On modified Picard and Gauss—Weierstrass singular integrals
We introduce a certain modification of the Picard and Gauss—Weierstrass singular integrals and prove approximation theorems for them.
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| author | Rempulska, L. Walczak, Z. Ремпульська, Л. Вальчак, З. |
| author_facet | Rempulska, L. Walczak, Z. Ремпульська, Л. Вальчак, З. |
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| description | We introduce a certain modification of the Picard and Gauss—Weierstrass singular integrals and prove approximation theorems for them. |
| first_indexed | 2026-03-24T02:47:32Z |
| format | Article |
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UDC 517.9
L. Rempulska, Z. Walczak ( Inst. Math. Poznań Univ. Technol., Poland)
ON MODIFIED PICARD AND GAUSS – WEIERSTRASS
SINGULAR INTEGRALS
PRO MODYFIKOVANI SYNHULQRNI INTEHRALY
PIKARA TA HAUSSA – VEJ{RÍTRASSA
We introduce certain modification of the Picard and Gauss – Weierstrass singular integrals and we prove approx-
imation theorems for them.
Vvedeno deqku modyfikacig synhulqrnyx intehraliv Pikara ta Haussa – Vej[rßtrassa, a takoΩ dove-
deno aproksymacijni teoremy dlq cyx intehraliv.
1. Introduction. 1.1. Let Lp ≡ Lp(R), with fixed 1 ≤ p ≤ ∞, be the space of all
real-valued functions, Lebesgue integrable with p -th power over R := (−∞,+∞) if
1 ≤ p < ∞ and uniformly continuous and bounded on R if p = ∞. We define the
norm in Lp, as usual, by the formula
‖f‖p ≡ ‖f (·) ‖p :=
(∫
R
|f(x)|pdx
)1/p
if 1 ≤ p <∞,
sup
x∈R
|f(x)| if p = ∞,
(1)
where
∫
R
≡
∫ +∞
−∞
.
Denote (as usual) by ω1(f ;Lp; ·) and ω2(f ;Lp; ·) the modulus of continuity and the
second modulus of smoothness of f ∈ Lp, respectively, i.e.,
ωi(f ;Lp; t) := sup
0≤h≤t
‖∆i
hf(·)‖p, t ≥ 0, i = 1, 2, (2)
where ∆1
hf(x) = f(x+ h) − f(x) and ∆2
hf(x) = f(x+ h) + f(x− h) − 2f(x).
It is known [1] that for f ∈ Lp, 1 ≤ p ≤ ∞, and i = 1, 2 the following conditions
are satisfied:
i) ωi(f ;Lp;λt) ≤ (1 + λ)iωi(f ;Lp; t) for λ, t ≥ 0;
ii) limt→0+ ωi(f ;Lp; t) = 0;
iii) ω2(f ;Lp; t) ≤ 2ω1(f ;Lp; t) for t ≥ 0.
1.2. Let Pr(f ; ·) and Wr(f ; ·) be the Picard singular integral and the
Gauss – Weierstrass singular integral of function f ∈ Lp, respectively, i.e.,
Pr(f ;x) :=
1
2r
∫
R
f(x+ t) exp
(
−|t|
r
)
dt, (3)
Wr(f ;x) :=
1√
4πr
∫
R
f(x+ t) exp
(
− t
2
4r
)
dt (4)
for x ∈ R, r > 0 and r → 0 + . It is known [1] that these singular integrals are well
defined on every space Lp and Pr(f), Wr(f) with every fixed r > 0 are linear positive
operators from the space Lp to Lp.
The fundamental approximation property of integrals (3) and (4) gives the following
theorem.
c© L. REMPULSKA, Z. WALCZAK, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11 1577
1578 L. REMPULSKA, Z. WALCZAK
Theorem A [1 – 3]. Let f ∈ Lp, 1 ≤ p ≤ ∞. Then
‖Pr(f ; ·) − f(·)‖p ≤ 5
2
ω2(f ;Lp; r),
‖Wr(f ; ·) − f(·)‖p ≤ 7
2
ω2(f ;Lp;
√
r )
for all r > 0.
