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DOI: 10.26485/0459-6854/2018/68.2/4 PL ISSN 0459-6854
B U L L E T I N
DE LA SOCIE´TE´ DES SCIENCES ET DES LETTRES DE  LO´DZ´
2018 Vol. LXVIII
Recherches sur les de´formations no. 2
pp. 45–52
Dedicated to the memory of
Professor Yurii B. Zelinskii
Anatoly S. Serdyuk and Tetiana A. Stepanyuk
LEBESGUE-TYPE INEQUALITIES
FOR THE FOURIER SUMS ON CLASSES OF
GENERALIZED POISSON INTEGRALS
Summary
For functions from the set of generalized Poisson integrals Cα,rβ Lp, 1 ≤ p <∞, we obtain
upper estimates for the deviations of Fourier sums in the uniform metric in terms of the best
approximations of the generalized derivatives fα,rβ of functions of this kind by trigonometric
polynomials in the metric of the spaces Lp. Obtained estimates are asymptotically best
possible.
Keywords and phrases: Lebesgue-type inequalities, Fourier sums, generalized Poisson inte-
grals, best approximations
Let Lp, 1 ≤ p < ∞, be the space of 2pi–periodic functions f summable to the
power p on [0, 2pi), in which the norm is given by the formula ‖f‖p =
( 2pi∫
0
|f(t)|pdt
) 1
p
;
L∞ be the space of measurable and essentially bounded 2pi–periodic functions f with
the norm ‖f‖∞ = ess sup
t
|f(t)|; C be the space of continuous 2pi–periodic functions
f , in which the norm is specified by the equality ‖f‖C = max
t
|f(t)|.
Denote by Cα,rβ Lp, α > 0, r > 0, β ∈ R, 1 ≤ p ≤ ∞, the set of all 2pi–periodic
[45]
46 A. S. Serdyuk, T. A. Stepanyuk
functions, representable for all x ∈ R as convolutions of the form (see, e.g., [1, p. 133])
f(x) =
a0
2
+
1
pi
pi∫
−pi
Pα,r,β(x− t)ϕ(t)dt, a0 ∈ R, ϕ ⊥ 1, (1)
where ϕ ∈ Lp and Pα,r,β(t) are fixed generated kernels
Pα,r,β(t) =
∞∑
k=1
e−αk
r
cos
(
kt− βpi
2
)
, α, r > 0, β ∈ R. (2)
The kernels Pα,r,β of the form (2) are called generalized Poisson kernels. For r = 1
and β = 0 the kernels Pα,r,β are usual Poisson kernels of harmonic functions.
If the functions f and ϕ are related by the equality (1), then function f in this
equality is called generalized Poisson integral of the function ϕ. The function ϕ in
equality (1) is called as generalixed derivative of the function f and is denoted by
fα,rβ .
The set of functions f from Cα,tβ Lp, 1 ≤ p ≤ ∞, such that fα,rβ ∈ B0p , where
B0p = {ϕ : ||ϕ||p ≤ 1, ϕ ⊥ 1} , 1 ≤ p ≤ ∞,
we will denote by Cα,rβ,p .
Let En(f)Lp be the best approximation of the function f ∈ Lp in the metric of
space Lp, 1 ≤ p ≤ ∞, by the trigonometric polynomials tn−1 of degree n− 1, i.e.,
En(f)Lp = inftn−1
‖f − tn−1‖Lp .
Let ρn(f ;x) be the following quantity
ρn(f ;x) := f(x)− Sn−1(f ;x), (3)
where Sn−1(f ; ·) are the partial Fourier sums of order n− 1 for a function f .
Least upper bounds of the quantity ‖ρn(f ; ·)‖C over the classes Cα,rβ,p , we denote
by En(Cα,rβ,p )C , i.e.,
En(Cα,rβ,p )C = sup
f∈Cα,rβ,p
‖f(·)− Sn−1(f ; ·)‖C , r > 0, α > 0, 1 ≤ p ≤ ∞. (4)
Asymptotic behaviour of the quantities En(Cα,rβ,p )C of the form (4) was studied in
[2]–[10].
