Ukrainian Mathematical Journal, Vol. 55, No. 1, 2003
CHARACTERIZATION OF THE POINTS OF ϕ-STRONG SUMMABILITY OF FOURIER–LAPLACE SERIES FOR FUNCTIONS OF THE CLASS L p (S m ) , p > 1 R. A. Lasuriya
UDC 517.51
We consider the behavior of the ϕ-strong means of Fourier – Laplace series for functions that belong to L p ( S m ) , p > 1, on a set of points of full measure on an m-dimensional sphere S m.
1. Let S m, m ≥ 3, be the unit sphere centered at the origin in the Euclidean space R m and let Lp( S m ), 1 ≤ p < ∞, L 1 ( S m ) = L ( S m ), be the space of functions f (x ) with the norm
f
m
Lp (S )
=
∫
S
m
f ( x ) p dS( x )
1/ p
.
Further, let ∞
∑
n=0
( n + λ )Γ ( λ ) f ( y) Pn( λ ) (cos γ ) dS( y) ∫ λ +1 2π Sm
(1)
m where Γ ( λ ) is the Euler gamma function, be the Fourier – Laplace series of a function f ( x ) ∈ L ( S ), let
σ(nλ ) ( f ; x ) =
1 Anλ
n
∑ Anλ−−k1 Sk(λ ) ( f ; x )
k =0
be the Cesàro ( C, λ )-means of series (1), let Φ(nλ ) (cos γ )
1 = λ An
n
∑ (k + λ ) Anλ− k Pk(λ ) (cos γ ) ,
λ =
k =0
m−2 , 2
m let cos γ = ( x, y ) be the inner product of vectors x = ( x 1 , … , x m ) and y = (y 1 , … , y m ) , x, y ∈ S , let
Sn( λ ) ( f ; x ) be a partial sum of series (1), and let Pn( λ ) (t ) be the Gegenbauer polynomials (ultraspherical polynomials). A point x ∈ S m is called a Dp-point of a function f ( x ) ∈ L p ( S m ) , p ≥ 1 (see, e.g., [1, p. 262]) if h
∫
p
∫
[ f ( y) − f ( x ) ] dt ( y) d γ = o(h2 λp +1 ) ,
h → 0.
(2)
0 ( x , y ) = cos γ
Abkhazia University, Sukhumi. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 55, No. 1, pp. 45–54, January, 2003. Original article submitted June 21, 2001. 0041–5995/03/5501–0055 $25.00
© 2003
Plenum Publishing Corporation
55
56
R. A. LASURIYA
A point x ∈ S m is called a D*p -point of a function f if both x and x* are Dp-points of the function f ; here, x* is the point opposite to x. If a point x is a Dp-point for a function f, then it is also a Dp′ -point for any p ′ < m
p. It is well known (see, e.g., [1, p. 263]) that if f (x ) ∈ L p( S ), p > 1, then almost every point x ∈ S m is a D*p -point of this function. m
It was established in [2] that if f ( x ) ∈ L p ( S ) , p > 1, then, at every D*p -point of the function f, i.e., almost everywhere on S m, one has 1 n ∑ σ(kλ ) ( f ; x ) − f ( x ) n→ ∞n k =1 lim
q
= 0
∀ q > 0.
(3)
Following Totik [3], we introduce the following types of strong means: 1/ q
q 1 r hn( q, r), λ ( f ; x ) = ∑ ρ(kλj ) ( f ; x ) r j =1
,
(4)
ρ(kλj ) = σ(kλj ) ( f ; x ) − f ( x ) , n ≤ k 1 < k 2 < … < k r ≤ 2 n – 1, Hnϕ, λ ( f ; α; x ) =
n ∈ N,
∞
∑ α k (u) ϕ( ρ(kλ ) ( f ; x ) ) ,
(5)
k =n
where ϕ ( u ) is a certain function nonnegative on [ 0, + ∞ ), and α = (α k (u)) , k ∈ N, is a certain sequence of nonnegative functions defined on a set U containing at least one limit point. In the present paper, we consider the behavior of quantities (4) and (5) at the D*p -points [see (2)] of a funcm
tion f ( x ) ∈ L p ( S ), p > 1; as a corollary, we establish equality (3). 2. For what follows, we need two auxiliary statements, one of which is given here without proof. m
m
Lemma 1. Let f (x ) ∈ L p( S ) and m ≥ 3. Then, at every D1* -point x ∈ S , i.e., almost everywhere m
on S , the following relation is true: ρ(kλ ) ( f ; x ) = σ(kλ ) ( f ; x ) − f ( x ) = o(ln k ) ,
k → ∞.