The limit properties (as r → 0+ ) of these integrals were given in many papers and
monographs (e.g., [1 – 3]).
1.3. The order of approximation given in Theorem A can be improved by certain
modification of formulas (3) and (4).
Let N := {1, 2, . . . } and N0 := N ∪ {0}. For fixed n ∈ N0 and 1 ≤ p ≤ ∞,
we denote by Lp,n the set of all f ∈ Lp which derivatives f ′, . . . , f (n) belong also to
Lp. The norm in these Lp,n (n ∈ N0, 1 ≤ p ≤ ∞) is defined by (1), i.e., for f ∈ Lp,n
we have ‖f‖p,n ≡ ‖f‖p, where ‖f‖p is defined by (1). Moreover, for f ∈ Lp,n there
exists norms ‖f (k)‖p, 0 ≤ k ≤ n, defined analogously to (1). Clearly, Lp,0 ≡ Lp.
Definition. Let f ∈ Lp,n with fixed n ∈ N0 and 1 ≤ p ≤ ∞. We define the
modified Picard and Gauss – Weierstrass singular integrals by formulas
Pr;n(f ;x) :=
1
2r
∫
R
n∑
j=0
f (j) (t)
j!
(x− t)j exp
(
−|t− x|
r
)
dt, (5)
Wr;n(f ;x) :=
1√
4πr
∫
R
n∑
j=0
f (j) (t)
j!
(x− t)j exp
(
− (t− x)2
4r
)
dt, (6)
for x ∈ R and r > 0.
In particular, we have Pr;0(f ; ·) ≡ Pr(f ; ·) and Wr;0(f ; ·) ≡Wr(f ; ·) for f ∈ Lp.
In Section 2, we shall give some elementary properties of integrals (5) and (6). In
Section 3, we shall prove two approximation theorems.
2. Auxiliary results. It is obvious that formulas (5) and (6) can be written in the
following form:
Pr;n(f ;x) =
n∑
j=0
(−1)j
j!
1
2r
∫
R
f (j) (t+ x) tje−|t|/rdt, (5′)
Wr;n(f ;x) :=
n∑
j=0
(−1)j
j!
1√
4πr
∫
R
f (j) (t+ x) tje−t2/4rdt (6′)
for every f ∈ Lp,n, x ∈ R, and r > 0.
By elementary calculations, we can prove the following lemma.
Lemma 1. For every n ∈ N0 and r > 0, we have
In :=
1
r
+∞∫
0
tne−t/rdt = n!rn, (7)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
ON MODIFIED PICARD AND GAUSS – WEIERSTRASS SINGULAR INTEGRALS 1579
I∗n :=
1√
4πr
+∞∫
0
tne−t2/4rdt =
=
1
2
if n = 0,
2k−1(2k − 1)!!rk if n = 2k ≥ 2,
4kk!rk+1/2
√
π
if n = 2k + 1 ≥ 1,
(8)
where (2k − 1)!! = 1 · 3 · 5 · · · · · (2k − 1) for k ∈ N.
Applying Lemma 1, we shall prove the main lemma.
Lemma 2. Let n ∈ N0 and 1 ≤ p ≤ ∞ be fixed numbers. Then for every
f ∈ Lp,n and r > 0 we have
‖Pr;n(f ; ·)‖p ≤
n∑
j=0
‖f (j)‖p
j!
Ij =
n∑
j=0
rj‖f (j)‖p, (9)
‖Wr;n(f ; ·)‖p ≤
n∑
j=0
‖f (j)‖p
j!
I∗j , (10)
where Ij and I∗j are given by (7) and (8).
Formulas (5) and (6) and inequalities (9) and (10) show that the integrals Pr;n(f)
and Wr;n(f), with fixed n ∈ N0 and r > 0, are linear operators from the space Lp,n
into Lp.
