CC BY
31Probl. Anal. Issues Anal. Vol. 10 (28), No 2, 2021, pp. 79-98 79
DOI: 10.15393/j3.art.2021.9635
UDC 517.5, 519.651
a. testici, d. m. israfilov
APPROXIMATION BY MATRIX TRANSFORMS IN MORREY SPACES
Abstract. In this work, approximation properties of matrix transforms constructed via the Fourier and Faber series in the subspaces of Morrey spaces are investigated.
Key words: Morrey space, Morrey-Smirnov class, matrix transform, Faber series, approximation, Lipschitz class 2010 Mathematical Subject Classification: 30E10, 41A25, 46E30, 42A10
1. Introduction and main results. Let r C C be a rectifiable Jordan curve. For a given 0 < A ^ 2 and 1 ^ p < rc, the Morrey space Lp'x (r) is defined as the set of all functions f E L/oc (r), such that
1 r
LP,Hr) ,-^sup -——— J |f (z)|P \dz^ < rc, 1 1 Bnr
where | B n r| denotes the Lebesgue measure of B n r and the supremum is taken over all disks B C C centered on r. Let T := {w : |w| = 1} or T := [0, 2n]. In the case of r = T, the Morrey space Lp'x (T) can be defined as the set of all functions f E L/oc (0, 2n) for which
If W) = If llL^(0,2n) := {^p—Ir/ If (eid )\P |d»| } ^ < rc,
where the supremum is taken over all subintervals I C (0, 2n). Lp'x (r), 0 < A ^ 2 and 1 ^ p < rc, becomes a Banach space equipped with the norm ||"|lp'a(f). If we choose A — 2, then Lp'2 (r) coincides with the Lebesgue space Lp (r); also, if we choose A = 0, then Lp'0 (r) coincides with (r). Moreover, LP'Xl (r) C LP'Aa (r) as soon as 0 ^ A i ^ A2 ^ 2.
( Petrozavodsk State University, 2021
Let G C C be a bounded Jordan domain with rectifiable boundary r and let G- := Extr. Denoting by Ep (G) the classical Smirnov class of analytic functions in G, we define the Morrey-Smirnov class Ep'x (G) as
Ep'x (G) := {/ G E1 (G) : f G Lp'x (r)} .
Then Ep'x (G), 0 < A ^ 2 and 1 ^ p < x>, becomes a Banach space equipped with the norm ||f ||ep,a(g) := ||f \\LP,\(r). If we choose A = 2, then Ep'2 (G) coincides with the classical Smirnov class Ep (G). Moreover, it can be easily seen that Ep'Xl (G) C Ep'Xa (G) iff 0 ^ A1 ^ A2 ^ 2.
If G := D := {w : |w| < 1}, then we obtain the Morrey-Hardy space Hp'x (D) : = Ep'x (D), defined on D.
Morrey spaces were introduced by Morrey in [24] and have important applications in differential equations. They are commonly used for study of local behavior of solutions of the elliptic differential equations, especially. Many authors have considered the fundamental problems of potential theory, maximal and singular operator theory in these spaces (see for instance: [1-3], [25], [8], [22]). Also, problems of approximation theory in Morrey spaces have been studied; in particular, in the papers [13], [14], [18-20], [5] the direct and inverse theorems of approximation theory in the Morrey spaces Lp'x (T) and also in the Morrey-Smirnov classes Ep'x (G) were obtained.
In this work, we study approximation properties of matrix transforms constructed via the Fourier and Faber series, in the subclasses of Morrey spaces and Morrey-Smirnov classes of analytic functions, respectively. Let us give some definitions needed to formulate the main results obtained in this work.
Let f G L1 (T) and let f (x) ~ a0/2 + (ak cos kx + bk sin kx) be its
k= i
Fourier series representation with the Fourier coefficients
n n
ak := — f (t) cos (kt) dt and bk := — f (t) sin (kt) dt.
Let also Sn (f) (x) = uk (f) (x), n = 0,1, 2,... be the n-th partial k=0
sums of the Fourier series of f, where u0 (f) (x) := a0 and uk (f) (x) := (ak cos kx + bk sin kx), k =1, 2,...