(6)
m
Lemma 2. Let f (x ) ∈ L p( S ), 1 < p ≤ 2, n ∈ N, j = 1, … , r, and n ≤ k 1 < k 2 < … < k r ≤ 2 n – 1. Then, at every D*p -point x ∈ S m, i.e., almost everywhere on S m , the following relation holds uniformly in r ∈ N: ∀q > 0
lim
n→∞
hn( q, r), λ ( f ; x ) ln (ne / r )
= 0.
(7)
CHARACTERIZATION OF THE POINTS OF ϕ-S TRONG SUMMABILITY OF FOURIER–LAPLACE S ERIES
57
Proof. We set
∫
gx ( γ ) =
[ f ( y) − f ( x ) ] dt ( y) .
( x , y ) = cos γ
By using the Hölder inequality, one can show that hn( q, r), λ ( f ; x ) ≤ hn( q, r′ ), λ ( f ; x ) ,
0 < q ≤ q ′,
and, therefore, it is sufficient to prove (7) for an arbitrarily large power q ′ ≥ 2. We choose q ′ so large that the corresponding conjugate power p ′ does not exceed p. Then the condition f ∈ Lp ′ ( S m ) , 1 / p ′ + 1 / q ′ = 1, is a consequence of the assumption that f ∈ L p ( S m ). In this case, a D*p -point x ∈ S function f. Furthermore, we have 1 r hn( q, r′ ), λ ( f ; x ) = ∑ ρ(kλj ) ( f ; x ) r j =1
q′
1/ q′
q′ 1/ q′
0
r 1 + C( λ ) ∑ r j =1
q′ 1/ q′
gx ( γ ) Φ(kλj ) (cos γ ) d γ
r 1 + C( λ ) ∑ r j =1
q′ 1/ q′
π / ( 2( k j +1))
∫
q′ 1/ q′
q′ 1/ q′
( λ ) ( ) ( ) ( ) − f y f x dt y [ ] Φ k j (cos γ ) d γ ∫ ( x , y ) = cos γ
π r Γ (λ ) 1 = ∑ g ( γ ) Φ(kλj ) (cos γ ) d γ λ +1 ∫ x r j =1 2 π 0
r 1 ≤ C( λ ) ∑ r j =1
is a also a D*p′ -point of the
1 r Γ (λ ) = ∑ f ( y) Φ(kλj ) (cos γ ) dS( y) − f ( x ) λ +1 ∫ r j =1 2 π Sm
1 r Γ (λ ) = ∑ [ f ( y) − f ( x ) ] Φ(kλj ) (cos γ ) dS( y) ∫ λ + 1 r 2π Sm j =1 π r Γ (λ ) 1 = ∑ λ +1 ∫ r j =1 2 π 0
m
q′ 1/ q′
π − π / ( 2( k j +1))
∫
π / ( 2( k j +1))
π
∫
gx ( γ ) Φ(kλj ) (cos γ ) d γ
q′ 1/ q′
gx ( γ ) Φ(kλj ) (cos γ ) d γ
π − π / ( 2( k j +1))
= In(1, )r ( f ; x ) + In( 2, r) ( f ; x ) + In(3, )r ( f ; x ) ,
n ∈ N.
