Internat. J. Math. & Math. Sci.
Vol. 9 No. 4 (1986) 753-756
753
ON THE NON-EXISTENCE OF
SOME INTERPOLATORY POLYNOMIALS
C.H. ANDERSON and J. PRASAD
Department of Mathematics
California State University
Los Angeles, California 90032 U.S.A.
(Received May 20, 1985)
x2)Tn(X)
ABSTRACT. Here we prove that if x k, k
1,2,...,n + 2 are the zeros of (i
where T (x) is the Tchebycheff polynomial of first kind of degree n,
aj
n
n + 2 and
are any real numbers there does not exist
1,2
n +
j
yj, j 2,3
Bj
Q3n+3(x)
a unique polynomial
Q3n+3(xj)
n
+ 1.
aj, Q3n+3(xj)
<
of degree
Bj,
1,2
j
3n + 3
n
satisfying the conditions:
+ 2
and
Q3n+3(xj) yj,
P n (x)
nodes of interpolation where
j
is
2,3
x2)Pn(X)
(1
of
the Legendre polynomial
degree
Similar result is also obtained by choosing the roots of
as the
n.
KEY WORDS AND PHRASES. Roots, interpolatory polynomials, non-existence, nodes.
1980 AMS SUBJECT CLASSIFICATION CODE. 41A25.
INTRODUCTION.
In [i] R.B. Saxena considered an interesting problem of (0,1,3) interpolation by
where Pn_2(x) is the Legendre polynomial of
taking the roots of (i
2, as the nodes of interpolation. By (0,1,3) interpolation, Saxena meant
degree n
i.
x2)Pn_2(x),
{aj}nl, {Bj}n-12
that for the collections
of
(i
x2)Pn_2(x
-i
a polynomial
Xn
n
{Yj}I
of real numbers and the zeros
arranged so that
< x2 < Xl
< Xn_l <
R n (x)
of degree
R n (x)
a
R’(xj).
Bj;
j
and
j
j
2
j
2,3
<
3
3n
can be constructed so that
n,
I,
n
and
R n (x) =y
j
j
j
12
n
Saxena proved that such a polynomial exists uniquely if n is even and for n odd
there does not exist a unique polynomial R (x) satisfying the above conditions.
n
Later Varma [2] obtained the following result in this direction:
1,2
THEOREM
(VARMA). Given a positive integer n and real numbers ak(k
n + I) there is, in general no polynomial, F3n+l(X) of
n + 2), Bk,Yk(k
2,3,
such that F3n+l(Xk)
n + 2, F3n+1(x k)
degree J 3n +
ak; k 1,2
x
oj
C.H. ANDERSON AND J. PRASAD
754
k
2,3
zeros of
+
n
and
x2)Tn(X
(1
F3n+l(Xk)
2,3
k
k;
Tn (x)
where
n
+
provided
are the
XkS
is Tchebycheff polynomial of first kind and -if
there exists such a polynom+/-al then there is an -inf-in-ity of them,
MAIN RESULTS.
In connection with the above results we shall prove the following.
THEOREM 2. For any positive integer n, with
>
>
>
>
-i the zeros of (1
where P (x) is the Legendre polynomial of degree
n
n, there is in general no polynomial R3n+l(X
3n +
of degree
such that, for
2.
I 2
x2)Pn(X
arbitrary real numbers
{a.} n+2
{
3
}n+l
and
j 2
n+l
{Yj}2
n+l n+2
the conditions:
RBn+1(j) aj;
R3n+l( j) Bj;
j
1,2
n
+ l,n + 2,
(2.1)
j
2,3
n
+
(2.2)
R3n+l( j)
j
2,3
n
+
(2.3)
and
yj;
If there does exist such a polynomial then there are infinitely many
are satisfied.
of them.
We’also prove the following result for Tchebycheff nodes:
For any positive integer
THEOREM 3.
>
Xn+ 2
Q3n+3(x)
and
-i
the zeros of
of degree
{yj}n+l
2
_<
n
(x)
3n + 3
2
x
(I
n, with
)Tn(X
x
>
x
2
>
>
x
n
>
x
n+l
there is in general no polynomial
such that for arbitrary real numbers
{aj}+2-_ {j}n+21
the conditions:
Q3n+3(xj) aj;
Q3n+3(xj) j;
j
1,2
n
+ 1,n + 2,
(2.4)
j
1,2
n
+ 1,n + 2
(2.5)
j
2,3
n
+
(2.6)
and
Q3n+B(Xj) yj;
If there does exist such a polynomial then there are infinitely many
are satisfied.
of them.