Let A = (an,fc) be an infinite lower-triangular regular matrix with non-
negative elements and let s^ = n=0 an,k be its n-th row sum for n = 0,1, 2,... We say that the matrix A = (an,k) has almost monotone increasing (decreasing) rows, if there is a constant K1 (K2), depending only on A, such that an,k ^ K1an>TO (an,m ^ K2an,k), where 0 ^ k ^ m ^ n. The matrix transform of Fourier series of f G Lp'x (T) with respect to A = (an,k) is defined as
n
(f) (x) = £ an,kSk (f) (x). k=0
If an,k := pn-k/Pn, for a given sequence (pn) of positive numbers, where Pn = Pk, then the matrix transform Tt.A) (f) coincides with
the Norlund mean
1n
Nn (f) (x) = Pn-kSk (f) (x),
P n , „
k=0
which reduces to the Cesaro means
n + 1
^ (f ) (x) = n-+T £ Sk (f ) (x)
fc=Q
in the case pn = 1 for all n = 0,1, 2,... Let us define the modulus of smoothness Q (f, -)pA : [0, œ) ^ [0, œ) defined as
fi(/,i)p,A := sup ||f (■ + t) - f (•)|£p,,(T) , 5> 0.
We use the relation f = O (g), which means that f ^ cg for a positive constant c, independent of f and g.
Definition 1. Let f G Lp,A (T), 0 <A ^ 2, 1 < p < œ and 0 < a ^ 1. We say that f G LipP,A(T, a) if Q (f, 5)pA = O (5a) for 0 ^ 5.
Firstly, we study the approximation properties of the matrix transforms T,1A) (f ) in the subspaces Lipp,A(T,a), 0 < a ^ 1, and then extend the obtained results to the subclasses of Ep,A (G). Our main results are following:
Theorem 1. Let 0 < A ^ 2 and 1 < p < œ. Let f G Lipp,A (T,a), 0 < a < 1, and let A = (an,k) be a lower-triangular matrix with |siA) — 1| = O (n-a). If one of the conditions:
(i) A has almost monotone decreasing rows and (n + 1)an>0 = O(1),
(ii) A has almost monotone increasing rows and (n + 1)an>k = O(1), where k is the integer part of n/2,
holds, then
||f - T(A) (f)|U(T) = O (n-a) .
Let (pn) be a sequence of positive numbers. If (pn) is almost monotone decreasing, then the matrix A = (an>k) with an>k := pn-k/Pn has almost monotone increasing rows and
/ ^ ^(n + 1) Pfc ^ K (k + 1) Pfc „m (n + 1) a„;fc ^ K---= Ki---= O (1),
Pn Pk
where k = [n/2]. Thus A satisfies the condition ( ii) of Theorem 1. If (pn) is almost monotone increasing and (n + 1)pn = O (Pn), then A has almost monotone decreasing rows and
(n + 1K,0 ^ (n + 1)-n = -P-O (Pn) = O(1).
-n -n
Therefore, A satisfies the condition (i) of Theorem 1 and, hence, we have
Corollary 1. Let 0 < A ^ 2 and 1 < p < rc. Let also f E LipP,A (T, a) for 0 < a < 1 and let (pn) be a sequence of positive numbers. If one of the conditions:
(i) (pn) is almost monotone increasing and (n + 1)pn = O (Pn) ,
(ii) (pn) is almost monotone decreasing
holds, then If - Nn (f )hlp,a(t) = O (n-a).
Theorem 2. Let 0 < A ^ 2 and 1 < p < rc. Let f E Lip^ (T,1) and let A = (an>k) be a lower-triangular matrix satisfying the relation \snA) - 1\ = O (n-i). I^n-Kn-k)|an,fc-i — an,fc| = O(1), then ||f-T^f)|LAm = O(n-i).
■ik=1\llJ rV|u-n,k-i | — ||j j.n W7||lp,a(T)
Since ^n-ii (n - k) |an,k-i - «n,k| ^ n K.k-i - «n,k|, by Theo-
rem 1 we immediately have Corollary 2.
Corollary 1. Let 0 < A ^ 2 and 1 < p < rc. Let f E Lip^ (T, 1) and let A = (an>k) be a lower-triangular matrix satisfying the relation |snA) - 1| = O(n-i).
