517.51

..
, , E-mail: Agafonovanu@info.sgu.ru
- - [0,1) L1 [0,1). - .
: , , , , -.
On Uniform Convergence of Transformations of Fourier Series on Multiplicative Systems N.Yu. Agafonova
Necessary and suffiecient conditions for uniform -summability of Fourier - Vilenkin series of Functions from Orlicz spaces [0,1) and L1 [0,1) are obtained. Some corollaries for matrices with generalized monotone coeffiecients are given.
Key words: multiplicative systems, Fourier - Vilenkin series, uniform convergence, multipliers, uniform -summability.

{pn}JJ=1 , 2 < pn < N n G N. mQ = 1, mn = p1.. .pn n G N. x G [0,1)
x = J^ xn/mn, 0 < xn < pn, xn G Z.
(1)
n=1
(1) , = /1, ,1 G N < 1, . G [0,1) (1),

0 = = X! /, , = , + (), 0 < < ,
=1
G Z. . G Z+

= mj-1, 0 < < pj, - G Z, (2)
3=1
G [0,1)
Xk(x) = 2ni J2xj kj /pJ
= 1
, {xk(x)}^=Q , L[0,1)
.. , 2009
3
[1, 1.5] ( 0 ) = ()%() .. G [0,1) G [0,1) G Z+.
G Z+. {()}=0
/ (n) = /(t)Xn(t) n e Z+ Sn(/)(x) = /(k)Xk(x), n e N.
n 1
X) =
k=0
n 1
xk(x) =: Dn(x) n- . Lp[0,1), 1 < p < ,
k=0
1
1 p
, ||/||p = (/ |f (t)|pdt) .
0
Lp . () , [0, ) ,
(0) = 0, lim = + lim = 0. (3)
^ u ^0 u
(-) = sup(uv ()) , , () -
>0
. L[0,1) ,
1
||/|| = < sup
I /()() dx
: f (^()|) dx) < 1 >. () = up/p, 1 < p < ,
0
L[0,1) Lp[0,1)( ). . [2]. , / Lp[0,1), 1 < p < , ||Sn(/)||p < C||/||p , , lim ||/ Sn(/)||p = 0 (. , [3]).
^
L[0,1) - , 2-: (2u) < C^), (2u) < C(u), u [0, ). [4] ,
||Sn(/)|| < C||/|| lim ||/ Sn(/)|| =0. (4)
n^
(), () , 2-, / L[0,1), g L^[0,1) [2, 9, 9.3]
||/g||i <||/||
(4) , / L[0,1), g L^[0,1) -
/(t)g(t) dt = / (i)g(i).
i=0
B[0,1) [0,1) II/|| = sup |/(x)| C*[0,1) /(x) lim ||/( 0 h) - /()^ = 0
x[0,1) h^0
( ||/||^). UC[0,1) / e L1 [0,1), {xn}n=0 . (An}=0 -

(X, Y), / e X[0,1) An/(n)xn()
n=0
Y[0,1). {Akn}n=0 . n e Z+

Akn/(k)xk() gn(), gn(), ,
k=0
(), , / ˗.
- [0,1) L1 [0,1). [5], / e L2n [6].
1
1
1.
, . [7].
1 [8, 8]. 1) () +, (3). {} G (,)
||
n || :-------
n1
i=0
= 0(1).
2) (Ak} (L1,UC)
n1
Ai Xi
i=0
= 0(1).
2 [9]. () f G [0,1) ,
=0 1
.

l|Sm
akXk(x) k=0
3. () R+, (3)

