In mathematics , the multiple orthogonal polynomials (MOPs) are orthogonal polynomials in one variable that are orthogonal with respect to a finite family of measures . The polynomials are divided into two classes named type 1 and type 2 .[1]
In the literature, MOPs are also called
d
{\displaystyle d}
-orthogonal polynomials , Hermite-Padé polynomials or polyorthogonal polynomials . MOPs should not be confused with multivariate orthogonal polynomials.
Multiple orthogonal polynomials
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Consider a multiindex
n
→
=
(
n
1
,
…
,
n
r
)
∈
N
r
{\displaystyle {\vec {n}}=(n_{1},\dots ,n_{r})\in \mathbb {N} ^{r}}
and
r
{\displaystyle r}
positive measures
μ
1
,
…
,
μ
r
{\displaystyle \mu _{1},\dots ,\mu _{r}}
over the reals. As usual
|
n
→
|
:=
n
1
+
n
2
+
⋯
+
n
r
{\displaystyle |{\vec {n}}|:=n_{1}+n_{2}+\cdots +n_{r}}
.
MOP of type 1
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Polynomials
A
n
→
,
j
{\displaystyle A_{{\vec {n}},j}}
for
j
=
1
,
2
,
…
,
r
{\displaystyle j=1,2,\dots ,r}
are of type 1 if the
j
{\displaystyle j}
-th polynomial
A
n
→
,
j
{\displaystyle A_{{\vec {n}},j}}
has at most degree
n
j
−
1
{\displaystyle n_{j}-1}
such that
∑
j
=
1
r
∫
R
x
k
A
n
→
,
j
d
μ
j
(
x
)
=
0
,
k
=
0
,
1
,
2
,
…
,
|
n
→
|
−
2
,
{\displaystyle \sum \limits _{j=1}^{r}\int _{\mathbb {R} }x^{k}A_{{\vec {n}},j}d\mu _{j}(x)=0,\qquad k=0,1,2,\dots ,|{\vec {n}}|-2,}
and
∑
j
=
1
r
∫
R
x
|
n
→
|
−
1
A
n
→
,
j
d
μ
j
(
x
)
=
1.
{\displaystyle \sum \limits _{j=1}^{r}\int _{\mathbb {R} }x^{|{\vec {n}}|-1}A_{{\vec {n}},j}d\mu _{j}(x)=1.}
[2]
Explanation
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This defines a system of
|
n
→
|
{\displaystyle |{\vec {n}}|}
equations for the
|
n
→
|
{\displaystyle |{\vec {n}}|}
coefficients of the polynomials
A
n
→
,
1
,
A
n
→
,
2
,
…
,
A
n
→
,
r
{\displaystyle A_{{\vec {n}},1},A_{{\vec {n}},2},\dots ,A_{{\vec {n}},r}}
.
MOP of type 2
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A monic polynomial
P
n
→
(
x
)
{\displaystyle P_{\vec {n}}(x)}
is of type 2 if it has degree
|
n
→
|
{\displaystyle |{\vec {n}}|}
such that
∫
R
P
n
→
(
x
)
x
k
d
μ
j
(
x
)
=
0
,
k
=
0
,
1
,
2
,
…
,
n
j
−
1
,
j
=
1
,
…
,
r
.
{\displaystyle \int _{\mathbb {R} }P_{\vec {n}}(x)x^{k}d\mu _{j}(x)=0,\qquad k=0,1,2,\dots ,n_{j}-1,\qquad j=1,\dots ,r.}
[2]
Explanation
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If we write
j
=
1
,
…
,
r
{\displaystyle j=1,\dots ,r}
out, we get the following definition
∫
R
P
n
→
(
x
)
x
k
d
μ
1
(
x
)
=
0
,
k
=
0
,
1
,
2
,
…
,
n
1
−
1
{\displaystyle \int _{\mathbb {R} }P_{\vec {n}}(x)x^{k}d\mu _{1}(x)=0,\qquad k=0,1,2,\dots ,n_{1}-1}
∫
R
P
n
→
(
x
)
x
k
d
μ
2
(
x
)
=
0
,
k
=
0
,
1
,
2
,
…
,
n
2
−
1
{\displaystyle \int _{\mathbb {R} }P_{\vec {n}}(x)x^{k}d\mu _{2}(x)=0,\qquad k=0,1,2,\dots ,n_{2}-1}
⋮
{\displaystyle \vdots }
∫
R
P
n
→
(
x
)
x
k
d
μ
r
(
x
)
=
0
,
k
=
0
,
1
,
2
,
…
,
n
r
−
1
{\displaystyle \int _{\mathbb {R} }P_{\vec {n}}(x)x^{k}d\mu _{r}(x)=0,\qquad k=0,1,2,\dots ,n_{r}-1}
Literature
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Ismail, Mourad E. H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable . Cambridge University Press. pp. 607–647. ISBN 9781107325982 .
López-Lagomasino, G. (2021). An Introduction to Multiple Orthogonal Polynomials and Hermite-Padé Approximation. In: Marcellán, F., Huertas, E.J. (eds) Orthogonal Polynomials: Current Trends and Applications. SEMA SIMAI Springer Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-56190-1_9
References
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^ López-Lagomasino, G. (2021). An Introduction to Multiple Orthogonal Polynomials and Hermite-Padé Approximation. In: Marcellán, F., Huertas, E.J. (eds) Orthogonal Polynomials: Current Trends and Applications. SEMA SIMAI Springer Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-56190-1_9
^ a b Ismail, Mourad E. H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable . Cambridge University Press. pp. 607–608. ISBN 9781107325982 .