Pál Type Interpolation Problems with Additional Value Nodes
Poornima Tiwari
Department of Mathematics and Statistics, The Bhopal School of Social Sciences, Bhopal, M.P., India
Keywords: PTIP, Regularity, Roots of Unity, Value Nodes, 2020 Mathematics Subject Classification: 41A05.
Abstract: The author termed Pál type interpolation problems as 𝑃𝑇𝐼𝑃. In this paper the regularity of

0, 1

−𝑃𝑇𝐼𝑃 and

0, 2

−𝑃𝑇𝐼𝑃, with addition of two non-zero complex nodes ±𝜁 or two real nodes ±1 at value nodes for pairs
of considered polynomials is evaluated.
1 INTRODUCTION
L. G. Pál 1975, introduced a new kind of
Interpolation on zeros of two different Polynomials.
It involves of finding a polynomial of degree (𝑚+
𝑛−1), that has prescribed values at 𝑚 pairwise
distinct nodes and prescribed values for 𝑟

derivative at 𝑛 pairwise distinct nodes. These nodes
are called value nodes and derivative nodes
respectively.
Let
𝜋

be the set of polynomials of degree less than
or equal to 𝑛 with complex coefficients. Let 𝐴(𝑧) ∈
𝜋

and 𝐵
(
𝑧
)
∈𝜋

, then for a given positive integer
𝑟 the problem of
(
0, 𝑟
)
−𝑃𝑇𝐼𝑃 on the pair {𝐴
(
𝑧
)
,
𝐵(𝑧)}, is to determine a polynomial 𝑃
(
𝑧
)
∈𝜋

,
which assumes arbitrary prescribed values at the
zeros of 𝐴
(
𝑧
)
and arbitrary prescribed values of the
𝑟

derivative at the zeros of 𝐵
(
𝑧
)
. The problem is
regular if and only if any 𝑃(𝑧) satisfying
𝑃
(
𝑦

)
=0; where 𝐴
(
𝑦

)
=0 ; 𝑖=1,2,…,𝑛,
𝑃
()
𝑧

=0; where 𝐵𝑧

=0 ; 𝑗=1,2,…,𝑚,
vanishes identically. Here the zeros of 𝐴
(
𝑧
)
, 𝐵
(
𝑧
)
are
assumed to be simple.
(De Bruin and Sharma 2003) observed regularity of
0, 𝑚

,…,𝑚

−𝑃𝑇𝐼𝑃 on the zeros of (𝑧

−𝛼


),
(𝑧

−𝛼


), … , (𝑧

−𝛼


) with 0<𝛼

< 𝛼

<, … , <
𝛼

.
(De Bruin 2005) explored necessary and sufficient
condition for regularity of
(
0, 𝑟
)
−𝑃𝑇𝐼𝑃 with respect
to exchanging value-nodes and derivative-nodes.
(De Bruin and Dikshit 2005) examined regularity of
(
0, 𝑟
)
−𝑃𝑇𝐼𝑃 on the pair
{
(
𝑧

−1
)(
𝑧−𝜁
)
,
(
𝑧

−
1
)
}
, where 𝑚 and 𝑛 are given positive integers and
𝜁
is not a zero of the polynomial
(
𝑧

−1
)
. They
determined largest domain for
𝜁, which ensures
regularity of the problem. They observed that
(
0, 𝑟
)
−𝑃𝑇𝐼𝑃 on the pair
{
(
𝑧

−1
)(
𝑧−𝜁
)
,
(
𝑧

−
1
)
}
, for positive integers 𝑚 and 𝑛 are not regular, if
𝑟> 𝑚+1. For the case, 𝑟≤𝑚+1 and on the basis
of relationship between the positive integers 𝑚 and 𝑛,
they explored
(
0, 𝑟
)
− on some different pairs and
found those problems are regular under certain
conditions.
(Dikshit 2003) considered 𝑃𝑇𝐼𝑃 involving
Möbius transform of zeros of (𝑧