If E1 |an>fc_i - ara>fc| = O(n-1), then ||/ - TÍA)(/}|lp,a(t) = O(n-1). fc=i
If Efc-1 P-Pk+i| = O(Pra/n), we have En_í -a„)fc| = O(n-1),
where a„)fc := pra_fc/Pra and Pra = E/c=0Pk (see, [16]). Hence, Corollary 1 implies
Corollary 2. Let 0 < A ^ 2 and 1 < p < ro. Let also / e Lipp'A (T, 1) and let (pn) be a sequence of positive numbers.
n_ 1
If E |pk - Pk+1| = O(Pn/n), then ||/ - Nn (/)||LP,A(T) = O(n-1). fc=1
Note that similar results in classical and variable Lebesgue spaces were proved in [4], [23], [21], [11], [15]. We extend the results obtained above to the subclasses of Ep'A (G). Moreover, similar results in weighted Orlicz space were proved in [17]. Therefore, we need to give some definitions and auxiliary results.
Definition 2. Let r be a smooth Jordan curve and let #(s) be the angle between the tangent and the positive real axis expressed as a function of arclength s. If 0(s) has a modulus of continuity s) satisfying the Dini smoothness condition f^ [w(0, s)/s] ds < ro, ó > 0, then we say that r is a Dini smooth curve.
We denote the set of Dini smooth curves by D.
Let be the conformal mapping of G_ onto D_, normalized by the conditions (ro) = ro and lim ^ (z) /z > 0 , and let := be its
inverse. Since r is a rectifiable Jordan curve, the derivatives ^ and have definite nontangential boundary values a. e. on r and T, and the boundary functions are integrable with respect to the Lebesgue measure on r and T, respectively [10, p. 419-438]. On the other hand, if r e D, then, by [27], there are positive constants c > 0, i =1, 2, 3, 4, such that
0 < c1 ^ (w) | ^ c2 < ro, 0 < c3 ^ (z) | ^ c4 < ro, (1)
a. e. on T and r, respectively. Using (1), it is easily to see that if r e D, then there exist some positive constants c2 and c'3, such that for any arc Y C r the relation c2 |y| ^ |^(y)| ^ c3' |y|, where |y| and |<^(y)| are the linear Lebesgue measures of y and its image under the conformal mapping <£, holds. Then, denoting /0 (w) := (/ o (w), w e T, we have the implication
/ e Lp'A (r) ^ /0 e Lp'A (T). (2)
For a given f E LP'A (r), the Cauchy-type integral
(w):=21i / *** dT, w E D 2ni j t - w
T
is analytic in D. If f E LP'A (r), then, by (2) f0 E LP'A (T) and by Corollary 1 proved in [13], we have f+ E HP'A (D).
Let f E Ep'a(g), 0 < A ^ 2, 1 <p< rc and 0 <a ^ 1. Denoting ^(f,^)G,P,A := n(f0+,i)P;A, we say that
f E LipP'A(G, a) if n(f,5)G>p>A = O(5a) for 0 ^ 5.
Now let Fk, k = 0,1, 2,..., be the Faber polynomials for G, defined by the series representation (see, [26]):
(w) ^ ^, w E D- and z E G. (3)
^ (w) - z k=0 w
On the other hand, by the Cauchy integral formula:
f / mdz = -1 / dw, z G G,
J Z — z J ^ (w) — z
r T
for every f G Ep'A (G). Comparing this formula with (3), we have
<x
f (z) - £ afcFk (z) ,z G G, (4)
fc=0
where
ak = ak (f) : = ^ J ^k^ k = 0, 1 2,... (5)
T
The series (4) is called the Faber series of f G Ep'A (G) and the coefficients ak, k = 0,1, 2,..., are the Faber coefficients of f G Ep'A (G). For f g Ep'A (G), we define the n-th partial sums of series (4) as
SG (f)(z):= £ afc (f) Ffc (z) , n = 1, 2, 3,... fc=0
and n-th matrix transform
n
TgA (f) (z) := £ an,kSG (f) (z) , n =1, 2, 3,...
fc=Q
of the Faber series with respect to the infinite lower-triangular regular matrix A = (an>k) with non-negative elements an>k. If an>k := pn-k/Pn for a given sequence (pn) of positive numbers, where Pn = En=QPk, then the matrix transforms T^^ (f), n = 0,1,..., coincide with the Norlund means
1n
NG (f) (z) = — £ Pn-kSG (f) (z)
Pn
k=Q
of Faber series. Now, let s-a) = En=Q an,k , n = 0,1,... be the n-th row sum of the matrix A.