In [11]–[13] it was found the analogs of the Lebesgue inequalities for functions
f ∈ Cα,rβ Lp in the case r ∈ (0, 1) and p = ∞, and also in the case r ≥ 1 and
1 ≤ p ≤ ∞, where the estimates for the deviations ‖f(·)−Sn−1(f ; ·)‖C are expressed
in terms of the best approximations En(f
α,r
β )Lp . Namely, in [11] it is proved that the
following best possible inequalitiy holds
‖f(·)− Sn−1(f ; ·)‖C ≤
( 4
pi2
lnn1−r +O(1)
)
e−αn
r
En(f
α,r
β )L∞ , (5)
where O(1) is a quantity uniformly bounded with respect to n, β and f ∈ Cα,rβ L∞.
Lebesgue-type inequalities 47
The present paper is a continuation of [11], [12], and is devoted to getting asymp-
totically best possible analogs of Lebesgue-type inequalities on the sets Cα,rβ Lp,
r ∈ (0, 1) and p ∈ [1,∞). This case was not considered yet. Let formulate the
results of the paper.
By F (a, b; c; d) we denote Gauss hypergeometric function
F (a, b; c; z) = 1 +
∞∑
k=1
(a)k(b)k
(c)k
zk
k!
,
(x)k :=
x
2
(x
2
+ 1
)(x
2
+ 2
)
...
(x
2
+ k − 1
)
.
For arbitrary α > 0, r ∈ (0, 1) and 1 ≤ p ≤ ∞ we denote by n0 = n0(α, r, p) the
smallest integer n such that
1
αr
1
nr
+
αrχ(p)
n1−r
≤

1
14 , p = 1,
1
(3pi)3 · p−1p , 1 < p <∞,
1
(3pi)3 , p =∞,
(6)
where χ(p) = p for 1 ≤ p <∞ and χ(p) = 1 for p =∞.
The following statement holds.
Theorem 1. Let 0 < r < 1, α > 0, β ∈ R and n ∈ N. Then in the case 1 < p <∞
for any function f ∈ Cα,rβ Lp and n ≥ n0(α, r, p), fthe following inequality is true:
‖f(·)− Sn−1(f ; ·)‖C ≤ e−αnrn
1−r
p
( ‖ cos t‖p′
pi
1+ 1
p′ (αr)
1
p
F
1
p′
(1
2
,
3− p′
2
;
3
2
; 1
)
+
+γn,p
((
1 +
(αr)
p′−1
p
p′ − 1
) 1
n
1−r
p
+
(p)
1
p′
(αr)1+
1
p
1
nr
))
En(f
α,r
β )Lp ,
1
p
+
1
p′
= 1, (7)
where F (a, b; c; d) is Gauss hypergeometric function, and in the case p = 1 for any
function f ∈ Cα,rβ L1 and n ≥ n0(α, r, 1), the following inequality is true:
‖f(·)−Sn−1(f ; ·)‖C ≤ e−αnrn1−r
( 1
piαr
+γn,1
( 1
(αr)2
1
nr
+
1
n1−r
))
En(f
α,r
β )L1 . (8)
In (7) and (8), the quantity γn,p = γn,p(α, r, β) is such that |γn,p| ≤ (14pi)2.
Proof of Theorem 1. Let f ∈ Cα,rβ Lp, 1 ≤ p ≤ ∞. Then, at every point x ∈ R the
following integral representation is true:
ρn(f ;x) = f(x)− Sn−1(f ;x) = 1
pi
pi∫
−pi
fα,rβ (t)P
(n)
α,r,β(x− t))dt, (9)
where
P
(n)
α,r,β(t) :=
∞∑
k=n
e−αk
r
cos
(
kt− βpi
2
)
, 0 < r < 1, α > 0, β ∈ R. (10)
48 A. S. Serdyuk, T. A. Stepanyuk
The function P
(n)
α,r,β(t) is orthogonal to any trigonometric polynomial tn−1 of
degree not greater than n− 1. Hence, for any polynomial tn−1 from we obtain
ρn(f ;x) =
1
pi
pi∫
−pi
δn(t)P
(n)
α,r,β(x− t)dt, (11)
where
δn(x) = δn(α, r, β, n;x) := f
α,r
β (x)− tn−1(x). (12)
Further we choose the polynomial t∗n−1 of the best approximation of the function
fα,rβ in the space Lp, i.e., such that
‖fα,rβ − t∗n−1‖p = En(fα,rβ )Lp , 1 ≤ p ≤ ∞,
to play role of tn−1 in (11). Thus, by using the inequality∥∥∥∥
pi∫
−pi
K(t− u)ϕ(u)du
∥∥∥∥
C
≤ ‖K‖p′‖ϕ‖p, (13)
ϕ ∈ Lp, K ∈ Lp′ , 1 ≤ p ≤ ∞, 1
p
+
1
p′
= 1
(see, e.g., [14, p. 43]), we get
‖f(·)− Sn−1(f ; ·)‖C ≤ 1
pi
‖P (n)α,r,β‖p′En(fα,rβ )Lp . (14)
For arbitrary υ > 0 and 1 ≤ s ≤ ∞ assume
Is(υ) :=
∥∥ 1√
t2 + 1
∥∥
Ls[0,υ]
, (15)
where
‖f‖Ls[a,b] =

(
b∫
a
|f(t)|sdt
) 1
s
, 1 ≤ s <∞,
ess sup
t∈[a,b]
|f(t)|, s =∞.