(8)
58
R. A. LASURIYA
By virtue of the well-known inequality [4, p. 133]
Φ(kλ ) (cos γ ) < Ck 2 λ +1 , k ∈ N, 0 ≤ γ ≤ π, at a D*p′ -
point x we have π / ( 2( k j +1)) r 1 2 λ +1 (1) In , r ( f ; x ) ≤ C ( λ ) ∑ k j gx ( γ ) d γ ∫ r j =1 0
q′ 1/ q′
1/ q′
q′ 1 r 1 ≤ C(λ ) ∑ k (j 2 λ +1)q ′ o 2 λ +1 kj r j =1
= o(1) ,
n → ∞.
(9)
Furthermore, r 1 In(3, )r ( f ; x ) ≤ C(λ) ∑ r j =1
π
∫
π − π / ( 2( k j
r 1 + C( λ ) ∑ r j =1
[
]
(λ ) ∫ f ( y) − f ( x* ) dt( y)Φ k j (cos γ )dγ +1)) ( x, y) = cos γ
π
∫
π − π / ( 2( k j
1 q′ q′ /
f ( x * ) − f ( x ) dt ( y) Φ(kλj ) (cos γ ) dγ ∫ +1)) ( x , y ) = cos γ
[
]
q′ 1/ q′
.
Taking into account that Φ(kλj ) (cos γ ) < Ck λ , π / 2 ≤ γ ≤ π, we get π 1 r gx * ( γ ) d γ In(3, )r ( f ; x ) ≤ C(λ ) ∑ k λj ∫ r j =1 π − π / ( 2( k j +1))
f ( x * ) − f ( x ) q ′ + C(λ) r 1 r ≤ C( λ ) ∑ r j =1
q′ 1/ q′
π 2λ sin γ d γ ∑ k λj ∫ j =1 π − π / (2( k j +1)) r
q ′ 1 / q ′
q′ 1/ q′
λ 1 k j o 2 λ +1 kj
+ o(1)
′
′ q′ 1/ q 1 r 1 C(λ) 1 r 1 / q ≤ C(λ) ∑ o λ +1 + o(1) = o(1) . + o(1) ≤ λ +1 ∑ o(1) n r j =1 r j =1 k j
By virtue of the representation [4]
(10)
CHARACTERIZATION OF THE POINTS OF ϕ-S TRONG SUMMABILITY OF FOURIER–LAPLACE S ERIES
Φ(kλ ) (cos γ ) =
59
λ sin [ (k + (3λ + 1) / 2)γ − λπ ] M 1 + λ +1 λ λ +1 λ λ + 1 k + 1 4 (sin γ ) (sin ( γ / 2)) (sin γ ) ( sin ( γ / 2 ))
= Φ(kλ,1) (cos γ ) + Φ(kλ,2) (cos γ ) , π kπ −γ < , 2 2( k + 1) where M is a value uniformly bounded with respect to k and γ, we get π − π / (2( k j +1))
r 1 In( 2, r) ( f ; x ) ≤ C(λ) ∑ r j =1
∫
π / (2( k j +1))
r 1 + C(λ) ∑ r j =1
gx (γ )Φ(kλj,)1 (cos γ ) d γ
′ q ′ 1 / q
q′
π − π / (2( k j +1))
∫
π / (2( k j +1))
gx (γ )Φ(kλj,)2 (cos γ ) d γ
1/ q′
= In(,2r) ( f ; x ) + In(,2r) ( f ; x ) .(11) The Minkowski inequality yields r 1 In(,2r) ( f ; x ) ≤ C(λ) ∑ r j =1
π − π / (2( k j +1))
∫
π / 2(2( k j +1))
1 r + C( λ ) ∑ r j =1
gx ( γ )
sin k j γ cos((3λ + 1)γ / 2 − λπ) (sin γ )λ (sin (γ / 2))λ +1
π − π / ( 2( k j +1))
∫
gx ( γ )
π / ( 2( k j +1))
′ q′ 1/ q
dγ
cos k j γ sin((3λ + 1)γ / 2 − λπ) (sin γ )λ (sin ( γ / 2))λ +1
q′ 1/ q′
dγ
= A n, r ( f; x ) + B n, r ( f; x ).
