Universal series by trigonometric system in weighted $L^1_{\mu}$ spaces

0.15 MB·English

by  Sergo A. Episkoposian

UNIVERSAL SERIES BY TRIGONOMETRIC SYSTEM
IN WEIGHTED L1 SPACES
µ
S. A. EPISKOPOSIAN
Abstract. In this paper we consider the question of existence
5 of trigonometric series universal in weighted L1[0,2π] spaces with
µ
1
0 respect to rearrangementsand in usual sense.
2

n
a
J 1. Introduction

5

]
A
F Let X be a Banach space.
.
h
at Definition 1.1. A series
m
∞
[ (1.1) f , f X
k k ∈
1 k=1
X
v
is said to be universal in X with respect to rearrangements, if for any
7
2
f X the members of (1.1) can be rearranged so that the obtained
8 ∈
∞
0
series f converges to f by norm of X.
0 σ(k)
.
1 Xk=1
0
Definition 1.2. The series (1.1) is said to be universal (in X) in the
5
1
usual sense, if for any f X there exists a growing sequence of natural
:
v ∈
i numbers nk such that the sequence of partial sums with numbers nk of
X
the series (1.1) converges to f by norm of X.
r
a
Definition 1.3. The series (1.1) is said to be universal (in X) con-
cerning partial series, if for any f X it is possible to choose a partial
∞ ∈
series f from (1.1), which converges to the f by norm of X.
nk
k=1
X
1991 Mathematics Subject Classification. AMS Classification 2000 Primary
42A20 .
Keywords and phrases. Universalseries,trigonometricsystem,weightedspaces.
The author was supported in part by Grant- 01-000 from the Government of
Armenia .
1
2 S. A.EPISKOPOSIAN
Note, that many papers are devoted (see [1]- [9]) to the question on
existenceofvariousetypesofuniversalseriesinthesenseofconvergence
almost everywhere and on a measure.
The first usual universal in the sense of convergence almost every-
where trigonometric series were constructed by D.E.Menshov [1] and
V.Ya.Kozlov [2]. The series of the form
∞
1
(1.2) + a coskx+b sinkx
k k
2
k=1
X
was constructed just by them such that for any measurable on [0,2π]
function f(x) there exists the growing sequence of natural numbers
n such that the series (1.2) having the sequence of partial sums with
k
numbers n converges to f(x) almost everywhere on [0,2π]. (Note
k
here, that in this result, when f(x) L1 , it is impossible to replace
∈ [0,2π]
convergence almost everywhere by convergence in the metric L1 ).
[0,2π]
This result was distributed by A.A.Talalian on arbitrary orthonor-
mal complete systems (see [3]). He also established (see [4]), that if
φ (x) ∞ - the normalized basis of space Lp ,p > 1, then there
{ n }n=1 [0,1]
exists a series of the form
∞
(1.3) a φ (x), a 0.
k k k
→
k=1
X
which has property: for any measurable function f(x) the members of
series (1.3) can be rearranged so that the again received series converge
on a measure on [0,1] to f(x).
W. Orlicz [5] observed the fact that there exist functional series that
are universal with respect to rearrangements in the sense of a.e. con-
vergence in the class of a.e. finite measurable functions.
It is also useful to note that even Riemann proved that every con-
vergent numerical series which is not absolutely convergent is universal
with respect to rearrangements in the class of all real numbers.
Let µ(x) be a measurable on [0,2π] function with 0 < µ(x) 1,x
≤ ∈
[0,2π] and let L1[0,2π] be a space of mesurable functions f(x), x
µ ∈
[0,2π] with
2π
f(x) µ(x)dx < .
| | ∞
Z0
UNIVERSAL SERIES BY TRIGONOMETRIC SYSTEM IN WEIGHTED L1 SPACES3
µ
M.G.Grigorian constructed a series of the form (see [6]),
∞ ∞
C eikx with C q < , q > 2
k k
| | ∞ ∀
k=−∞ k=−∞
X X
whichisuniversalinL1[0,2π]concerningpartialseriesforsomeweighted
µ
function µ(x), 0 < µ(x) 1,x [0,2π].
≤ ∈
In [9] it is proved that for any given sequence of natural numbers
λ ∞ with λ ∞ there exists a series by trigonometric system of
{ m}m=1 m ր
the form
∞
(1.4) C eikx, C = C ,
k −k k
k=1
X
with
m
C eikx λ , x [0,2π], , m = 1,2,...,
k m
(cid:12) (cid:12) ≤ ∈
(cid:12)Xk=1 (cid:12)
so that for e(cid:12)ach ε > 0(cid:12)a weighted function µ(x),
(cid:12) (cid:12)
(cid:12) (cid:12)
0 < µ(x) 1, x [0,2π] : µ(x) = 1 < ε
≤ |{ ∈ 6 }|
can be constructed, so that the series (1.4) is universal in the weighted
space L1[0,2π] with respect simultaneously to rearrangements as well
µ
as to subseries.