+1) and (𝑧

−
1) with one or two extra derivative nodes.
(De Bruin 2005) investigated regularity of
(
0, 𝑚
)
−𝑃𝑇𝐼𝑃 on zeros of the pair
{𝑤

(

)
(
𝑧
)
, 𝑤

(

)
(
𝑧
)
}, where
𝛼 be a complex number
with
𝛼

, 𝛼

, 𝛼

, 𝛼

≠1 ; 𝑛, 𝑚≥1.
The method of considering non-uniformly
distributed nodes on unit disk is generalized, by
involving the Möbius transform of zeros of (𝑧

−
𝜌

) on the circle
|
𝑧
|
= 𝜌

(Mandoli and Pathak
2008).
(
0, 1
)
−𝑃𝑇𝐼𝑃 are found to be regular for
following pairs, where 𝑎

(𝑧) ∈𝜋

and 𝑏

(𝑧) ∈𝜋

with simple zeros, 𝐴

(𝑧) and 𝐵

(𝑧) are the sets of
zeros of the polynomials 𝑎

(𝑧) and 𝑏

(𝑧)
respectively such that 𝐵

(𝑧) ⊆𝐴

(𝑧) (Modi et al
2012)
●
{
𝑎

(
𝑧
)
,
(
𝑧−𝜁
)
𝑏

(
𝑧
)
}
.
●
{
(
𝑧−𝜁
)
𝑎

(
𝑧
)
, 𝑏

(
𝑧
)
}
.
●
{
𝑎

(
𝑧
)
,
(
𝑧−𝜁

)(
𝑧−𝜁

)
𝑏

(
𝑧
)
}
;
𝜁

≠𝜁

.
182
Tiwari, P.
Pál Type Interpolation Problems with Additional Value Nodes.
DOI: 10.5220/0012609300003739
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 1st International Conference on Artificial Intelligence for Internet of Things: Accelerating Innovation in Industry and Consumer Electronics (AI4IoT 2023), pages 182-184
ISBN: 978-989-758-661-3
Proceedings Copyright © 2024 by SCITEPRESS – Science and Technology Publications, Lda.

● 𝑎

(
𝑧
)
,
∏


(
𝑧−𝜁

)
𝑏

(
𝑧
)
 ;
𝜁

are
pairwise distinct.
●
{
𝑎

(
𝑧
)
, 𝛹(𝑡)𝑏

(
𝑧
)
}
;
𝛹
(
𝑡
)
∈𝜋
𝑡
(𝑡≥2) be a
polynomial with simple zeros.
●
{
(
𝑧−𝜁

)
𝑎

(
𝑧
)
,
(
𝑧−𝜁

)
𝑏

(
𝑧
)
}
.
The author (Pathak and Tiwari 2019, Pathak and
Tiwari 2018) revisited regularity of Pál type Birkhoff
interpolation and have introduced a new class of
𝑃𝑇𝐼𝑃. Also, the author (Pathak and Tiwari 2018,
Pathak and Tiwari 2020) examined the regularity of
‘incomplete’ type 𝑃𝑇𝐼𝑃 on non-uniformly distributed
nodes by omitting real and complex nodes and
studied ‘Incomplete’ type
𝑃𝑇𝐼𝑃 on zeros of
polynomials with complex coefficients.
2 MAIN RESULTS
The author considered the polynomials 𝑎

(𝑧) ∈𝜋

and 𝑏

(𝑧) ∈𝜋

with simple zeros. 𝐴

(𝑧) and 𝐵

(𝑧)
are the sets of zeros of the polynomials 𝑎

(𝑧) and
𝑏

(𝑧) respectively such that 𝐵

(𝑧) ⊆𝐴

(𝑧). Section
2.1 deals with
(
0, 1
)
−𝑃𝑇𝐼𝑃, while section 2.2 deals
with
(
0, 2
)
−𝑃𝑇𝐼𝑃.
(
0,1
)
−𝑃𝑇𝐼𝑃 with two Additional Value Nodes
Theorem 2.1: Let 𝑚, 𝑛≥1, then
(
0,1
)
−𝑃𝑇𝐼𝑃 on
{
(
𝑧