(12)
Let us represent A n, r ( f; x ) as a sum of three terms and use the Minkowski inequality. As a result, we get 1 r A n, r ( f; x ) ≤ C(λ ) ∑ r j =1
π −1 / ( 2 n )
∫
gx ( γ )
sin k j γ cos((3λ + 1)γ / 2 − λπ)
1 / (2n)
1 r + C( λ ) ∑ r j =1
λ
(sin γ ) (sin ( γ / 2))
π / ( 2( k j +1))
∫
1 / (2n)
gx ( γ )
λ +1
q′ 1/ q′
dγ
sin k j γ cos((3λ + 1)γ / 2 − λπ) λ
(sin γ ) (sin ( γ / 2))
λ +1
q′ 1/ q′
dγ
60
R. A. LASURIYA π −1 / ( 2 n )
1 r + C( λ ) ∑ r j =1
∫
gx ( γ ) π − π / ( 2( k j +1))
sin k j γ cos((3λ + 1)γ / 2 − λπ)
= An(1, )r ( f ; x ) + An( 2, r) ( f ; x ) + An(3, )r ( f ; x ) ,
λ
(sin γ ) (sin ( γ / 2))
λ +1
q′ 1/ q′
dγ
n ∈ N.
(13)
Taking into account the inequality r ≤ n, we obtain
An(1, )r (
1/ (2r )
sin k j γ cos((3λ + 1)γ / 2 − λπ) 1 r f ; x ) ≤ C( λ ) ∑ ∫ gx ( γ ) dγ (sin γ )λ (sin ( γ / 2))λ +1 r j =1 1 / ( 2 n ) 1 r + C( λ ) ∑ r j =1
π −1 / ( 2 n )
∫
gx ( γ )
q′ 1/ q′
sin k j γ cos((3λ + 1)γ / 2 − λπ) λ
(sin γ ) (sin ( γ / 2))
1 / (2r )
λ +1
q′ 1/ q′
dγ
= An(1, r) ( f ; x ) + An(,1r) ( f ; x ) .(14) Integrating by parts, we get 1 r 1 / ( 2 r ) dγ (1) An, r ( f ; x ) ≤ C(λ ) ∑ ∫ gx ( γ ) 2 λ +1 r γ j =1 1 / ( 2 n )
q′ 1/ q′
1/ (2r ) 1/ (2r ) dγ = C(λ ) (2r )2 λ +1 ∫ gx ( τ) d τ + (2λ + 1) ∫ o( γ 2 λ +1 ) 2 λ + 2 γ 1 / (2n) 1 / (2n)
{
1 n n = C(λ ) r 2 λ +1o 2 λ +1 + o(1)ln ≤ C(λ ) o(1) + o(1)ln r r r
} = o(1)ln ner .
Introducing the auxiliary function cos(( 3λ + 1) γ / 2 − λπ ) , g x ( γ ) λ +1 (sin γ ) λ (sin ( γ / 2)) Gx ( γ ) = 0,
γ ∈[ 1 / (2r ), π − 1 / (2r ) ], γ ∈[ − π, 1 / (2r ) ) U ( π − 1 / (2n ), π ],
Gx ( γ + 2 π) = Gx ( γ ) ∈ Lp ′ and using the Hausdorff–Young theorem [5, p. 153], we obtain
( gx ( γ ) ∈ L p ′ ) ,
(15)
CHARACTERIZATION OF THE POINTS OF ϕ-S TRONG SUMMABILITY OF FOURIER–LAPLACE S ERIES
π −1 / ( 2 n ) gx (γ ) (1) −1 / q ′ An , r ( f ; x ) ≤ C ( λ )r ∫ λ +1 sin λ γ 1 / ( 2 r ) γ
p′
61
1/ p′
dγ
,
1 / p ′ + 1 / q ′ = 1.