2- . akXk(x)
k=0
/ L[0,1) , j|Smn|| .
. , / L[0,1) lim ||/ Sn(/)|| = 0,
n
j|Smn ||. , j|Smn || < M. / L[0,1) {n^=1 i __________________________________________
N, lim J (STOn. (x) /(x)) g(x) dx = 0 g L^[0,1).
i o n
00
g(x) = Xj(x), j Z+, , aj = /(j), ajXj(x)
j=0
/ L[0,1).
4 [10, 10]. {Akn}kn=0 , lim Akn = 1
Kn
AinXi
i=0
= 0(1). / C*[0,1)
Ain/(i)xi(x) f /(x).
i=0 n=0
2.
1. (x) 3. / L [0,1) -, :
1) lim Akn Z+;
n
2) n Z+ Kn(t) L^ [0,1) , Kn (i) = Ain, i Z+,
||Kn || = 0(1).
. . n Z+ {Ain }i=0 ^,UC).
i1
1 , ||Kin||^ < Mn < , Kin(x) := AjnXj(x), i N, n Z+.
j=0
K
< 3, , , ()
j=0
G [0,1). , [0,1)
1 00
[0,1) . / f ( )(0 ^ = ')()
x [0,1). L [0,1):
j=0
ln (/) = / /(0 0 t)Kn (t) dt = Ajn / (j).

(5)
j =0

OO
CXD

1
1
n
, (5)
n ^ . L[0,1) L^[0,1), ||Kn|| . ,
1 ________________
ln(Xk) = / Kn(t)Xk(t) dt = Akn n ^ . , 1) 2) .
0
. / L^0,1), ||Kn|| < M {Akn}^=0 Z+. ,

(4), ||Kin|| < M1 i N, n Z+. 1 Ajn/(j)Xj(x)
j=0
an (x) n Z+. (5) / L. ||Kn || < M ln . , ln (Xi) lim Ain, ln L [0,1) P
n
{X^i=0. {ln(/)}n 0 / L^0,1).
Ta/(t) = /(t a). , ln(Ta/) = an(a). , P L[0,1) , lim ||Ta/ /|| = 0 / L[0,1). , ln(Ta/)
0
a [0,1), . = 1/mk , 0 < h < |Th/ /|| < . a [0,1) [i/mk, (i + 1)/mk), i Z+ [0,mk). |ln(Ta/) lm(Ta/)| < |ln(Ta/) ln(Ti/mfc/)| +
+ |ln (Ti/mk /) lm (Ti/mk /)| + |lm (Ti/mk /) lm(Ta/)| = ^1 + ^2 + ^3. ||ln || = ||Kn || < M,
11 + 13 < 2M||Ta/ Ti/mk/|| < 2Me. mk , n0(e), n,m > n i Z+ [0,mk) |ln(Ti/mfc/) lm(Ti/mfc/)|< |ln(Ta/) lm(Ta/)| < (2M + 1).
on \
{ln(Ta/)}n 0 a, {an(a)}n=0 a. .
2. / L1 [0,1) --, :
1) lim Akn N;
n
2) |Kin (t)| < Mn t [0,1) i N;
3) Kn(t), n Z+, Kn(i) = Ain, i Z+,
!KnJL = (1)-
. . / L[0,1) --

. 1 2). 2 , AinXi(x)
i=0

Kn B[0,1),n Z+. Ain/(i)Xi(x) ,
i=0
(Kn * /)(i) := Kn(i)/(i) = Ain/(i), i Z+ ,
1
Kn * /. ln(/) = J /(0t)Kn(t) dt = Ain/(i).
0 i=0
{ln(/)}n=0 / L[0,1), - |Kn|oo < M. ln(Xi) = Kn(i) = Ain n ^ i Z+. , 1)-3) .
. 1)-3). 1 , n N

^2 Ain/(i)Xi(x) . 1 1) -
i=0
ln(/) P , , / L1[0,1). , ln(Ta /) = an(a) lim ||Ta/ /II. ^ 0 / L1[0,1), , 1, -
11 111
a ln (/). .
1 . ak( + 1) > 0 lim ak = 0, {ak)k-0 AT. ak
k k=0
> 0 lim ak = 0, {ak}, _ A. , lim ak = 0
k k=0 k
| | |
|akak+1| < Can n Z+, {ak}k=0 RBVS.
k=n
.. [11] , .. [12] . [13].
1. 1 < p < , 1/p + 1/q = 1,
:
1) lim Akn Z+;
n |
2) n Z+ {Akn}0On=0 , R,

RBVS, Akn( + 1)q2 < Mq n Z+.
k=0 kn
/ Lp [0,1) -. . [14, 8, 9] , 2)