Theorem 3. Let r G D, 0 < A ^ 2, 1 < p < w. Let f G LipP'A(G, a), 0 < a < 1, and let A be a lower-triangular matrix with |s-in^^ —1| = O(n-a),
A = (an,fc).
If one of the following conditions:
(i) A has almost monotone decreasing rows and (n + 1) an>Q = O (1),
(ii) A has almost monotone increasing rows and (n +1) an>k = O (1), where k is the integer part of n/2,
holds, then
f — T^ (f) = O (n-a) .
Corollary 1. Let r G D, 0 <A ^ 2, 1 <p< w. Let also f G Lipp'A(G,a), 0 < a < 1, and let (pn) be a sequence of positive numbers. If one of the following conditions:
(i) (pn) is almost monotone increasing and (n + 1) pn = O (Pn),
(ii) (pn) is almost monotone decreasing, holds, then ||f — NG (f) ||LP,A(r) = O (n-a).
Theorem 4. Let r G D, 0 <A ^ 2 and 1 < p < w. Let also f G Lipp'A(G, 1) and let A = (an>k) be a lower-triangular matrix satisfying the relation |s-a) — 1| = O (n-1).
If En-i (n — k) |on,fc-1 — an,fc | = O (1), then
f — TGn (f) = O(n-1).
J G>n W y LP,A(r) V y
Corollary 1. Let r G D, 0 < A ^ 2 and 1 < p < ro. Let f G LipP'A(G, 1) and let A = (an,k) be a lower-triangular matrix satisfying the condition |siA) - 1| = O (n-1).
If EI-1 |an,k-i - | = O(n-1), then f -T^ (f) = O(n-1).
Corollary 2. Let r G D, 0 <A ^ 2 and 1 < p < ro. Let f G Lipp'A (G, 1) and let (pn) be a sequence of positive numbers. If En-1 |Pk - Pk+i| = O (Pn/n) , then ||f - NG (f )^LP,A(r) = O (n-1).
2. Auxiliary results.
Lemma 1. Let 0 < A ^ 2 and 1 < p < ro. Then there exists a constant c such that for every f e Lp'A (T) the inequality ||Sn (f) ||lp,a(T) ^ c |f Hlp,A(T) holds.
Proof. Let I be any subinterval of T with the characteristic function X/. By [7], the maximal function belongs to A1 (T), i.e., almost
everywhere on T the inequality M (Mx/) ^ cMx/ holds. Considering the boundedness [12] of Sn (f) in the weighted Lebesgue space, we have
J |Sn (f) (x)|p dx = J |Sn (f) (x)|p x/ (x) dx ^
/ T
|Sn (f) (x)|p MX/ (x) dx ^ c^ |f (x)|p MX/ (x) dx.
TT
Then, by the equivalence (see, also: [8])
Mx/ (x) - X/ (x) + ^ 2-2kX(2k+i/\2k/) (x)
we obtain
suP-)~x I |Sn (f )(x)|P dx ^
/ |111-* /
1 ' ' " (x)|p ( X/ (x) + > 2-2kX(2fc+m2k/) (x) )dx ^
^ c6 SUP-^ |f (x)|P (x/ (x) + ^ 2 2k X(2k+l/\2k /) (x) ) dx
/ |111-2 T fc=o
^ C6 sup—^ I |f (x)|p dx +
/ |111- 2 J
1 r ^
1 I , „,
+C6 sup—|/ (x)|p£ 2 2kX(2k+i/\2k/) (x) dx
1 111 2 T fc=°
+ £ 2-2k sup-^-k=0 1 |i|1-7 J
c4 II/IIL^(T) ^2"2k sup—- |/ (x)|p (x) dx ^
^ cj II/ILp,7(T) + £ 2-2k sup—^ i |/(x)|p (x) dx) ^ V ( ) k=0 1 |111-7 2k+1J j
^ cJ II/i!Lt.A(T) +
Lp,7(t)
+ V 2"2fc+(fc+1)(1"7) sup-1-7 [ |/(x)|p (x) dx) ^
k=0 1 |2k+1/|1"7 J iI J
oo
^ c/II/!Lp,7(t) + £ 2"2fc+(fc+1)(1"7) sup—^ i |/(x)|p (x)dx) ^
V ( ) k=0 1 |111 2 J '
7
I ■
I
lLP,A(T) '
because of £ 2-2k+(k+1)(1-2) < w. □
fc=Q
Let f G L1 (T) and let f be its conjugate function defined as
f (x) := 1 [ f(t)) ) dt.