−𝜁

)
𝑎

(
𝑧
)
, 𝑏

(
𝑧
)
}
; ±𝜁∉𝐴

(
𝑧
)
, 𝐵

(𝑧) ⊆
𝐴

(𝑧) is regular.
Proof: Here, we have total (𝑚+ 𝑛+2) interpolation
points.
We need to determine a polynomial 𝑃(𝑧) ∈𝜋

with
𝑃
(
𝑦

)
=0 ; 𝑦

∈𝐴

(
𝑧
)
; 𝑖=1,2,…,𝑚,
𝑃
(
±𝜁
)
=0 ; ±𝜁∉𝐴

(
𝑧
)
,
𝑃

𝑧

=0 ; 𝑧

∈𝐵

(𝑧) ; 𝑗=1,2,…,𝑛.
Let 𝑃
(
𝑧
)
=
(
𝑧

−𝜁

)
𝑎

(
𝑧
)
𝑄
(
𝑧
)
; where 𝑄
(
𝑧
)
∈
𝜋

.
Thus 𝑃
(
𝑧
)
∈𝜋

.
The posed problem will be regular, if 𝑃(𝑧) ≡0.
Since 𝑃

𝑧

=0, we get
𝑧


−𝜁

𝑎

𝑧

𝑄

𝑧

+ 𝑄𝑧

𝑧


−
𝜁

𝑎


𝑧

+2𝑧

𝑎

𝑧

=0.
As 𝑧

∈𝐵

(𝑧) ⊆𝐴

(𝑧), we have
𝑧


−𝜁

𝑎


𝑧

𝑄𝑧

=0.
Since ±𝜁∉𝐴

(
𝑧
)
and 𝑎

(𝑧) has simple zeros, the
polynomial and its derivative cannot vanish
simultaneously at the same point, we have
𝑄𝑧

=0.
Since 𝑧

has 𝑛 values, we get
𝑄
(
𝑧
)
= 𝐶𝑞

(
𝑧
) (
2.1
)
According to our assumption 𝑄(𝑧) ∈𝜋

and
therefore on account of equation (2.1), we get
𝑄
(
𝑧
)
≡0.
Corollary 2.1: Let 𝑚, 𝑛≥1, then
(
0, 1
)
−𝑃𝑇𝐼𝑃 on
{
(
𝑧

−1
)
𝑎

(
𝑧
)
, 𝑏

(
𝑧
)
}
; ±1 ∉𝐴

(
𝑧
)
, 𝐵

(𝑧) ⊆
𝐴

(𝑧) is regular.
(
0,2
)
−𝑃𝑇𝐼𝑃 with two Additional Value Nodes
Theorem 2.2: Let 𝑚, 𝑛≥1, then
(
0, 2
)
−𝑃𝑇𝐼𝑃 on
{
(
𝑧

−𝜁

)
𝑎

(
𝑧
)
, 𝑏

(
𝑧
)
}
; ±𝜁∉𝐴

(
𝑧
)
, 𝐵

(𝑧) ⊆
𝐴

(𝑧) is regular.
Proof: Here, we have total (𝑚+ 𝑛+2) interpolation
points.
We need to determine a polynomial 𝑃(𝑧) ∈𝜋