(16)
It follows from the definition of a D*p -point x that, for any ε > 0, there exists h 0 ( ε ) > 0 such that h
Φ p ′ (h) =
∫
gx ( γ )
p′
dγ < εh 2 λp ′+1 ,
0 < h ≤ h0 <
0
π . 2
(17)
First, assume that 1 / ( 2 r ) < h 0 . Then, taking (17) into account and integrating by parts, we get
An(,1r) (
f ; x ) ≤ C ( λ )r
≤ C ( λ )r
−1 / q ′
−1 / q ′
h0 gx (γ ) ∫ λ +1 (sin γ )λ 1 / ( 2 r ) γ
+
∫
∫
dγ +
h0
( λ +1) p ′ 1 p ′−1 C ( q ) ε ( 2 r ) + ′ h0
≤ C ( λ, q ′ )r −1 / q ′ εr p ′−1 +
π −1 / ( 2 n )
π −1 / ( 2 n )
p′
π −1 / ( 2 n )
∫
π −1 / ( 2 n )
∫
h0
∫( x , y ) = cos γ [
{
dγ 1/ p′
]
f ( y ) − f ( x * ) dt ( y )
p′
(sin γ )λp ′
≤ C(λ, q ′)r − 1 / q ′ εr p ′−1 + K
1/ p′
p′
d γ (sin γ )λ p ′ gx (γ )
(sin γ )λ p ′
h0
∫( x , y ) = cos γ [ f ( x ) − f ( x * ) ] dt ( y )
h0
gx (γ ) λ +1 γ (sin γ )λ
p′
dγ
1 / p ′ dγ
}/
ne ≤ C ( λ, q ′ ) ε + o ln , r
1 p′
n → ∞, 1 / p ′ + 1 / q ′ = 1.
(18)
Thus, by virtue of (15) and (18), relation (14) yields ne An(1,)r ( f ; x ) = o ln , r Let us estimate An( 2, r) ( f ; x ) . Integrating by parts, we get
n → ∞.
(19)
62
R. A. LASURIYA
1 r π / ( 2( k j +1)) gx ( γ ) (2) An, r ( f ; x ) ≤ C(λ ) ∑ d γ ∫ 2 λ +1−1 / ( 2 q ′ ) 1 / ( 2 q ′ ) γ r j =1 1 / ( 2 n ) γ n1 / 2 ≤ C( λ ) r n1 / 2 ≤ C( λ ) r
1 ∑ k 2j λ +1−1/ (2q ′)o k 2λ +1 + o j j =1 r
r
∑ o(
)
q′ k j−1 / ( 2 q ′ )
j =1
1/ q′
q′ 1/ q′
π / ( 2( k j +1))
∫
1 / (2n)
1−1 / ( 2 q ′ ) γ
n1 / 2 ≤ C( λ ) n −1 / 2 r
dγ
r
j =1
∑ o(1)
q′ 1/ q′
= o ( 1 ).
(20)
Integrating by parts again, we obtain π −1 / ( 2 n ) gx ( γ ) 1 r An(3, )r ( f ; x ) ≤ C(λ ) ∑ dγ ∫ λ r j =1 π − π 2( k +1) (sin γ ) /( j )
1 r π −1 / ( 2 n ) ≤ C( λ ) ∑ ∫ r j =1 π − π / ( 2( k j +1)) π −1 / ( 2 n )
∫
+
q′ 1/ q′
∫( x, y) = cos γ [ f ( y) − f ( x* ) ] dt( y) (sin γ )λ
∫( x, y) = cos γ [ f ( x* ) − f ( x ) ] dt( y) (sin γ )λ
π − π / ( 2( k j +1))
dγ
dγ
q′ 1/ q′
π −1 / ( 2 n ) 1 r π / ( 2( k j +1)) g ( γ ) x d γ + (sin γ )λ d γ ≤ C( λ ) ∑ ∫ ∫ λ r j =1 1 / ( 2 n ) (sin γ ) π − π / ( 2( k j +1))
q′ 1/ q′
1 r Φ* ( γ ) π / ( 2( k j +1)) π / ( 2( k j +1)) Φ* ( γ ) + ≤ C( λ ) ∑ λ ∫ γ λ +1 d γ r j =1 γ 1 / (2n) 1 / (2n) π / ( 2( k j +1))
+
∫
1 / (2n)
γ λ dγ
q′ 1/ q′
≤
C( λ ) = o(1) , n +1
ne , r
n → ∞.
n → ∞.