Proof. Inequalities (9) and (10) for n = 0 are given in [1].
If n ∈ N and p = ∞, then by (5′), (1), and (7) we get
‖Pr;n(f ; ·)‖∞ ≤
n∑
j=0
‖f (j)‖∞
j!
1
2r
∫
R
|t|je−|t|/rdt =
=
n∑
j=0
‖f (j)‖∞
j!
Ij =
n∑
j=0
‖f (j)‖∞rj , r > 0.
If n ∈ N and 1 ≤ p <∞, then by (5′), (1), (7), and by Fubini inequality [4] we get
‖Pr;n(f ; ·)‖p =
∥∥∥∥∥∥
n∑
j=0
(−1)j
j!2r
∫
R
f (j) (t+ ·) tje−|t|/rdt
∥∥∥∥∥∥
p
≤
≤
n∑
j=0
1
j!2r
∫
R
∣∣∣∣∣∣
∫
R
f (j) (t+ x) tje−|t|/rdt
∣∣∣∣∣∣
p
dx
1/p
≤
≤
n∑
j=0
1
j!2r
∫
R
|t|je−|t|/r
∫
R
∣∣∣f (j) (t+ x)
∣∣∣pdx
1/p
dt =
=
n∑
j=0
‖f (j)‖p
j!
Ij =
n∑
j=0
rj‖f (j)‖p, r > 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1580 L. REMPULSKA, Z. WALCZAK
Hence, the proof of (9) is completed. The proof of inequality (10) is analogous.
3. Theorems. 3.1. First, we shall prove the theorem on the order of approximation
of f ∈ Lp,n by Pr;n(f) and Wr;n(f).
Theorem 1. Suppose that f ∈ Lp,n with fixed n ∈ N and 1 ≤ p ≤ ∞. Then
‖Pr;n(f ; ·) − f(·)‖p ≤ (n+ 2)rnω1
(
f (n);Lp; r
)
, (11)
‖Wr;n(f ; ·) − f(·)‖p ≤ 2
n!
(
I∗n + r−1/2I∗n+1
)
ω1
(
f (n);Lp; r1/2
)
≤
≤M1(n)rn/2ω1
(
f (n);Lp; r1/2
)
for all r > 0, (12)
where I∗n is given by (8), M1(n) is positive constant depending only on n, and ω1
(
f (n);
Lp; ·
)
is defined by (2).
Proof. We shall prove only (11), because by (5) and (6) and Lemma 1 the proof of
(12) is analogous.
We shall apply the following modified Taylor formula of f ∈ Lp,n with n ∈ N :
f(x) =
n∑
j=0
f (j) (t)
j!
(x− t)j +
+
(x− t)n
(n− 1)!
1∫
0
(1 − u)n−1
{
f (n) (t+ u (x− t)) − f (n) (t)
}
du (13)
for a fixed t ∈ R and every x ∈ R.
Since
∫
R
e−|t−x|/rdt = 2r for r > 0 and x ∈ R, we have by (13) and (5):
f(x) =
1
2r
∫
R
f(x)e−|t−x|/rdt = Pr;n (f ;x) +
+
1
2r
∫
R
(x− t)n
(n− 1)!
1∫
0
(1 − u)n−1∆1
u(x−t)f
(n) (t) du
e−|t−x|/rdt (14)
for x ∈ R and r > 0.
1. Let p = ∞. Then by (2) and the properties of ω1 (f ;L∞; ·) we have
|∆1
u(x−t)f
(n) (t) | ≤ ω1
(
f (n);L∞; |u(x− t)|
)
≤
≤ ω1
(
f (n);L∞; |t− x|
)
≤
(
1 + r−1 |t− x|
)
ω1
(
f (n);L∞; r
)
for 0 ≤ u ≤ 1 and t, x ∈ R. From this and by (14) we get
|f(x) − Pr;n (f ;x) | ≤
≤ ω1
(
f (n);L∞; r
)
n!2r
∫
R
(
|t− x|n + r−1 |t− x|n+1
)
e−|t−x|/rdt =
=
ω1
(
f (n);L∞; r
)
n!