In this paper we prove the following results.
Theorem 1.4. There exists a series of the form
∞ ∞
(1.5) C eikx with C q < , q > 2
k k
| | ∞ ∀
k=−∞ k=−∞
X X
such that for any number ε > 0 a weighted function µ(x), 0 < µ(x) 1,
≤
with
(1.6) x [0,2π] : µ(x) = 1 < ε
|{ ∈ 6 }|
can be constructed, so that the series (1.5) is universal in L1[0,2π] with
µ
respect to rearrangements .
Theorem 1.5. There exists a series of the form (1.5) such that for any
number ǫ > 0 a weighted function µ(x) with (1.6) can be constructed,
so that the series (1.5) is universal in L1[0,2π] in the usual sense .
µ
4 S. A.EPISKOPOSIAN
2. BASIC LEMMA
Lemma 2.1. For any given numbers 0 < ε < 1, N > 2 and a step
2 0
function
q
(2.1) f(x) = γ χ (x),
s · ∆s
s=1
X
where ∆ is an interval of the form ∆(i) = i−1, i , 1 i 2m and
s m 2m 2m ≤ ≤
2π (cid:2) −1 (cid:3)
(2.2) γ ∆ < ǫ3 8 f2(x)dx , s = 1,2,...,q.
s s
| |· | | · ·
(cid:18) Z0 (cid:19)
p
there exists a measurable set E [0,2π] and a polynomial P(x) of the
⊂
form
P(x) = C eikx
k
N0≤X|k|<N
which satisfy the conditions:
E > 2π ε,
| | −
P(x) f(x) dx < ε,
| − |
ZE
C 2+ε < ε, C = C
k −k k
| |
N0≤X|k|<N
max C eikx dx < ε+ f x) dx,
N0≤m<NZe(cid:12)(cid:12)(cid:12)N0≤X|k|≤m k (cid:12)(cid:12)(cid:12)  Ze| ( |
for every measurable s(cid:12)ubset e of E. (cid:12) 
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof Let 0 < ǫ < 1 be an arbitrary number.
2
Set
ε π 3ε π
(2.3) g(x) = 1, if x [0,2π] · , · ;
∈ \ 2 2
(cid:20) (cid:21)
2 ε π 3ε π
g(x) = 1 , if x · , · ;
− ε ∈ 2 2
(cid:20) (cid:21)
We choose natural numbers ν and N so large that the following in-
1 1
equalities be satisfied:
1 2π ε
(2.4) g (t)e−iktdt < , k < N ,
1 0
2π 16 √N | |
(cid:12)Z0 (cid:12) · 0
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
UNIVERSAL SERIES BY TRIGONOMETRIC SYSTEM IN WEIGHTED L1 SPACES5
µ
where
(2.5) g (x) = γ g(ν x) χ (x).
1 1 · 1 · · ∆1
(By χ (x) we denote the characteristic function of the set E.) We put
E
(2.6) E = x ∆ : g (x) = γ ,
1 s s s
{ ∈ }
By (2.3), (2.5) and (2.6) we have
(2.7) E > 2π (1 ǫ) ∆ ; g (x) = 0, x / ∆ ,
1 1 1 1
| | · − ·| | ∈
2π 2
(2.8) g2(x)dx < γ 2 ∆ .
1 ǫ ·| 1| ·| 1|
Z0
Since the trigonometric system eikx ∞ is complete in L2[0,2π], we
{ }k=−∞
can choose a natural number N > N so large that
1 0
2π ε
(2.9) C(1)eikx g (x) dx ,
(cid:12) k − 1 (cid:12) ≤ 8
Z0 (cid:12)(cid:12)0≤X|k|<N1 (cid:12)(cid:12)
(cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) (cid:12)
1 2π
(2.10) C(1) = g (t)e−iktdt.
k 2π 1
Z0
Hence by (2.4),(2.5) and (2.9) we obtain
(2.11)
1
2
2π ε ε
C(1)eikx g (x) dx + C(1) 2 < ;
(cid:12) k − 1 (cid:12) ≤ 8  | k |  4
Z0 (cid:12)(cid:12)N0≤X|k|<N1 (cid:12)(cid:12) 0≤X|k|<N0
(cid:12) (cid:12)  
Now as(cid:12)sume that the numbers ν(cid:12) < ν < ...ν , N < N < ... <
1 2 s−1 1 2
(cid:12) (cid:12)
N , functions g (x),g (x),...,g (x) andthesets E ,E ,....,E are
s−1 1 2 s−1 1 2 s−1
defined. We take sufficiently large natural numbers ν > ν and
s s−1
N > N to satisfy
s s−1
1 2π ε
(2.12) g (t)e−iktdt < , 1 s q, k < N ,
s s−1
2π 16 √N ≤ ≤ | |
(cid:12)Z0 (cid:12) · s−1
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
2π ε
(2.13) C(s)eikx g (x) dx ,
(cid:12) k − s (cid:12) ≤ 4s+1
Z0 (cid:12)(cid:12)0≤X|k|<Ns (cid:12)(cid:12)
(cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) (cid:12)
1 2π
(2.14) g (x) = γ g(ν x) χ (x), C(s) = g (t)e−iktdt.
s s · s · · ∆s k 2π s
Z0
6 S. A.EPISKOPOSIAN
Set
(2.15) E = x ∆ : g (x) = γ ,
s s s s
{ ∈ }
Using the above arguments (see (2.16)-(2.18)), we conclude that the
function g (x) and the set E satisfy the conditions:
s s
(2.16) E > 2π (1 ǫ) ∆ ; g (x) = 0, x / ∆ ,
s s s s
| | · − ·| | ∈
2π 2
(2.17) g2(x)dx < γ 2 ∆ .
s ǫ ·| s| ·| s|
Z0
2π ε
(2.18) C(s)eikx g (x) dx < .
(cid:12) k − 1 (cid:12) 2s+1
Z0 (cid:12)(cid:12)Ns−1X≤|k|<Ns (cid:12)(cid:12)
(cid:12) (cid:12)
Thus, byinduction(cid:12)wecandefinenaturalnumb(cid:12)ersν < ν < ...ν , N <
(cid:12) (cid:12) 1 2 q 1
N < ... < N , functions g (x),g (x),...,g (x) and sets E ,E ,....,E
2 q 1 2 q 1 2 q
such that conditions (2.14)- (2.16) are satisfied for all s, 1 s q.
≤ ≤
We define a set E and a polynomial P(x) as follows:
q
(2.19) E = E ,
s
s=1
[
(2.20)
q
P(x) = C eikx = C(s)eikx , C = C ,
k  k  −k k
N0≤X|k|<N Xs=1 Ns−1X≤|k|<Ns
 