REMARK I. The comparison of our Theorem 2 with the above mentioned result of
Saxena shows that -if we do not prescribe the third derivative at +/then there does
not exist a unique polynomial regardless whether
is even or odd.
n
In
an earlier
work [3] we have shown that along with the conditions (2.1), (2.2) and (2.3) if we also
prescribe the first derivative at
does not exist.
derivative at
+/-
a unique polynomial of degree
a unique polynomial of degree
<
the nodes of interpolation are different from that of
REMARK 2.
3n + 3
does not exist although
[3].
We shall give here the proof of Theorem 3 only.
2 can be obtained along the same lines.
PROOF OF THEOREM 3. We will show that if all of
e.
O; j
n + l,n + 2,
1,2
j
O; j
1,2
n
+ l,n + 2,
y.
O; j
2,3
n
+
3
<
3n + 3 still
It is also evident from Theorem 3 that even if we prescribe the first
+/-
The proof of Theorem
(2.7)
755
NON-EXISTENCE OF SOME INTERPOLATORY POLYNOMIALS
Q3n+3(x)
then there exists a polynomial
_<
of degree
zero, but satisfies (2.4), (2.5) and (2.6).
3n + 3
which is not identically
The desired result then follows immediately
From the definition of 0 (x) and conditions
(2.4), (2.5) and (2.6), together with the requirements (2.7), it is clear that the
from the theory of linear equations.
desired polynomial must be of the form
where
Q3n+3(x)
An_l(X) is
required
an unknown polynomial of degree
O; for
Q3n+3(xj)
(i
x
2
x2)2Tn2(X)n_ 1(x)
(I
3Xn_l(X)
)*n-l(x)
for unknown real constant
(2.9)
CTn(X
cos 0
x
Letting
c.
Since we have also
i.
n
n + 1, simple calculate_on prove.des
2,3
j
<_
and
n-i
n-] (x)
akcos
k=O
kO
we obtaln
(i
2
x
n-i
)n_l(X)
k=l
akk
sin k0 sin
0.
Thus (2.9) becomes
n-i
c cos
nO
.[k
k=O
From this,
.
sink0 sin 0- 3cos k@ cos 0].
obtain on simplification
n-1
2 c cos nO
k=O
ak[(k-
3)cos(k- 1)0- (k + 3)cos (k + 1)0],
cos kO, for
from which, by collecting the coefficients of
(6a 0 +
-2a
a2)cosO- 4alcos20
n-2
{(k-
+
2)ak+
k=3
1)an_2Cos(n
-(n +
(k +
I)0
2)ak_l}COS
(n +
k0
2)an_iCOS
n0
2c cos nO.
This, in turn, leads to the following system of equations
-2a
0
-(6a 0 +
-4a
O,
(k
2)ak+
+ l)an_ 2
+ 2)an_
-(n
-(n
If
n
O,
a2)
is even, then
a
0
a
2
a
4
(k +
O; k
2)ak_
3,4
n
O,
2c.
........ an_
2
O;
a
0
2,
k
O,l,...,n, we may write
-
C. H. ANDERSON AND J. PRASAD
756
but
an-l-2j
n- 2
k=O
2-
is not necessarily zero.
If
is odd, then
n
a
a
a
3
an_ 2
5
for
j
(n- 4)/2
0,
0,
while
(=___n-ll/?
-2c
a2j -n-
2
II
k=j
2k2k + 3’ j
1,2
with the special case
a
0
-a2/6
which are not necessarily zero.
Hence regardless whether
there does not exist a unique polynomial
Q3n+3(x)
n
of degree
is even or odd, in
_<
3n + 3
general,
satisfying
(2.4), (2.5) and (2.6) and there are infinitely many if they exist.
This completes the proof of Theorem 3. For a complete history on lacunary
interpolation we refer to a paper by J. Balzs [4].
REFERENCES
I.
SAXENA, R.B., On some interpolatory properties of Legendre polynomials III,
Academic Bulgare Des Sciences Bull. De L’Institut__D_e_____M@_th.._ 8 (1964),
2.
VARMA, A.K., Non-existence of interpolatory polynomials, 2ubl. Math. (Debre_c_an) 15
(1968), 75-77.
PRASAD, J. AND ANDERSON, C.H., Some remarks on non-existence of interpolation
polynomials, Math. Notae Vol. 24 (1981-1982), 67-72.
BALIZS, J., Slyozott (0,2) lnterpolacio Ultraszferikus polinomok gySkeim, Magyar
Tud. Akad. Mat. Fiz. Oszt. K6zl. ll (3) (1961), 305-338.
63-94.
3.
4.