(21)
Comparing (19), (20), and (21) with (13), we obtain A n, r ( f ; x ) = o(1)ln
(22)
CHARACTERIZATION OF THE POINTS OF ϕ-S TRONG SUMMABILITY OF FOURIER–LAPLACE S ERIES
63
By analogy, we can estimate B n, r ( f ; x ) : B n, r ( f ; x ) = o(1)ln
ne , r
n → ∞.
(23)
ne , r
n → ∞.
(24)
By virtue of (22) and (23), it follows from (12) that In(,2r) ( f ; x ) = o(1)ln
Let us estimate In(,2r) ( f ; x ) . Integrating by parts and taking into account the definition of the kernels Φ(kλ,i) (cos γ ) , i = 1, 2, , we get In(,2r) (
π /2 gx ( γ ) 1 r 1 dγ f ; x ) ≤ C( λ ) ∑ q ′ ∫ 1 + λ r j =1 k j π 2( k +1) (sin γ ) (sin ( γ / 2))λ +1 /( ) j
π − π / ( 2( k j +1))
∫
+
π /2
gx ( γ ) dγ λ +1 λ +1 (sin γ ) (sin ( γ / 2))
q′ 1/ q′
π /2 gx ( γ ) 1 r 1 dγ ≤ C( λ ) ∑ q ′ ∫ r j =1 k j π 2( k +1) γ 2 λ + 2 /( j ) π /2
∫
+
π / ( 2( k j +1))
gx * ( γ ) γ λ +1
dγ +
π /2
∫
π / ( 2( k j +1))
(sin γ )2 λ d γ (sin γ )λ +1
q′ 1/ q′
r r q′ q ′ 1 / q ′ 1 1 1 / 1 1 = o(1) , ≤ C(λ ) ∑ q ′ (o( k j ) + K ) ≤ C(λ ) ∑ q ′ o k qj ′ r j =1 k j r j =1 k j
( )
n → ∞.
(25)
By virtue of (24) and (25), relation (11) yields In( 2, r) ( f ; x ) = o(1)ln
ne , r
n → ∞.
(26)
Taking (8), (90, (10), and (26) into account, we conclude that hn( q, r′ ),λ ( f ; x ) = o(1)ln at every D*p -point x ∈ S that 1 / ( 2 r ) < h 0 .
m
m
ne , r
n → ∞,
(27)
of the function f ∈ L p ( S ), and relation (27) is satisfied uniformly in r, provided
64
R. A. LASURIYA
If 1 / ( 2 r ) ≥ h 0 , then, by virtue of (6), we have ε ln k ln (2 h0 )
ρ(kλ ) ( f ; x ) <
for all k greater than a certain n 0 . Setting n ≥ n 0 ( k j ≥ n 0 ∀ j = 1, … , r ), we get ρ(kλj ) ( f ; x ) <
ε ε ln n + ln 2r = o(1)ln ne , ln k j < ln (2 h0 ) r ln (2 h0 ) r
j = 1, … , r.