where
(s)
(2.21) C = C for N k < N , s = 1,2,...,q, N = N 1.
k k s−1 ≤ | | s q−
By Bessel’s inequality and (2.3), (2.14) for all s [1,q] we get
∈
1
2 1
2π 2
(2.22) C(s) 2 g2(x)dx
 | k |  ≤ s ≤
Ns−1X≤|k|<Ns (cid:20)Zo (cid:21)
 
2
γ ∆ , s = 1,2,...,q.
s s
≤ √ε ·| |· | |
p
From (2.3), (2.12) and (2.13) it follows that
(2.23) E > 2π ε.
| | −
UNIVERSAL SERIES BY TRIGONOMETRIC SYSTEM IN WEIGHTED L1 SPACES7
µ
Taking relations (2.1), (2.3), (2.10), (2.12), (2.18) - (2.21) we obtain
(2.24)
q
P(x) f(x) dx C(s)eikx g (x) dx < ε
| − | ≤  (cid:12) k − s (cid:12) 
ZE Xs=1 ZE(cid:12)(cid:12)Ns−1X≤|k|<Ns (cid:12)(cid:12)
 (cid:12) (cid:12) 
By (2.1), (2.2), (2.20) -(2.21)for(cid:12)any k [N ,N] we have (cid:12)
0
(cid:12) ∈ (cid:12)
N
(2.25) C 2+ǫ max C ǫ C 2
k k k
| | ≤ N0≤k≤N| | · | | ≤
N0≤X|k|<N kX=N0
q
8
max γ ∆ C(s) 2
≤ 1≤s≤q"rǫ ·| s|· | s|#·  | k |  ≤
p Xs=1 Ns−1X≤|k|<Ns
 q 
8 8
max γ ∆ γ 2 ∆
s s s s
≤ 1≤s≤q"rǫ ·| |· | |#· ǫ · | | ·| | ≤
s=1
p X
8 8 1
max γ ∆ f2(x)dx < ǫ;
s s
≤ 1≤s≤q"rǫ ·| |· | |#· ǫ ·(cid:20)Z0 (cid:21)
p
That is, the statements 1) - 3) of Lemma are satisfied. Now we will
check the fulfillment of statement 4) of Lemma. Let N m < N,
0
≤
then for some s , 1 s q, (N m < N ) we will have (see
0 ≤ 0 ≤ s0 ≤ s0+1
(2.20) and (2.21))
(2.26)
s0
C eikx = C(s)eikx + C(s0+1)eikx.
k  k  k
N0≤X|k|≤m Xs=1 Ns−1X≤|k|<Ns Ns0−X1≤|k|≤m
 
