Search Results for “universal taylor series”

Source: Universal Taylor series

A universal Taylor series is a formal power series

∑

n
=
1

∞

a

n

x

n

{\displaystyle \sum _{n=1}^{\infty }a_{n}x^{n}}

, such that for every continuous function

h

{\displaystyle h}

on

[
−
1
,
1
]

{\displaystyle [-1,1]}

, if

h
(
0
)
=
0

{\displaystyle h(0)=0}

, then there exists an increasing sequence

(

λ

n

)

{\displaystyle \left(\lambda _{n}\right)}

of positive integers such that

lim

n
→
∞

‖

∑

k
=
1

λ

n

a

k

x

k

−
h
(
x
)

‖

=
0

{\displaystyle \lim _{n\to \infty }\left\|\sum _{k=1}^{\lambda _{n}}a_{k}x^{k}-h(x)\right\|=0}

In other words, the set of partial sums of

∑

n
=
1

∞

a

n

x

n

{\displaystyle \sum _{n=1}^{\infty }a_{n}x^{n}}

is dense (in sup-norm) in

C
[
−
1
,
1

]

0

{\displaystyle C[-1,1]_{0}}

, the set of continuous functions on

[
−
1
,
1
]

{\displaystyle [-1,1]}

that is zero at origin.