m
Therefore, at every D*p -point x ∈ S , we have ne q ′ 1 / 1 hn( q, r′ ),λ ( f ; x ) ≤ ∑ o(1)ln r r j =1 r
q′
= o(1)ln
ne , r
n → ∞, 1 / p ′ + 1 / q ′ = 1. Lemma 2 is proved. 3. Following the definition from [6], we denote by Φ the set of functions ϕ ( ⋅ ) continuous on [ 0, + ∞ ) and such that ϕ ( 0 ) = 0, ϕ ( u ) > 0 ∀ u > 0, ϕ ( u ) ≤ e
Bu
∀ u ∈ [ 0, + ∞ ) , and ϕ ( 2 u ) ≤ A ϕ (u ) ∀ u ∈ [ 0, 1 ], q
u
where A = A ( ϕ ). This set contains, e.g., the functions ϕ (u ) = u , q > 0, ϕ ( u ) = e – 1, etc. Further, let 1 r ρ(kλj ) ( f ; x ) ∑ j = 1 . (lλ ) ( x ) = (lλ ) ( f ; x ) = sup r ln (ne / r ) n≥l
(28)
1≤ r ≤ n
Then the following statement is true: m
m
Theorem 1. Let f ( x ) ∈ L p( S ), m ≥ 3, 1 < p ≤ 2, and ϕ ( ⋅ ) ∈ Φ. Then, at every D*p -point x ∈ S , m
i.e., almost everywhere on S , the following assertions are true: (i) Vnϕ, λ ( f ; x ) =
(ii)
1 2 n −1 ∑ ϕ ρ(kλ ) ( f ; x ) n k =n
(
)≤
(
)
Cϕ (nλ ) ( x ) , C > 0;
lim Vnϕ, λ ( f ; x ) = 0.
n →∞
Proof. By virtue of Lemma 2, for q = 1 we have
(29)
(30)
CHARACTERIZATION OF THE POINTS OF ϕ-S TRONG SUMMABILITY OF FOURIER–LAPLACE S ERIES
65
lim (mλ ) ( x ) = 0
(31)
m →∞ m
at every D*p -point x ∈ S . We proceed further by analogy with the corresponding result in [6]. Let (nλ ) ( x ) = 0. Then ρ(kλ ) ( f ; x ) = 0 ∀ k ≥ n and, hence, taking into account the condition ϕ ( 0 ) = 0, we obtain the assertions of the theorem. Now let (nλ ) ( x ) > 0, σ ∈ N, Bn, σ ( x ) =
{ k ∈[ n, 2n − 1]: (σ − 1)(nλ ) ( x ) ≤ ρ(kλ ) ( f ; x ) ≤ σ (nλ ) ( x ) },
and let µ n, σ ( x ) be the total number of elements of the set Bn, σ ( x ) . Then Vnϕ, λ ( f ; x ) =
1 ∞ ∑ n σ =1
∑
(
ϕ ρ(kλ ) ( f ; x )
k ∈Bn , σ ( x )
)
it is assumed here that if Bn, σ ( x ) = { ∅ } for certain σ, then
≤
1 ∞ ϕ σ (nλ ) ( x ) µ n, σ ( x ); ∑ n σ =1
(
∑ k ∈B
n, σ ( x )
)
(32)
= 0.
Let µ n, σ ( x ) ≥ 1. By virtue of the definition (28) of (mλ ) ( x ) , we have 1 r ne ρ(kλj ) ( f ; x ) ≤ (nλ ) ( x )ln . ∑ r j =1 r
(33)
Setting r = µ n, σ ( x ), taking into account the definition of Bn, σ ( x ) , and using (39), we get (σ − 1)(nλ ) ( x ) ≤
1 ne ∑ ρ(λ) ( f ; x) ≤ (nλ) ( x)ln µ ( x) , µ n, σ ( x ) k ∈Bn, σ ( x ) k n, σ
whence µ n, σ ( x ) ≤ ne 2e −σ .
(34)
Using the inequality ∞
∑ ϕ(uσ)e− σ
≤ Cϕ(u)
σ =1
1 ∀ u ∈ 0, 2 AB
established in [6] and relations (34) and (32), we obtain Vnϕ, λ ( f ; x ) ≤ C
∞
∑ ϕ(σ n ( x)) e− σ
σ =1
≤ Cϕ( n ( x )) ,
(35)
provided that (nλ ) ( x ) < 1 / ( 2 A ) ( B = 1 ) and ϕ ( ⋅ ) ∈ Φ ( B = 1 ). However, by virtue of (31), there exists n 0 ∈ N such that (nλ ) ( x ) < 1 / (2 A ) ∀ n > n 0 , i.e., condition (35) is satisfied.