J v ; W 2 tan
-n
The conjugate operator f is bounded in the weighted Lebesgue space [12]. Applying the same method used in the proof of Lemma 1, we obtain
Lemma 2. Let 0 < A ^ 2 and 1 < p < w. Then there exists a constant c such that for every f G Lp'A (T) the inequality ||/||lp,a(T) ^ c If IIlp,a(t) holds.
Lemma 3. Let 0 < a ^ 1, 0 < A ^ 2, 1 < p < w. If f G LipP'A (T, a), r =1, 2,..., then |f — Sn (f)||LP,*(T) = O (n-a).
Proof. Let Tn , n = 1, 2,..., be the best-approximation trigonometric polynomial to f G Lipp'A (T,a) in nn, where nn is the set of trigonometric polynomials of degree not exceeding n. Then, from the direct theorem proved in [13], we have ||f — Tn||LP,A(T) = O (n-a). Hence, applying
Lemma 1, we have
(T)
tra 1 lp,a(T) + ||Tra Sn (/)|LP,A(T) ^ — tra 1 lp,a(T) + 11 (T„ — / )|lp,a(T) =
ö II/-T
n Hlp'a(T)
0(n
Thus, the lemma is proved. □
Let WfA(T): = {f : f be absolutely continuous and f' G Lp'A (T)} be the Sobolev-Morrey space defined on T.
Lemma 4. Let 0 < A ^ 2 and 1 < p < ro. Then f e Lipp'A (T, 1) ^ f e Wf'A(T).
Proof. Let f G LipP'A (T, 1). Since Lp'A (T, 1) C Lp (T), we have If IIlp(t) ^ c |f ||lp,a(t) and then n (f, ^ ^ cfi(f,i)p>A = O (i). This relation shows that f is an absolute continuous function on T and, moreover, f' G Lp (T). Since [f (x + t) - f (x)] /t ^ f' (x), t ^ 0, a. e. on T, we have
2 f |/(x + t) - / (x)
SJ t
¿/2
and, then, by the Fatou Lemma in Lp'A(T, 1)
dt ^ |/' (x)|, S ^ 0+,
1/ 'I
Lp,a(t)
\/(x + f) - / (x)| dt
¿^0+ S J t
¿/2
Lp,a (t)
^ lim inf
¿
2 n
¿/2
¿
4 1 i
S Ö
dt
lp,a (T)
lp,a(T)
^ c^m+ inf ^(f,^ = O (1);
hence, f' G Wf'A(T). Conversely, if f G Wf'A(T), then, by absolute
t
continuity of f, we have f (x + t) - f (x) = J f' (x + u) du. Considering
—a
t
the boundedness of the maximal operator in the Morrey spaces in [6], we have
fi(/,i)p.A = sup II/ (x + t) - / (x)ILP,A(T) ^
^ sup
|/' (x + u)| du
Lp,a(t)
1 j |/' (x + u)| du
Lp,a(t) ^ ci||/'^(l) ^
Thus f G Lipp'A (T, 1). □
Lemma 5. Let 0 < A ^ 2 and 1 < p < w. If f G LipP'A (T, 1), then I|Sn (f) — ^n (f)|Lp,,(T) = O (n-1).
Proof. Let f G LipP'A (T,1). By Lemma 4, we have f G WfA(T). If f
x
has the Fourier series E uk (f), then the Fourier series of the conjugate
fc=Q
x
function /' is E kuk (f). After simple computations, we have fc=1
Sfc (/) - ^ (/)
£ ufc (/)
I
n k
k=0
I
+ 1 E E Uv (/) = £ Uk (/) - -— £ Uv (/)
k=0 v=0 k=0 v=0
n + I — k \ ^^ k
1
fc=Q
- + I
:)uk(/H £ —y uk(/) k=0
and, hence,
k
Sn (/) - ^n (/) = £ -— uk (/) .