It follows from the paper [9] for arbitrary r ∈ (0, 1), α > 0, β ∈ R, 1 ≤ s ≤ ∞,
1
s +
1
s′ = 1, n ∈ N and n ≥ n0(α, r, s′) the following estimate holds
1
pi
‖P (n)α,r,β‖s =e−αn
r
n
1−r
s′
( ‖ cos t‖s
pi1+
1
s (αr)
1
s′
Is
(pin1−r
αr
)
+
+δ(1)n,s
( 1
(αr)1+
1
s′
Is
(pin1−r
αr
) 1
nr
+
1
n
1−r
s′
))
, (16)
where the quantity δ
(1)
n,s = δ
(1)
n,s(α, r, β), satisfies the inequality |δ(1)n,s| ≤ (14pi)2.
Substituting s = p′ =∞, from 14 and (16) we get (8).
Lebesgue-type inequalities 49
Further, according to [9] for n ≥ n0(α, r, s′), 1 < s <∞, 1s + 1s′ = 1, the following
equality takes place
Is
(pin1−r
αr
)
= F
1
s
(1
2
,
3− s
2
;
3
2
; 1
)
+
Θ
(1)
α,r,s′,n
s− 1
( αr
pin1−r
)s−1
, (17)
where |Θ(1)α,r,s′,n| < 2.
Let now consider the case 1 < p <∞.
Formulas (16) and (17) for s = p′ and n ≥ n0(α, r, p) imply
1
pi
∥∥Pα,r,n∥∥p′ = e−αnrn 1−rp ( ‖ cos t‖p′
pi
1+ 1
p′ (αr)
1
p
F
1
p′
(1
2
,
3− p′
2
;
3
2
; 1
)
+
+γ(1)n,p
( 1
p′ − 1
(αr)
p′−1
p
n(1−r)(p′−1)
+
p
1
p′
(αr)1+
1
p
1
nr
+
1
n
1−r
p
))
=
= e−αn
r
n
1−r
p
( ‖ cos t‖p′
pi
1+ 1
p′ (αr)
1
p
F
1
p′
(1
2
,
3− p′
2
;
3
2
; 1
)
+
+γ(2)n,p
((
1 +
(αr)
p′−1
p
p′ − 1
) 1
n(1−r)(p′−1)
+
p
1
p′
(αr)1+
1
p
1
nr
))
, (18)
where the quantities δ
(i)
n,p = δ
(i)
n,p(α, r, β), satisfiy the inequality |δ(i)n,p| ≤ (14pi)2, i =
1, 2. Estimate (7) follows from (14) and (18). Theorem 1 is proved. 
It should be noticed, that estimates (7) and (8) are asymptotically best possible
on the classes Cα,rβ,p , 1 ≤ p <∞.
If f ∈ Cα,rβ,p , then ‖fα,rβ ‖p ≤ 1, and En(fα,rβ )Lp ≤ 1, 1 ≤ p <∞. Considering the
least upper bounds of both sides of inequality (7) over the classes Cα,rβ,p , 1 < p <∞,
we arrive at the inequality
En(Cα,rβ,p )C ≤ e−αn
r
n
1−r
p
( ‖ cos t‖p′
pi
1+ 1
p′ (αr)
1
p
F
1
p′
(1
2
,
3− p′
2
;
3
2
; 1
)
+
+γn,p
((
1 +
(αr)
p′−1
p
p′ − 1
) 1
n
1−r
p
+
(p)
1
p′
(αr)1+
1
p
1
nr
))
En(f
α,r
β )Lp ,
1
p
+
1
p′
= 1. (19)
Comparing this relation with the estimate of Theorem 4 from [9] (see also [10]),
we conclude that inequality (7) on the classes Cα,rβ,p , 1 < p < ∞, is asymptotically
best possible.