Hence and from (2.1), (2.2), (2.3), (2.18), (2.19) and (2.22) for any
measurable set e E we obtain
⊂
C eikx dx
k
(cid:12) (cid:12) ≤
Ze(cid:12)(cid:12)Ns−1X≤|k|≤m (cid:12)(cid:12)
(cid:12) (cid:12)
s0 (cid:12) (cid:12)
(cid:12) (cid:12)
C(s)eikx g (x) dx +
≤  (cid:12) k − s (cid:12) 
Xs=1 Ze(cid:12)(cid:12)Ns−1X≤|k|<Ns (cid:12)(cid:12)
 (cid:12) (cid:12) 
s0 (cid:12) (cid:12)
+ g (x)(cid:12)dx+ C(s0+1(cid:12))eikx dx <
| s | (cid:12) k (cid:12)
Xs=1 Ze Ze(cid:12)(cid:12)Ns0−X1≤|k|≤m (cid:12)(cid:12)
s0 ε (cid:12)(cid:12) 2 (cid:12)(cid:12)
< + f(x) dx(cid:12) + γ ∆(cid:12) <
2s+1 | | √ε ·| s0+1|· | s0+1|
s=1 Ze
X p
8 S. A.EPISKOPOSIAN
< f(x) dx+ε.
| |
Ze
3. PROOF OF THEOREMS
Proof of Theorem 1.4 Let
(3.1) f (x),f (x),...,f (x), x [0,2π]
1 2 n
∈
be a sequence of all step functions, values and constancy interval end-
points of which are rational numbers. Applying Lemma consecutively,
we can find a sequence E ∞ of sets and a sequence of polynomials
{ s}s=1
(3.2) P (x) = C(s)eikx
s k
Ns−1X≤|k|<Ns
1 = N < N < ... < N < ...., s = 1,2,....,
0 1 s
which satisfy the conditions:
(3.3) E > 1 2−2(s+1), E [0,2π],
s s
| | − ⊂
(3.4) P (x) f (x) dx < 2−2(s+1),
s s
| − |
ZEs
2+2−2s
(3.5) C(s) < 2−2s, C(s) = C(s)
k −k k
Ns−1X≤|k|<Ns(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(3.6) max C eikx dx < 2−2(s+1) + f (x) dx,
k s
Ns−1≤p<NsZe(cid:12)(cid:12)(cid:12)Ns−X1≤|k|≤p (cid:12)(cid:12)(cid:12)  Ze| |
 (cid:12) (cid:12) 
for every measurab(cid:12)le subset e of Es(cid:12).
(cid:12) (cid:12)
Denote
∞ ∞
(3.7) C eikx = C(s)eikx ,
k  k 
kX=−∞ Xs=1 Ns−1X≤|k|<Ns
 
