Least squares approximation with constraints
Authors: Milovanović Gradimir V. , Wrigge Staffan 
Published: June 2010

In this paper we study two families of functions

F
e

{F_e}

and

F
o

{F_o}

, and show how to approximate the functions in the interval

[
−
1
,
1
]

[ - 1,1]

. The functions are assumed to be real when the argument is real. We define
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript e Baseline equals left-brace f colon f left-parenthesis negative x right-parenthesis equals f left-parenthesis x right-parenthesis comma f left-parenthesis 1 right-parenthesis equals 0 comma f element-of upper L squared left-bracket negative 1 comma 1 right-bracket right-brace">

F
e

=
{
f
:
f
(
−
x
)
=
f
(
x
)
,
f
(
1
)
=
0
,
f
∈

L
2

[
−
1
,
1
]
}

{F_e} = \{ f:f( - x) = f(x),f(1) = 0,f \in {L^2}[ - 1,1]\}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript o Baseline equals left-brace f colon f left-parenthesis negative x right-parenthesis equals minus f left-parenthesis x right-parenthesis comma f left-parenthesis 1 right-parenthesis equals 0 comma f element-of upper L squared left-bracket negative 1 comma 1 right-bracket right-brace period">

F
o

=
{
f
:
f
(
−
x
)
=
−
f
(
x
)
,
f
(
1
)
=
0
,
f
∈

L
2

[
−
1
,
1
]
}
.

{F_o} = \{ f:f( - x) = - f(x),f(1) = 0,f \in {L^2}[ - 1,1]\} .

\]
Let further

P

m

{\mathcal {P}_m}

be the set of all real polynomials of degree not higher than m such that the polynomials belong to the set

F
e

{F_e}

if m is even and to the set

F
o

{F_o}

if m is odd. We determine the least squares approximation for the function

f
∈

F
e

f \in {F_e}

(or

F
o

{F_o}

) in the class

P

2
n

{\mathcal {P}_{2n}}

(or

P

2
n
+
1

{\mathcal {P}_{2n + 1}}

), with respect to the norm

‖
f
‖

=

(
(
f
,
f
)

)

1

/

2

\left \| f \right \| = {((f,f))^{1/2}}

, where the inner product is defined by

(
f
,
g
)
=

∫

−
1

1

f
(
x
)
g
(
x
)
w
(
x
)
d
x

(f,g) = \smallint _{ - 1}^1f(x)g(x)w(x)dx

, with

f
,
g
∈

L
2

[
−
1
,
1
]
=

L
w
2

[
−
1
,
1
]

f,g \in {L^2}[ - 1,1] = L_w^2[ - 1,1]

and

w
(
x
)
=

(
1
−

x
2

)

λ
−
1

/

2

w(x) = {(1 - {x^2})^{\lambda - 1/2}}

. We also consider the general case when f is neither an even nor an odd function but

f
∈

L
2

[
−
1
,
1
]

f \in {L^2}[ - 1,1]

and

f
(
−
1
)
=
f
(
1
)
=
0

f( - 1) = f(1) = 0

. Using the theory of Gegenbauer polynomials we obtain the approximating polynomials in the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Subscript 2 n Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript k equals 1 Overscript n Endscripts d Subscript n comma k Baseline left-parenthesis 1 minus x squared right-parenthesis Superscript k Baseline when f element-of upper F Subscript e Baseline">

ϕ

2
n

(
x
)
=

∑

k
=
1

n

d

n
,
k

(
1
−

x
2

)

k

when

f
∈

F
e

{\phi _{2n}}(x) = \sum \limits _{k = 1}^n {{d_{n,k}}{{(1 - {x^2})}^k}\;{\text {when}}\,f \in {F_e}}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Subscript 2 n plus 1 Baseline left-parenthesis x right-parenthesis equals x sigma-summation Underscript k equals 1 Overscript n Endscripts e Subscript n comma k Baseline left-parenthesis 1 minus x squared right-parenthesis Superscript k Baseline when f element-of upper F Subscript o Baseline period">

ϕ

2
n
+
1

(
x
)
=
x

∑

k
=
1

n

e

n
,
k

(
1
−

x
2

)

k

when

f
∈

F
o

.

{\phi _{2n + 1}}(x) = x\sum \limits _{k = 1}^n {{e_{n,k}}{{(1 - {x^2})}^k}\;{\text {when}}\,f \in {F_o}.}

\]
We apply the general theory to the functions

f
(
x
)
=
cos
⁡
(
π
x

/

2
)

f(x) = \cos (\pi x/2)

and

f
(
x
)
=

J
0

(

a
0

x
)

f(x) = {J_0}({a_0}x)

, where

a
0

=
{
min
x
>
0
:

J
0

(
x
)
=
0
}

{a_0} = \{ \min x > 0:{J_0}(x) = 0\}

.