(
In + r−1In+1
)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
ON MODIFIED PICARD AND GAUSS – WEIERSTRASS SINGULAR INTEGRALS 1581
for x ∈ R and r > 0, where In is given by (7). Applying equality (7), we obtain (11)
for p = ∞.
2. If 1 ≤ p <∞, then from (14) we deduce that
f(x) − Pr;n (f ;x) =
1
2r
∫
R
tn
(n− 1)!
1∫
0
(1 − u)n−1∆1
utf
(n) (x− t) du
e−|t|/rdt.
Applying (similarly as in the proof of Lemma 2) the Fubini inequality, we get
‖f(·) − Pr;n(f ; ·)‖p =
=
1
2r(n− 1)!
∫
R
∣∣∣∣∣∣
∫
R
tne−|t|/r
1∫
0
(1 − u)n−1∆1
utf
(n) (x− t) du
dt
∣∣∣∣∣∣
p
dx
1/p
≤
≤ 1
2r(n− 1)!
∫
R
|t|ne−|t|/r
∫
R
∣∣∣∣∣∣
1∫
0
(1 − u)n−1∆1
utf
(n) (x− t) du
∣∣∣∣∣∣
p
dx
1/p
dt ≤
≤ 1
2r(n− 1)!
∫
R
|t|ne−|t|/r
1∫
0
(1 − u)n−1
∫
R
∣∣∣∆1
utf
(n) (x− t)
∣∣∣p dx
1/p
du
dt ≤
≤ 1
2r(n− 1)!
∫
R
|t|ne−|t|/r
1∫
0
(1 − u)n−1ω1
(
f (n);Lp; |ut|
)
du
dt ≤
≤ 1
2rn!
ω1
(
f (n);Lp; r
) ∫
R
|t|ne−|t|/r
(
1 + r−1|t|
)
dt =
=
1
n!
(
In + r−1In+1
)
ω1
(
f (n);Lp; r
)
for r > 0,
where In is given by (7). Using (7), we immediately obtain (11) for 1 ≤ p <∞. Thus
the proof is completed.
From Theorem 1 and Theorem A we derive the following two corollaries.
Corollary 1. For every f ∈ Lp,n, n ∈ N0, 1 ≤ p ≤ ∞, we have
lim
r→0+
r−n‖Pr;n(f ; ·) − f(·)‖p = 0,
lim
r→0+
r−n/2‖Wr;n(f ; ·) − f(·)‖p = 0.
Corollary 2. Let f ∈ Lp,n, n ∈ N0, 1 ≤ p ≤ ∞, and let f (n) ∈ Lip (α;Lp)
with a fixed 0 < α ≤ 1, i.e., ω1
(
f (n);Lp; t
)
= O(tα), t > 0. Then
‖Pr;n(f ; ·) − f(·)‖p = O(rn+α),
‖Wr;n(f ; ·) − f(·)‖p = O(r(n+α)/2)
for r > 0.
Remark 1. Theorem 1 shows that the order of approximation of function f ∈ Lp,n,
with n ≥ 2 and 1 ≤ p ≤ ∞, by the integrals Pr;n(f) and Wr;n(f) is better than for
Pr(f) and Wr(f) given in Theorem A.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1582 L. REMPULSKA, Z. WALCZAK
3.2. Now we shall prove the Voronovskaya-type theorem for integrals Pr;n(f) and
Wr;n(f).
Theorem 2. Suppose that f ∈ L∞,n+2 with a fixed n ∈ N0. Then
Pr;n (f ;x) − f(x) =
(−1)n − 1
2
rn+1f (n+1)(x)+
+
1 + (−1)n
2
(n+ 1)rn+2f (n+2)(x) + o
(
rn+2
)
as r → 0+ (15)
and
Wr;n (f ;x) − f(x) =
(−1)n − 1
(n+ 1)!
