Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle
Authors: Ranga A. 
Published: March 2015

The principal objective here is to look at some algebraic properties of the orthogonal polynomials

Ψ
n

(
b
,
s
,
t
)

\Psi _n^{(b,s,t)}

with respect to the Sobolev inner product on the unit circle
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle f comma g mathematical right-angle Subscript upper S Sub Superscript left-parenthesis b comma s comma t right-parenthesis Subscript Baseline equals left-parenthesis 1 minus t right-parenthesis mathematical left-angle f comma g mathematical right-angle Subscript mu Sub Superscript left-parenthesis b right-parenthesis Subscript Baseline plus t ModifyingAbove f left-parenthesis 1 right-parenthesis With bar g left-parenthesis 1 right-parenthesis plus s mathematical left-angle f prime comma g Superscript prime Baseline mathematical right-angle Subscript mu Sub Superscript left-parenthesis b plus 1 right-parenthesis Subscript Baseline comma">

⟨
f
,
g

⟩

S

(
b
,
s
,
t
)

=
(
1
−
t
)

⟨
f
,
g

⟩

μ

(
b
)

+
t

f
(
1
)

¯

g
(
1
)
+
s

⟨

f

′

,

g

′

⟩

μ

(
b
+
1
)

,

\langle f,g\rangle _{S^{(b,s,t)}} = (1-t)\, \langle f,g\rangle _{\mu ^{(b)}} + t\, \overline {f(1)}\,g(1) + s\, \langle f^{\prime },g^{\prime }\rangle _{\mu ^{(b+1)}},

\]
where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle f comma g mathematical right-angle Subscript mu Sub Superscript left-parenthesis b right-parenthesis Subscript Baseline equals StartFraction tau left-parenthesis b right-parenthesis Over 2 pi EndFraction integral Subscript 0 Superscript 2 pi Baseline ModifyingAbove f left-parenthesis e Superscript i theta Baseline right-parenthesis With bar g left-parenthesis e Superscript i theta Baseline right-parenthesis left-parenthesis e Superscript pi minus theta Baseline right-parenthesis Superscript script upper I m left-parenthesis b right-parenthesis Baseline left-parenthesis sine squared left-parenthesis theta slash 2 right-parenthesis right-parenthesis Superscript script upper R e left-parenthesis b right-parenthesis Baseline d theta period">

⟨
f
,
g

⟩

μ

(
b
)

=

τ
(
b
)

2
π

∫

0

2
π

f
(

e

i
θ

)

¯

g
(

e

i
θ

)

(

e

π
−
θ

)

I

m
(
b
)

(

sin

2

⁡
(
θ

/

2
)

)

R

e
(
b
)

d
θ
.

\langle f,g\rangle _{\mu ^{(b)}} = \frac {\tau (b)}{2\pi } \int _{0}^{2\pi }\overline {f(e^{i\theta })} \,g(e^{i\theta })\,(e^{\pi -\theta })^{\mathcal {I}m(b)} (\sin ^{2}(\theta /2))^{\mathcal {R}e(b)} d\theta .

Here,

R

e
(
b
)
>
−
1

/

2

\mathcal {R}e(b) > -1/2

,

0
≤
t
>
1

0 \leq t > 1

,

s
>
0

s > 0

and

τ
(
b
)

\tau (b)

is taken to be such that

⟨
1
,
1

⟩

μ

(
b
)

=
1

\langle 1,1\rangle _{\mu ^{(b)}} = 1

. We show that, for example, the monic Sobolev orthogonal polynomials

Ψ
n

(
b
,
s
,
t
)

\Psi _n^{(b,s,t)}

satisfy the recurrence
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Psi Subscript n Superscript left-parenthesis b comma s comma t right-parenthesis Baseline left-parenthesis z right-parenthesis minus beta Subscript n Superscript left-parenthesis b comma s comma t right-parenthesis Baseline normal upper Psi Subscript n minus 1 Superscript left-parenthesis b comma s comma t right-parenthesis Baseline left-parenthesis z right-parenthesis equals normal upper Phi Subscript n Superscript left-parenthesis b comma t right-parenthesis Baseline left-parenthesis z right-parenthesis comma">

Ψ
n

(
b
,
s
,
t
)

(
z
)
−

β
n

(
b
,
s
,
t
)

Ψ

n
−
1

(
b
,
s
,
t
)

(
z
)
=

Φ
n

(
b
,
t
)

(
z
)
,

\Psi _n^{(b,s,t)}(z) - \beta _n^{(b,s,t)} \Psi _{n-1}^{(b,s,t)}(z) = \Phi _n^{(b,t)}(z),

n
≥
1

n \geq 1

, where

Φ
n

(
b
,
t
)

\Phi _n^{(b,t)}

are the monic orthogonal polynomials with respect to the inner product
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle f comma g mathematical right-angle Subscript mu Sub Superscript left-parenthesis b comma t right-parenthesis Baseline equals left-parenthesis 1 minus t right-parenthesis mathematical left-angle f comma g mathematical right-angle Subscript mu Sub Superscript left-parenthesis b right-parenthesis Baseline plus t ModifyingAbove f left-parenthesis 1 right-parenthesis With bar g left-parenthesis 1 right-parenthesis">

⟨
f
,
g

⟩

μ

(
b
,
t
)

=
(
1
−
t
)

⟨
f
,
g

⟩

μ

(
b
)

+
t

f
(
1
)

¯

g
(
1
)

\langle f,g\rangle _{\mu ^{(b,t)}} = (1-t)\, \langle f,g\rangle _{\mu ^{(b)}} + t\, \overline {f(1)}\,g(1)

. Some related bounds and asymptotic properties are also given.