let x e L q, x ^ 0, where g .= 21/(1+1) + £, 1Д /+ 1) ^ p < 1//, .

def '

/ = 1 , 2 , . . . , ее(0, 2Д/+1)), <р(м) = wp for и ^ 0, 0 < p < 1, <p( + oo) = + oo.

1 We shall estimate the next expression

ь b

Jm(x) = j <P_ 1 {J i C „ ( u ) < p [ a ' f F xw ( u , tf\du}dt,

a a

where

И^"(м, t) — m axj cp 1 <И ^ггу x(w + t) ) +

+ Xm(r)(p(b'|x(r + w + r)-x(M4-t)|)di;

bt - т гb — 1- 7 +

(9)

Modular spaces and approximation problems, III 169

Ф- 1

о

K m(v)(p(x{v + u + t))dv+ J K m{v)(p(b'\x{v + u + t ) -

a b

Ф_ 1 ( ] K m(v)(p(x(v + u + t))dv^j

— x(u + t)\)dv m, n = 1 , 2 , . . . , a' > 0, b' > 1.

Let us write

b' b

Ui x{u + t), u2 = (p K m(v)(p(x(v + u + t))dv],

О l a

b

vo — K m(v)(p(b'\x{v + u + t) — x(u + t)\)dv~\.

a

In the sequel, for p we choose the natural number / such that 1/(1+1) ^ p < 1 //. Then

W?(u, t) = max {(ttf + rg)1/p — ut}

i = 1,2

^ max {(ull{l+i)+ vll{l+l))l+i— щ}

i = 1,2

= max |у 0+ X Г ! 1)"?

i=l,2 ( fc=l \ f t J

k/(l+ 1) . j.(l + 1 — k)/{l + 1)

v0 and

(p(a' W™(u, t)) ^ a'p max jüg + Y u\km+1))p-v(^l+1 k)/(l+ 1))p|,

b b

i = 1 КЛи) yo +

+ 1 ( l *1) l' “!w+1),'’‘"8,+1' ‘>,<,+1,>' ) ‘,1(

1/P

dt.

In the following we shall apply the generalized Minkowski inequality of the form

b d d b

(3) j ( j IF (x, y) I dy)v dx ^ [ j ( j IF (x, y)\v dx)i/v dy]v

a c c a

for measurable function F in the rectangle <a, b> x <c, d}, v ^ 1.

(10)

Since (p 1 is a convex function, so for / = 1 ,2 we have

a a

« (/+1)1,P_1 | j [ | K„(u)t>s*/.]1"’dt+

( a a

+ £ K n(u)u?l<l + l '>r-vÿ+1- t>l<‘ + "» d uy i''d t\.

k = 1 V к / a a )

Using inequality (3), we obtain

ь ь ь b

J [J K n(u)v%du]1/pdt < b' {J /Cm(^)[f |x(*;-l-s)-.x(s)|ds]pdü}1/p

a a a a

and

L x = Щ K n(u)uflil+1))p- v ÿ +1- k)lil + 1))pdu]llpdt

a a

b b

^ b ' { b ' - i y kl(l+1) J [x(s)]k/(f+1)[{ K m(v)\x{v + s ) -x (s )\ pd v Y +l~kml+1>p)ds.

a a

Next, we apply Holder inequality with exponents P = p{\ + l)(\+co), Q = P / { P - 1), where (1 + y)/p(l +y + yl) < co + 1 < (1 + y)/lp for ( / - l ) / ( / + l )

< y < 1. Then

L i ^ b '( b '- l ) ~ k/(l+1) {J [ x ( s ) T Kl+1))Qds}llQ x

a

X {J [J + — x(s)\p dv\({l+1 ~k)l(l+ l)p)P ds}llP.

a a

Since (k/(l+ 1))Q < 1 + y for к = 1, x e L 1 + ?((a, Ь>, I , p), so there exists a constant M such that for к = 1, /

(4) {{ [x(s)](k/(,+ 1))Qds}llQ ^ M.