I∗n+1f
(n+1)(x)+
+
((−1)n + 1) (n+ 1)
(n+ 2)!
I∗n+2f
(n+2)(x) + o
(
r1+n/2
)
as r → 0+ (16)
for every x ∈ R, where I∗n is given by (8).
Proof. Fix x ∈ R and f ∈ L∞,n+2. Then f (j) ∈ L∞,n+2−j , 0 ≤ j ≤ n, and by
the Taylor formula we can write
f (j)(t) =
n+2−j∑
i=0
f (j+i) (x)
i!
(t− x)i + ϕj(t;x) (t− x)n+2−j (17)
for t ∈ R, where ϕj(t) ≡ ϕj(t;x) is a function such that ϕj(t)tn+2−j belongs to L∞
and limt→x ϕj(t) = ϕj(x) = 0 for every 0 ≤ j ≤ n. Using (17) to formula (5), we get
Pr;n(f ;x) =
1
2r
∫
R
e−|t−x|/r
n∑
j=0
(−1)j
j!
n+2−j∑
i=0
f (j+i) (x)
i!
(t− x)j+i
dt+
+
1
2r
∫
R
e−|t−x|/r(t− x)n+2
n∑
j=0
(−1)j
j!
ϕj(t)dt := Ar;n(x) +Br;n(x), r > 0. (18)
Further, by elementary calculations we have
Ar;n(x) =
1
2r
∫
R
e−|t−x|/r
n∑
j=0
(−1)j
j!
n∑
l=j
f (l)(x)
(l − j)! (t− x)
l +
+
f (n+1)(x)(t− x)n+1
(n+ 1)!
n∑
j=0
(
n+ 1
j
)
(−1)j+
+
f (n+2)(x)(t− x)n+2
(n+ 2)!
n∑
j=0
(
n+ 2
j
)
(−1)j
dt
and
n∑
j=0
(−1)j
j!
n∑
l=j
f (l)(x)
(l − j)! (t− x)
l =
=
n∑
l=0
f (l)(x)(t− x)l
l!
l∑
j=0
(
l
j
)
(−1)j = f(x) for n ∈ N0,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
ON MODIFIED PICARD AND GAUSS – WEIERSTRASS SINGULAR INTEGRALS 1583
because
l∑
j=0
(
l
j
)
(−1)j =
{
1 if l = 0,
0 if l ≥ 1.
Moreover, for n ∈ N0 we have
n∑
j=0
(
n+ 1
j
)
(−1)j = (−1)n,
n∑
j=0
(
n+ 2
j
)
(−1)j = (n+ 1)(−1)n.
Consequently,
Ar;n(x) = f(x)
1
2r
∫
R
e−|t−x|/rdt+
+
(−1)nf (n+1) (x)
(n+ 1)!
1
2r
∫
R
(t− x)n+1e−|t−x|/rdt+
+
(−1)n(n+ 1)f (n+2) (x)
(n+ 2)!
1
2r
∫
R
(t− x)n+2e−|t−x|/rdt.
Applying (7), we get for q ∈ N0 :
1
2r
∫
R
(t− x)qe−|t−x|/rdt =
1
2r
∫
R
tqe−|t|/rdt =
=
1 + (−1)q
2
Iq =
1 + (−1)q
2
q!rq.
From the above we obtain
Ar;n(f ;x) = f(x) +
(−1)n − 1
2
rn+1f (n+1)(x)+
+
1 + (−1)n
2
(n+ 1)rn+2f (n+2)(x), r > 0. (19)
Denoting by
Φn(t) :=
n∑
j=0
(−1)j
j!