' - + I
k=1
Using here Lemmas 1 and 2, we obtain
Sn (/) - ^n (/)
I Sn (/'
k
= II V ^^Uk (/) k=1
- + I
c
^ -
LP,7(l) -
Ilp>a(T)
LP,7(l)
C f-"1
Thus, the lemma is proved. □
5
t
Lemma 6. [11] Let A = (an>k) be infinite lower-triangular matrix and 0 < a < 1. If one of the conditions:
(i) A has almost monotone decreasing rows and (n + 1) an>0 = O (1),
(ii) A has almost monotone increasing rows and (n + 1)an>r = O (1), where r is the integer part of n/2 and siA) - 1 = O (n-a),
holds, thenYln=1 k—aan>k = O (n-a).
Lemma 7. [16] If A = (an>k) is an infinite lower-triangular matrix with non-negative elements an>k, then for every positive integer r and n such that 1 ^ r ^ n - 1, the equality
n— 1 k n— 1
( |an,m—1 - = ^^ (n - k) |an,k—1 - an,k|
k=r m=r k=r
holds.
Let P be the set of all polynomials with no restrictions on the degree and P (D) be the trace of all members of P on D. We define the operator Y : P (D) ^ EpA (G) as
T (P) (z) := -L f PM f' dw = ^ f PMi»*, z G G.
v M ; 2ni J f (w) - z , - z
T r
n
If P(w) := bkwk, then, by (3), we have k=0
y(E bkwk) = 5Le £ bk / ggM dw = £ bkFk (z).
k=0 k=0 T k=0
If r G D, 0 < A ^ 2, and 1 < p < ro, then the linear operator
Y : P (D) ^ Ep'A (G) is bounded (see [14]). Hence, extending the operator
Y from P (D) to Hp'A (D) as a linear and bounded operator, we obtain the extended operator Y : Hp'A (D) ^ Ep'A (G) with the representation
Y (f )(z ):=-L f ZMf^ dw, z G G, f G (D).
2ni J f (w) - z
T
Theorem 5. [14] Let r G D, 0 < A ^ 2 and 1 < p < ro. Then the operator Y (P) : HP'A (D) ^ Ep'A (G) is linear, bounded, one to one, and onto. Moreover, Y = f for every f G Ep'A (G).
3. Proofs of the Main Results.
Proof of Theorem 1. Let f G Lip^T, a), 0 < a < 1, 0 < A ^ 2, 1 < p < w and let A = (an>k) be a lower-triangular matrix with |s—a) — 1| = O (n-a). Suppose that one of the conditions (i) and (ii) holds. By definition of TlA) (f) and snA), we have
n
T!A)(f)(x) — f (x) = £ «n,fcSk(f)(x) — f (x) =
k=Q
n
= £ an,kSk(f )(x) — f (x) + snA)f (x) — snA)f (x) =
k=Q n
= £ «n,k [Sk (f) (x) — f (x)] + (s-a) — 1) f (x).
k=0
Since |s- ) — 1| = O (n a), by Lemmas 3 and 6 we obtain
||f — (f )|| lp,A(t) ^ an,Q ||Sq (f) f |lp,A(t) +
n
+ £ «n,k IISk (f) — f ilp,A(T) + |snA) — 1| If ilp,A(T) ^ k=1
1 n
^ O (—y ) + c £ an,kk-a + O (n-a) = O (n-a). □ n + k=1
Proof of Theorem 2. Let f G Lip^ (T, 1), 0 < A ^ 2, p G (1, w), A = (an>k) be a lower-triangular matrix, |s-a) — 1| = O(n-1). By Lemma 3, we have:
||f — T(A) (f)|Lp,,(T) ^ ||Sn (f) — T(A) (f)|Lp,,(T) + If — Sn (f)IU(T) =
= ||Sn (/) - TiA) (/)|LP,A(i) + O (--1) . (7)
If An,k := Em=k On,m, then
n n k
TnA) (/) (x) = £ an,kSk (/)(x) = £ an,^ £ um (/) (x)) =
k=0 k=0 m=0
n n n
= £ (£ an,k) uk(/) (x) = £ An,k uk(/) (x).