In the same way, the asymptotical sharpness of the estimate (8) on the class Cα,rβ,1
follows from comparing inequality
En(Cα,rβ,p )C ≤ e−αn
r
n1−r
( 1
piαr
+ γn,1
( 1
(αr)2
1
nr
+
1
n1−r
))
En(f
α,r
β )L1 (20)
and formula (9) from [8].
50 A. S. Serdyuk, T. A. Stepanyuk
Acknowledgements
Second author is supported by the Austrian Science Fund FWF project F5503 (part
of the Special Research Program (SFB) Quasi-Monte Carlo Methods: Theory and
Applications)
References
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[2] A. I. Stepanets, Deviation of Fourier sums on classes of infinitely differentiable func-
tions, Ukr. Mat. J. 36:6 (1984), 567–573.
[3] S. M. Nikolskii, Approximation of functions in the mean by trigonometrical polyno-
mials, (in Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 10 (1946), 207–256.
[4] S. B. Stechkin, An estimate of the remainder term of Fourier series for differentiable
functions, (in Russian) Tr. Mat. Inst. Steklova 145 (1980) 126–151.
[5] S. A. Telyakovskii, Approximation of functions of high smoothness by Fourier sums,
Ukr. Math. J. 41:4 (1989), 444–451.
[6] A. I. Stepanets, Deviations of Fourier sums on classes of entire functions, Ukr. Math.
J. 41:6 (1989), 672–677.
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uniform metric, Ukr. Math. J. 57:8 (2005), 1079–1096.
[8] A. S. Serdyuk and T. A. Stepanyuk, Uniform approximations by Fourier sums on
the classes of convolutions with generalized Poisson kernels, Dop. Nats. Akad. Nauk.
Ukr. No. 11 (2016), 10–16.
[9] A. S. Serdyuk, T. A. Stepanyuk, Approximations by Fourier sums of classes of gen-
eralized Poisson integrals in metrics of spaces Ls, Ukr. Mat. J. 69:5 (2017), 811–822.
[10] A. S. Serdyuk, T. A. Stepanyuk, Uniform approximations by Fourier sums on classes
of generalized Poisson integrals, Analysis Mathematica, 2018 (accepted for publica-
tion)
[11] A. I. Stepanets, On the Lebesgue inequality on classes of (ψ, β)-differentiable func-
tions, Ukr. Math. J. 41:4 (1989), 435–443.
[12] A. I. Stepanets, A. S. Serdyuk, Lebesgue inequalities for Poisson integrals, Ukr. Math.
J. 52:6 (2000), 798–808.
[13] A. S. Serdyuk, A. P. Musienko, The Lebesgue type inequalities for the de la Vall?e
Poussin sums in approximation of Poisson integrals, Zb. Pr. Inst. Mat. NAN Ukr.
7:1 (2010), 298–316.
[14] N. P. Korneichuk, Exact Constants in Approximation Theory, Vol. 38, Cambridge
Univ. Press, Cambridge, New York 1990.
Graz University of Technology
Kopernikusgasse 24/II 8010, Graz
Austria
E-mail: tania stepaniuk@ukr.net
Lebesgue-type inequalities 51
Institute of Mathematics
National Academy of Sciences of Ukraine
Tereshchenkivska str. 3, UA-01004, Kyiv
Ukraine
E-mail: serdyuk@imath.kiev.ua
Presented by Adam Paszkiewicz at the Session of the Mathematical-Physical Com-
mission of the  Lo´dz´ Society of Sciences and Arts on May 14, 2018.
NIERO´WNOS´C´ TYPU LEBESGUE’A DLA SUM FOURIERA
NA KLASACH UOGO´LNIONYCH CA LEK POISSONA
S t r e s z c z e n i e
Dla funkcji ze zbioru uogo´lnionych ca lek Poissona Cα,rβ Lp, 1 ≤ p <∞, otrzymujemy
go´rne oszacownie dla odchylen´ sum Fouriera w jednostajnej metryce w terminach najlepszej
aproksymacji uogo´lnionych pochodnych fα,rβ funkcji tego typu w metryce przestrzeni Lp.
Uzyskane oszacowania sa¸ asymptotycznie najlepsze z moz˙liwych.
S lowa kluczowe: niero´wnos´ci typu Lebesgue’a, sumy Fouriera, uogo´lnione ca lki Poissona,
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