ϕj(t), t ∈ R,
we have Φn ∈ L∞ and limt→x Φn(t) = Φn(x) = 0. Hence, by (18) we get
Br;n(x) =
1
2r
∫
R
Φn (t) (t− x)n+2e−|t−x|/rdt,
which by the Hölder inequality and (5) and (7) implies that
|Br;n(x)| ≤
{
1
2r
∫
R
(t− x)2n+4e−|t−x|/rdt
}1/2 {
Pr;0
(
Φ2
n;x
)}1/2 ≡
≡ {I2n+4}1/2 {
Pr;0
(
Φ2
n;x
)}1/2
, r > 0.
Applying Corollary 1 and the properties of Φn(·), we can write
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1584 L. REMPULSKA, Z. WALCZAK
lim
r→0+
Pr;n(Φ2
n;x) = Φ2
n(x) = 0,
uniformly on R. From the above and (7) we deduce that
Br;x(x) = o
(
rn+2
)
as r → 0+ (20)
uniformly for x ∈ R.
Collecting (18) – (20), we obtain the desired assertion (15).
The proof of (16) is analogous.
From Theorem 2 we derive the following corollary.
Corollary 3. Let f ∈ L∞,n+2 with n ∈ N0. Then for every x ∈ R we have:
lim
r→0+
r−n−2{Pr;n (f ;x) − f(x)} = f (n+2) (x) , if n is even number,
and
lim
r→0+
r−n−1{Pr;n (f ;x) − f(x)} = −f (n+1) (x) , if n is odd number.
The similar equalities hold for Wr;n(f).
1. Butzer P. L., Nessel R. J. Fourier analysis and approximation. – Basel: Birkhauser and New York: Acad.
Press, 1971. – Vol. 1.
2. Firlej B., Rempulska L. On some singular integrals in Hölder spaces // Math. Nachr. – 1994. – 170. –
P. 93 – 100.
3. Mohapatra R. N., Rodriguez R. S. On the rate of convergence of singular integrals for Hölder continuous
functions // Ibid. – 1990. – 149. – P. 117 – 124.
4. Zygmund A. Trigonometric series. – Moscow, 1965. – Vol. 1 (in Russian).
Received 09.06.2003,
after revision — 08.07.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
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| id | umjimathkievua-article-3710 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:47:32Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cd/edbbc90248ba0efeb5a54f1bcd7210cd.pdf |
| spelling | umjimathkievua-article-37102020-03-18T20:02:37Z On modified Picard and Gauss—Weierstrass singular integrals Про модифіковані сингулярні інтеграли Пікара та Гаусса - Вейєрштрасса Rempulska, L. Walczak, Z. Ремпульська, Л. Вальчак, З. We introduce a certain modification of the Picard and Gauss—Weierstrass singular integrals and prove approximation theorems for them. Введено деяку модифікацію сингулярних інтегралів Пікара та Гаусса - Вейєрштрасса, a також доведено апроксимаційні теореми для цих інтегралів. Institute of Mathematics, NAS of Ukraine 2005-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3710 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 11 (2005); 1577–1584 Український математичний журнал; Том 57 № 11 (2005); 1577–1584 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3710/4145 https://umj.imath.kiev.ua/index.php/umj/article/view/3710/4146 Copyright (c) 2005 Rempulska L.; Walczak Z. |
| spellingShingle | Rempulska, L. Walczak, Z. Ремпульська, Л. Вальчак, З. On modified Picard and Gauss—Weierstrass singular integrals |
| title | On modified Picard and Gauss—Weierstrass singular integrals |
| title_alt | Про модифіковані сингулярні інтеграли Пікара та Гаусса - Вейєрштрасса |
| title_full | On modified Picard and Gauss—Weierstrass singular integrals |
| title_fullStr | On modified Picard and Gauss—Weierstrass singular integrals |
| title_full_unstemmed | On modified Picard and Gauss—Weierstrass singular integrals |
| title_short | On modified Picard and Gauss—Weierstrass singular integrals |
| title_sort | on modified picard and gauss—weierstrass singular integrals |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3710 |
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