On the other hand, since s^ = En=0 an,k, we get
Sn (f) (x) = Y1 um (f) (x) =
m=0
nn
= An,0 ^ Uk (f) (x) + (1 - An,0) ^ Uk (f) (x) k=0 k=0
n
= ^ An,0Uk (f) (x) + P1 - snA)) Sn (f) (x) .
k=0
Thus,
TiA) (f) (x) - Sn (f) (x) = J] (An,k - An,0) Uk (f) (x) +
k=1
+ PsnA) - 1) Sn (f) (x) .
By Lemma 1 and by the condition |snA) - 1| = O (n—1) :
Sn (f) - TiA) (f)
lp,a(T)
V (An,k - An,0) Uk (f) + O (n—1L)
Lp,A (T) v '
k=1
(8)
Setting 6n>k := n,kk n,°, k = 1, 2,..., n and applying the Abel transform, we have
n n n
y^ (An,k - An,0) Uk (f) = ^ bn,kkuk (f) = bn,n ^ mura (f) +
m=1
n-1 ( k )
+ ^^ (bn,k - bn,k+1^ ^^
k=1
k=1
mUm
k=1
m=1
and, then:
y^ (An,k - An,0) Uk (f)
k=1
|bn,n| || (f)
m=1
lp,a(T)
n-1 ( k
+ y] |bn,k - bn,k+1| (|| mUm (f)
LP,A(T)
+
k=1
m=1
LP>A(T)/
Now, using (6) and applying Lemma 5, we have
n
|| V mUm (f) ^ = (n +1) ||Sn (f) - ^n (f )|Lp,A(t) II z—' LP,A(T) l (^
m=1 w
P 1)
(n +1) O (n-^ = O (1); (10)
later, by the condition
S(A) - 1 1
O (n-1)
I, I _ |An,n — An,01 _ |ara,ra sk )|
|bra,ra|
nn
1 / ^ \ 1
= - (snA) - anJ ^ -snA) = O Pn—1) . (11)
nn
Since, by the relations (8)-(11),
( -1
||Sn (f) - T(A) (f)||lp,a(t) ^ OjV1 + Y - b(,k+il ), (12)
n
fc=i
n— 1
to complete the proof it remains to prove that |bn,k - bn>k+1| = O (n— 1).
k=1
After simple calculations, we have
bn,k bn,k+1
1
k(k + 1)
(k +1) (13)
m=0
and, later, iteration easily shows that for k = 1, 2, . . . , n,
y^ a»,m - (k + 1) a„,k ^^ m |a„,m—1 - a„,m|. (14)
m=0
m=1
If the condition Ek—1 (n - k) |a(,k-1 - a(,k| _ O (1) of Theorem 2 holds, then by (13) and (14) we get
k— 1 r 1 k
^^ |bn,k - bn,k+l| ^ k (k I 1) m |an,m—1 - ^ra,™1 +
k=1 k=1 ( )m=1
n— 1 1 k
+S k (k+1) X]m |a™
I , (15)
k=r m=1
for r := [n/2]. For the first term of the right-hand side, applying the Abel transform we have
r i k
E k (k I 1) E m |an,m-i - an,m| ^
1
(n — k)
T—1—^ E(n — k) K;fc-i — a„;fc| ^
(n — r) ' ^ ' fc=i
r r 1
^ £ |ara,fc-i — | = £ ( _ k) (n — k) |an,fc-i — a„)fc| ^
fc=i fc=i 1
^ O (1) = O (n-1) . (16)
(n — r)
For the second term, we can write
n— i 1
E k (k + 1) E m |an,m—i an,m| ^
fc=r m=i
n— i 1 r ^ E k(k I 1) E ^ |an,m—i — an,m| +
fc=r m=i
n—i 1 fc
1 ^ ^ / i 1 ^ ^ y m |an,m—i an,m| • ^ni 1 ^n2 .