66
R. A. LASURIYA
For n ≤ n 0, inequality (29) is satisfied by the proper choice of the constant. Passing to the limit and taking into account the conditions of the theorem, we arrive at relation (30). Theorem 1 is proved. Using Theorem 1, we establish the following statement: m
Theorem 2. Let f ( x ) ∈ L p( S ), m ≥ 3, 1 < p ≤ 2, ϕ ( ⋅ ) ∈ Φ, and let a sequence α = ( α k ( u ) ≥ 0 ) , m
u ∈ U, be nonincreasing with respect to k ∈ N for every fixed u. Then, at every D*p -point x ∈ S , the following relation is true: ∞
) ∑ α k (u)ϕ((kλ ) ( x )) ,
Hnϕ, λ ( f ; α; x ) ≤ K nα n (u)ϕ (nλ ) ( x ) +
(
(36)
k =n
where K is a positive constant independent of k ∈ N, u, and x. Proof. We represent Hnϕ, λ ( f ; α; x ) in the form Hnϕ, λ (
f ; α; x ) =
∞ 2 ν + 1 n −1
∑ ∑
ν= 0 k = 2 n ν
(
)
α k (u)ϕ ρ(kλ ) ( f ; x ) .
Using relation (29) and taking into account the conditions of the theorem, we obtain ∞ Hnϕ, λ ( f ; α; x ) ≤ K ∑ α 2 ν n (u)2 ν nϕ ( λν) ( x ) 2 n ν= 0
(
)
∞ ≤ K α n (u)nϕ (nλ ) ( x ) + 2 ∑ α 2 ν n (u)2 ν −1 nϕ ( λν) ( x ) 2 n ν =1
(
(
)
)
∞ ≤ K α n (u)nϕ (nλ ) ( x ) + ∑ α k (u)ϕ (kλ ) ( x ) . k =n
(
)
(
)
Theorem 2 yields the following statement: m
Corollary 1. Let f (x ) ∈ L p( S ), m ≥ 3, 1 < p ≤ 2, and ϕ ( ⋅ ) ∈ Φ . Also let α = { α (kn ) : α (kn ) ≥ 0; k, n ∈ N } be an infinite rectangular matrix of numbers such that, for any fixed n ∈ N, the numbers α (kn ) m
do not increase with respect to k. Then, at every D*p -point x ∈ S , the following relation is true: ∞
∑ α(kn) ϕ( ρ(kλ ) ( f ; x ) )
k =1
∞ ≤ K ∑ α (kn ) ϕ (kλ ) ( x ) . k =1
(
)
Inequality (37) follows immediately from (36) for n = 1 and α k (u) = α (kn ) .
(37)
CHARACTERIZATION OF THE POINTS OF ϕ-S TRONG SUMMABILITY OF FOURIER–LAPLACE S ERIES
67
REFERENCES 1. S. B. Topuriya, Fourier–Laplace Series on a Sphere [in Russian], Tbilisi University, Tbilisi (1987). 2. V. V. Khocholava, “On strong summability of Fourier – Laplace series for functions of the class Lp(S k ), p > 1,” Soobshch. Akad. Nauk Gruz. SSR, 97, No. 3, 573–576 (1980). 3. V. Totik, “On the strong approximation by the ( c, d )-means of Fourier series. I, II,” Anal. Math., 6, Nos. 1, 2, 57–85, 165–184 (1980). 4. E. Kogbetliantz, “Recherches sur la summabilité des series ultrasphériques par la méthode des moyennes arithmetiques,” J. Math. Pures Appl., 9, No. 3, 107–187 (1924). 5. A. Zygmund, Trigonometric Series [Russian translation], Vol. II, Mir, Moscow (1965). 6. V. Totik, “On the strong approximation of Fourier series,” Acta Math. Acad. Sci. Hung., 35, 157–172 (1980).