with
𝑃
(
𝑦

)
=0 ; 𝑦

∈𝐴

(
𝑧
)
; 𝑖= 1,2,…,𝑚,
𝑃
(
±𝜁
)
=0 ; ±𝜁∉𝐴

(
𝑧
)
,
𝑃

𝑧

=0 ; 𝑧

∈𝐵

(
𝑧
)
; 𝑗=1,2,…,𝑛.
Let 𝑃
(
𝑧
)
=
(
𝑧

−𝜁

)
𝑎

(
𝑧
)
𝑄
(
𝑧
)
; where 𝑄(𝑧) ∈
𝜋

.
Thus 𝑃
(
𝑧
)
∈𝜋

.
The posed problem will be regular, if 𝑃(𝑧) ≡0.
Since 𝑃

𝑧

=0, we get
𝑧


−𝜁

𝑎

𝑧

𝑄

𝑧


+2𝑧


−𝜁

𝑎


𝑧


+2𝑧

𝑎

𝑧

𝑄

𝑧


+ 𝑧


−𝜁

𝑎


𝑧

+4𝑧

𝑎


𝑧


+2𝑎

𝑧

𝑄𝑧

=0.
As 𝑧

∈𝐵

(𝑧) ⊆𝐴

(𝑧) and 𝑎

(𝑧) has simple zero,
the polynomial and its derivative cannot vanish
simultaneously at the same point, we have
Pál Type Interpolation Problems with Additional Value Nodes
183

2𝑧


−𝜁

𝑎


𝑧

𝑄

𝑧


+ 𝑧


−𝜁

𝑎


𝑧


+4𝑧

𝑎


𝑧

𝑄𝑧

=0.
Since 𝑄(𝑧) ∈𝜋

and 𝑧

has 𝑛 values, the
differential equation is given by
2
(
𝑧

−𝜁

)
𝑎


(
𝑧
)
𝑄

(
𝑧
)
+
{
(
𝑧

−𝜁

)
𝑎


(
𝑧
)
+
4𝑧𝑎


(
𝑧
)
}
𝑄
(
𝑧
)
= 𝐶

𝑏

(𝑧),
𝑄

(
𝑧
)
+ 
1
2
𝑎


(
𝑧
)
𝑎


(
𝑧
)
+
2𝑧
(𝑧

−𝜁

)
𝑄
(
𝑧
) (
2.2
)
= 𝐶
𝑏

(
𝑧
)
(
𝑧

−𝜁

)
𝑎


(
𝑧
)
,
for some constant 𝐶=



.
Integrating factor of differential equation (2.2) is
given by
𝜑
(
𝑧
)
= 𝑒𝑥𝑝


1
2
𝑎
𝑚
′′
(
𝑧
)
𝑎
𝑚
′
(
𝑧
)
+
2𝑧
(𝑧
2
−𝜁
2
)

𝑑𝑧,
𝜑
(
𝑧
)
=
(
𝑧
2
−𝜁
2
){
𝑎
𝑚
′
(
𝑧
)}
1
2
.
Set
𝜂
(
𝑧
)
=
{
𝑎
𝑚
′
(
𝑧
)}
1
2
.
Solution of differential equation (2.2) is given by
𝜑
(
𝑧
)
𝑄
(
𝑧
)
= 𝐶

𝜑
(
𝑡
)
𝑏
𝑛
(
𝑡
)

𝑡
2
−𝜁
2
𝑎
𝑚
′
(
𝑡
)
𝑑𝑡 ,
(
𝑧

−𝜁

)
𝜂
(
𝑧
)
𝑄
(
𝑧
)
=
𝐶








(

)


(

)
(




)



(

)
𝑑𝑡 ,
(
𝑧

−𝜁

)
𝜂
(
𝑧
)
𝑄
(
𝑧
)
= 𝐶


(

)


(

)



(

)
𝑑𝑡,
𝐶


(

)


(

)



(

)
𝑑𝑡 =0⇒𝐶=0.
Hence,
𝑄
(
𝑧
)
≡0.
Corollary 2.2: Let 𝑚, 𝑛≥1, then
(
0, 2
)
−𝑃𝑇𝐼𝑃 on
{
(
𝑧

−1
)
𝑎

(
𝑧
)
, 𝑏

(
𝑧
)
}
; ±1 ∉𝐴

(
𝑧
)
, 𝐵

(𝑧) ⊆
𝐴

(𝑧) is regular.
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