k-^ ^ r i i -')">
Since J2 |an,k—i — an,k| = O (n i), based on (16) we have fc=i
n-i 1
^ni ^ ^ ^ k | 1 |an,m—i ^n,™1
fc=r m=i
n—i 1 1 O (n—i) E fcTI = O (n—i) (n — r) ^ = O (n—i)
fc=r
Now we estimate In2. By Lemma 7
n— i 1 fc
^n2 ^ ^ ^ k | 1 ^ y |an,m—i ^n,™1 ^
fc=r m=r
1 n— 1 k 2 n—1 k
^ r + i / y / y |an,m—1 an,mjy ^ n / y / y |an,m—1 an,mjy ^
k=r m=r k=r m=r
2 n—1 2 n—1 ^ n £ (n - k) |an,k—1 -an,k| ^ - E (- - k) K,k—1 -an,k| = O (n—^ .
Thus
n1
E k (k1, 1) E m Km—i — an,m| ^ Ini + In2 = O (n i) . (17)
fc=r m=i
Hence, by the relations (15)-(17) we have
n-i
E |kn,fc — bn,fc+i| = O (n—i) . (18)
k=1
Now, the relations (7), (12), and (18) imply the desired inequality. □
Proof of Theorem 3. Let r G D, 0 < A ^ 2, 1 < p < œ,
and f G Lipp'A (G,a) for 0 < a < 1. Let A = (an,k) be a lower-triangular matrix with |sin^^ — 1| = O (n—a). Since f G Ep'A (G), we have f+ G Hp'A (D) C Hi (D), which implies that the boundary function of f+
belongs to Lp'A (T). Let E ^fc (f+) wk, w G D, be the Taylor-series expan-
k
'0 " ""
k=
sion of the function f+ on the unit disk D. By Theorem 3.4 in [9, p. 38], we get:
(f+) J^k (f0+) , k ^ 0,
ck fo+ =< . n 0, k < 0,
x
where E ck (f+) eikt is the Fourier series of the boundary function of k=
f0+ G Lp'A (T) C Li (T). Therefore, we have f + (w) = E cfc (f0+)
fc=—x
Assuming that f (w) = f+ (w) — f— (w) a. e. on T, we get:
(f) 1 ffo (w) 1 f f0+ (w) d 1 f f— (w) d
1 f0+ (w)
_ _o_
2ni J wk+1
T
-dw = ßk (fo+)
which shows that the Faber coefficients ak (f ), k = 0,1, 2,..., are the
fo+
Taylor coefficients of f+ at the origin, that is
fo+ (w) = Y ak (f ) wk,w G D. (19)
k=0
If E ak (f ) Fk (z) is the Faber-series expansion of f G Ep'A (G), then, by k=0
(19) and (3), we get
n
T( E Ck (fo+) = SnG (f ) (z) and Y (T(a) f)) = T^ (f ). (20)
k=0
Taking into account the conditions of Theorem 3, we have f+ G Lipp'x (T, a). Now, applying Theorem 1 for f+ and Theorem 5, we have:
f - Tga (f ) Lp,A(r) = ||Y (fo+) - Y (Tia) (f0+)) ^LP,A(r)
= || Y f - TnA) (fo+)) yLP,A(r) ^ c ||fo+ - TÎA) (fo+) IIlp^ = O (n-«) . □
Proof of Theorem 4. Let r G D, 0 < À ^ 2, 1 <p< œ. The condition f G Lip^'A (G, 1) of Theorem 4, by definition of classes Lipp'A (G, 1), means that f+ G Lipp'A (T, 1). Then, applying Theorem 2 for f+, by (20) and Theorem 5, we have
f - TÏÏ (f ) LP,A(r) = || Y (fo+) - Y (TlA) (fo+)) L.A(p)
LP>A(r)
= ||T (/+ - T(A) (/+)) Hl^d ^ c ||/0+ - T(A) (/+) Hlp^t) = O (n-1) . □
Acknowledgment. The authors would like to thank the anonymous referees for their remarks and suggestions, which helped to improve the manuscript.
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Received January 08, 2021. In revised form, April 18, 2021. Accepted April 21, 2021. Published online May 8, 2021.
Balikesir University, Balikesir 10145, Turkey Testici A. E-mail: testiciahmet@hotmail.com
Israfilov D. M. E-mail: mdaniyal@balikesir.edu.tr