a

Using inequalities (3) and (4), we have

ь b

L y ^ b ’ib’- l ) - m +1)M { \$ K m(v) Q \x(v + s)~

a a

~ * ( 5)|((l+1 l))PdsJll+ l)pm +1 “ *)P) dvyl+1~kmi* l)pK

(11)

Modular spaces and approximation problems, III 171

In an analogous manner we obtain

L 2 = j ( j K M u {i (l+1))p ■v(\$ +x~mi+1))p d u f lp dt

a a

< b'(f+1- fc)/(,+ 1) {J [em(s, x)-]W(l + 1))Qd s Ÿ IQ x

a b b

x {J[J K m(v)\x(v + s) — x(s)\pdv~\i(-l+1~k)lil+1)p)pds}1/p.

a a

Applying the generalized Minkowski inequality we obtain for к = 1, / and со ^ ( 1 /((/ + l)p — 1)) — 1 for p > 1/(1+1)

{j[J K m(v)(x(v + s))p dv~\kQli(l+ 1)p) ds}1/Q ^ M

a a

and

L 2 ^ b ,(l+1~kWl+1)M { j K m(v)x

a b

x [ J \x(v + s ) - x ( s ) f +1- kWl+1))Pd s f l+l)p)m+1~k)P)dvYl+1- km+l)p).

a

Therefore

J m(x) « a' b' [2 (/+ 1 ) ] 1 {2 [J K m(t>)(f \x [v+ s)- x (s)| d s f dv] "* +

a a

+ M У ( l + 1 ) [ ( b ’ - i r ^ + b ' - ^ ‘ > ] x

k = l v к )

x [ J (r)(j" Ix(v + s).

— x(s)|((i + 1 -*)/(*+ 1))p ^s)«l+ l)P)/(('+ 1 -k)P) dvy + 1 -k)/W+ l)p)J where

P = p ( /+ l) ( m + l) , 1 + y

p ( l + y + yt) < со + 1 < min 1 Ч~ у 1

Ip ’ p(l + 1)— 1 for p >

l + l for (/— ! ) /( /+ 1) < у < 1.

For <5e(0, j ( b — a)) and к = 1, / we can write { K » [ f \x(v + s ) - x ( s t l+l~k)^ l))Pd s f l+1)pm+1- k)P)dv

a + ô bô b

= I + J + I = T 3 + l4 + l5.

a a + ô b —ô

(12)

Then

L 3 ^ [ sup ] \ x (v + s ) - x ( s ) \ {{l+1- km + 1))Pd s f l+l)pmi+l- k)P)->0 a^v<a + ô a

with ô -> 0 (see [3]),

La ^ 2(p,l,k,P) J K m{v) {J [ x (v + s)]((/+1 -k)l{l + 1))Pds +

a + ô a

a

with m -> oo, where

/ 7+ 1 — к \ . " " ( ‘ ■ т г т : because x e L 1 + y(<a, b}, I , p). Similarly, L s ->0 with <5 ->0.

In an analogous manner we can estimate the integral (p, l, к, P) = (/+ 1)P

(,l + l - k ) P

ь b

j iCm(i/)(j |x(p + s) —x(s)| ds)pdv.

a a

Therefore for x e L q({a, b), I , p), where q = 27/(7+l) + e, when 7 is the natural number such that l/( /+ l) ^ p < 1/7, 0 < e < 2/(7+1), we have J m{x)

—► 0 with m —► oo. Using Theorem 3, we obtain

le«(% * ) - * ( • )le»tk->0 for m -» oo, where к (и) = u1,p for и ^ 0.

References

[1] J. M u s ie la k , J. P e e tr e , F-modular spaces, Functiones et Approximate I (1974), 67-73.

[2] J. M u s ie la k , Orlicz spaces and modular spaces, Springer, Berlin 1983.

[3] R. T a b e r s k i, Aproksymacja funkcji wielomianami trygonometrycznymi, Poznan 1979.

INSTITUTE O F MATHEMATICS A. MICKIEWICZ UNIVERSITY Poznan, Matejki 48/49

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