(s)
where C = C for N k < N , s = 1,2,....
k k s−1 ≤ | | s
Let ε be an arbitrary positive number. Setting
∞
Ω = E , n = 1,2,....;
n s
s=n
\
UNIVERSAL SERIES BY TRIGONOMETRIC SYSTEM IN WEIGHTED L1 SPACES9
µ
∞
(3.8) E = Ω = E , n = [log ε]+1;
n0 s 0 1/2
s=n0
\
∞ ∞
B = Ω = Ω Ω Ω .
n n0 n \ n−1
!
n=n0 n=n0+1
[ [ [
It is clear (see (3.3)) that B = 2π and E > 2π ε.
| | | | −
We define a function µ(x) in the following way:
(3.9) µ(x) = 1 for x E ([0,2π] B);
∈ ∪ \
µ(x) = µ for x Ω Ω , n n +1,
n n n−1 0
∈ \ ≥
where
n −1
(3.10) µ = 24n h ;
n s
·
" #
s=1
Y
h = f (x) + max C(s)eikx +1,
s || s ||C Ns−1≤p<Nsk k kC
Ns−X1≤|k|≤p
where
g(x) = max g(x) ,
C
|| || x∈[0,2π]| |
g(x) is a continuous function on [0,2π].
From (3.5),(3.7)-(3.10) we obtain
(A) – 0 < µ(x) 1,µ(x) is a measurable function and
≤
x [0,2π] : µ(x) = 1 < ε.
|{ ∈ 6 }|
∞
(B) – C q < , q > 2.
k
| | ∞ ∀
k=1
X
Hence, obviously we have
(3.11) lim C = 0.
k
k→∞
It follows from (3.8)-(3.10) that for all s n and p [N ,N )
0 s−1 s
≥ ∈
(3.12) C(s)eikx µ(x)dx =
(cid:12) k (cid:12)
Z[0,2π]\Ωs(cid:12)(cid:12)Ns−X1≤|k|≤p (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
∞ (cid:12) (cid:12)
= C(s)eikx µ dx
 (cid:12) k (cid:12) n  ≤
nX=s+1 ZΩn\Ωn−1 (cid:12)(cid:12)Ns−X1≤|k|≤p (cid:12)(cid:12)
 (cid:12) (cid:12) 
(cid:12) (cid:12)
(cid:12) (cid:12)
10 S. A.EPISKOPOSIAN
∞ 2π
2−4n C(s)eikx h−1dx < 2−4s.
≤  (cid:12) k (cid:12) s 
nX=s+1 Z0 (cid:12)(cid:12)Ns−X1≤|k|≤p (cid:12)(cid:12)
By (3.4), (3.8)-(3.10)for all(cid:12)s n we have (cid:12) 
(cid:12) 0 (cid:12)
(cid:12) ≥ (cid:12)
2π
(3.13) P (x) f (x) µ(x)dx = P (x) f (x) µ(x)dx+
s s s s
| − | | − |
Z0 ZΩs
+ P (x) f (x) µ(x)dx = 2−2(s+1)+
s s
| − |
Z[0,2π]\Ωs
∞
+ P (x) f (x) µ dx 2−2(s+1)+
s s n
| − | ≤
n=s+1(cid:20)ZΩn\Ωn−1 (cid:21)
X
∞ 2π
+ 2−4s f (x) + C(s)eikx h−1dx <
 | s | (cid:12) k (cid:12) s 
nX=s+1 Z0 (cid:12)(cid:12)Ns−1X≤|k|<Ns (cid:12)(cid:12)
  (cid:12) (cid:12) 
< 2−2(s+1) +(cid:12) 2−4s < 2−2s. (cid:12)
(cid:12) (cid:12)
Taking relations (3.6), (3.8)- (3.10) and (3.12) into account we obtain
that for all p [N ,N ) and s n +1
s−1 s 0
∈ ≥
2π
(3.14) C(s)eikx µ(x)dx =
(cid:12) k (cid:12)
Z0 (cid:12)(cid:12)Ns−X1≤|k|≤p (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
= C(s)eikx µ(x)dx+
(cid:12) k (cid:12)
ZΩs(cid:12)(cid:12)Ns−X1≤|k|≤p (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
+ (cid:12) C(s)ei(cid:12)kx µ(x)dx <
(cid:12) k (cid:12)
Z[0,2π]\Ωs(cid:12)(cid:12)Ns−X1≤|k|≤p (cid:12)(cid:12)
(cid:12) (cid:12)
s (cid:12) (cid:12)
< (cid:12) C(s)eikx(cid:12) dx µ +2−4s <
 (cid:12) k (cid:12) · n
n=Xn0+1 ZΩn\Ωn−1 (cid:12)(cid:12)Ns−X1≤|k|≤p (cid:12)(cid:12)
s  (cid:12) (cid:12) 
(cid:12) (cid:12)
< 2−2(s+1)(cid:12)+ f (x)(cid:12)dx µ +2−4s =
s n
| |
n=n0+1(cid:18) ZΩn\Ωn−1 (cid:19)
X
s
= 2−2(s+1) µ + f (x) µ(x)dx+2−4s <
n s
· | |
n=Xn0+1 ZΩs
2π
< f (x) µ(x)dx+2−4s.
s
| |
Z0
Let f(x) L1[0,2π] , i. e. 2π f(x) µ(x)dx < .
∈ µ 0 | | ∞
R