^ AknXk (x) Lq [0,1) Kn Lq [0,1) 11 Kn | < C1M. 1.
k=0 q |
2. 1 < p < , 1/p+ 1/q = 1,
:
1) lim Akn Z+;
n |
2) n Z+ , R,

RBVS, Akn^ + 1)1/q < M n Z+. k=0
/ Lp [0,1) -. . 2 [15] ,
( \ 1/q
5>kn(fc + 1)q < C1 Akn ( + 1)1/q < CM,
k=0 k=0
C1 N, 2 < pn < N, C RBVS. 1, 2.
C*[0,1) . {Akn}kn=0
n
, .. Ain/(i)Xi(x) = ^ Ain/(i)Xi(x) = Kn(x)
| i=0 i=0 | / L1 [0,1). Pn = {/ L1[0,1) : /(i) = 0, i > n}, En(/)p = inf{||/tn||p : tn Pn}.
3. pi = 2, / C* [0,1), en j 0, en < CeNn n N En(/) < Cen. ,
1) lim Akn = 1;
n
2) lim IlKnlL en = 0;
n 1
3) |S2m(Kn)||1 < M, m = [log2n] n N m =1 n = 0.
n
un(x) = Ain/(i)X(x) /(x).
i=0
. un (x) = / * Kn(x) = (/ S2m (/)) * Kn (x) + S2m (/) * Kn (x). .. [1, 10.5] ||/ S2m(/)|| < 2E2m(/) < C1e2m < C2en
- n Kn I
S2m(/) * Kn(x) = / * D2m * Kn(x) = S2m(Kn) * /(x). |Aknj -
kn k,n=0
Akn = Akn < 2m 1 Akn = 0 > 2m. , 4, S2m (Kn) * /(x) /(x). . . S2m (Kn) -
I
2) lim ||(/ S2m (/)) * Kn|00 < lim C2en |Kn|1 =0.
n n
= / *
Tn ^ ^ Kin
i=[ ^ ]
.. .
n

1. .., .., .. : . .: , 1987. 344 .
2. .., .. . .: , 1958.
3. Young W.S. Mean convergence of generalized Walsh -Fourier series // Trans. Amer. Math. Soc. 1976. V. 218, 2. P. 311-320.
4. Finet C., Tkebuchava G.E. Walsh - Fourier series and their generalizations in Orlicz spaces // J. Math. Anal. Appl. 1998. V. 221, 2. P. 405-418.
5. Karamata J., Tomic M. Sur la summation des series de Fourier des fonctions continues // Acad. Serbe Sci. Publ. Inst. Math. 1955. V. 8. P. 123-138.
6. Katayama M. Fourier series. XIII. Transformation of Fourier series // Proc. Japan Acad. 1957. V. 33, 3. P. 229-311.
7. Goes G. Multiplikatoren fur starke konvergenz von Fourier Reihen // Studia Math. 1958. V. 17. P. 299-311.
8. .., .. // , , : . . . . : - . -, 2005. . 3. . 3-23.
9. .. - // , , : . . . . : - . -, 2007. .4. . 3-10.
10. .., .., .., .. - . : , 1981.
11. .. // . . 1958. T. 44, 1. C. 53-84.
12. .. , // . . 1967. T. 74, 1. C. 100-118.
13. Leindler L. On the uniform convergence and boundedness of a certain class of sine series // Analysis Math. 2001. V. 27, 4. P. 279-285.
14. .. // Analysis Math. 2007. V. 33, 3. P. 227-246.
15. .. // Analysis Math. 2007. V. 33, 4. P. 247-262.
517.51

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E-mail: LukashovAL@info.sgu.ru, s_tyshkevich@yahoo.com
, , . .
: , , .
Extremal rational Functions on Several Arcs of the Unit Circle with Zeros on These Arcs
A.L. Lukashov, S.V. Tyshkevich
The solution of an extremal problem about rational function with fixed denominator and leading coefficient of nominator which is deviated least from zero on several arcs of the unit circle is given under restrictions on the location of zeros and additional conditions on mutual position of the arcs and zeros of denominator. The extremal function is represented in terms of the density of harmonic measure.
Keywords: best approximation, extremal rational function, harmonic measure.
, , .. 1853 . . [1]. [2].
. , .. , 2009