A general method of approximation. I
Authors: Wrigge Staffan , Fransén Arne 
Published: June 2010

In this paper we study two families of functions, viz. F and H, and show how to approximate the functions considered in the interval [0,1 ]. 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 equals left-brace f semicolon left-parenthesis i right-parenthesis f left-parenthesis one half plus x right-parenthesis equals f left-parenthesis one half minus x right-parenthesis comma left-parenthesis ii right-parenthesis f left-parenthesis 0 right-parenthesis equals f left-parenthesis 1 right-parenthesis equals 0 comma left-parenthesis iii right-parenthesis f left-parenthesis x right-parenthesis is analytic in a sufficiently large neighborhood of x equals 0 right-brace comma">

F
=
{
f
;
(

i

)

f

(

1
2

+
x

)

=
f

(

1
2

−
x

)

,
(

ii

)

f
(
0
)
=
f
(
1
)
=
0
,
(

iii

)

f
(
x
)

is analytic in a sufficiently large neighborhood of

x
=
0
}
,

F = \{ f;({\text {i}})\,f\left ( {\frac {1}{2} + x} \right ) = f\left ( {\frac {1}{2} - x} \right ),({\text {ii}})\,f(0) = f(1) = 0,({\text {iii}})\;f(x)\;{\text {is analytic in a sufficiently large neighborhood of}}\;x = 0\},

\]

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H equals left-brace h semicolon left-parenthesis j right-parenthesis h left-parenthesis one half plus x right-parenthesis equals minus h left-parenthesis one half minus x right-parenthesis comma left-parenthesis jj right-parenthesis h left-parenthesis 0 right-parenthesis equals h left-parenthesis 1 right-parenthesis equals 0 comma left-parenthesis jjj right-parenthesis h left-parenthesis x right-parenthesis is analytic in a sufficiently large neighborhood of x equals 0 right-brace period">

H
=
{
h
;
(

j

)

h

(

1
2

+
x

)

=
−
h

(

1
2

−
x

)

,
(

jj

)

h
(
0
)
=
h
(
1
)
=
0
,
(

jjj

)

h
(
x
)

is analytic in a sufficiently large neighborhood of

x
=
0
}
.

H = \{ h;({\text {j}})\;h\left ( {\frac {1}{2} + x} \right ) = - h\left ( {\frac {1}{2} - x} \right ),({\text {jj}})\;h(0) = h(1) = 0,({\text {jjj}})\;h(x)\;{\text {is analytic in a sufficiently large neighborhood of}}\;x = 0\}.

\]
The approximations are defined in the interval [0,1 ] by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="min integral Subscript 0 Superscript 1 Baseline left-parenthesis f left-parenthesis x right-parenthesis minus sigma-summation Underscript n equals 1 Overscript k Endscripts c Subscript n comma k Baseline left-bracket x left-parenthesis 1 minus x right-parenthesis right-bracket Superscript n Baseline right-parenthesis squared x Superscript q Baseline left-parenthesis 1 minus x right-parenthesis Superscript q Baseline d x">

min

∫
0
1

(

f
(
x
)
−

∑

n
=
1

k

c

n
,
k

[
x
(
1
−
x
)
]

n

)

2

x
q

(
1
−
x
)

q

d
x

\min \int _0^1 {{{\left ( {f(x) - \sum \limits _{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right )}^2}{x^q}{{(1 - x)}^q}\;dx}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="min integral Subscript 0 Superscript 1 Baseline left-parenthesis h left-parenthesis x right-parenthesis minus left-parenthesis 1 minus 2 x right-parenthesis sigma-summation Underscript n equals 1 Overscript k Endscripts c Subscript n comma k Baseline left-bracket x left-parenthesis 1 minus x right-parenthesis right-bracket Superscript n Baseline right-parenthesis squared x Superscript q Baseline left-parenthesis 1 minus x right-parenthesis Superscript q Baseline d x comma">

min

∫
0
1

(

h
(
x
)
−
(
1
−
2
x
)

∑

n
=
1

k

c

n
,
k

[
x
(
1
−
x
)
]

n

)

2

x
q

(
1
−
x
)

q

d
x

,

\min \int _0^1 {{{\left ( {h(x) - (1 - 2x)\sum \limits _{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right )}^2}{x^q}{{(1 - x)}^q}\;dx} ,

\]
where

q
∈
{
0
,
1
,
2
,
…
}

q \in \{ 0,1,2, \ldots \}

. The associated matrices are analyzed using the theory of orthogonal polynomials, especially the Jacobi polynomials

G
n

(
p
,
q
,
x
)

{G_n}(p,q,x)

. We apply the general theory to the basic trigonometric functions

sin
⁡
(
x
)

\sin (x)

and

cos
⁡
(
x
)

\cos (x)

.