WEIERSTRASS’ PROOF OF THE WEIERSTRASS
APPROXIMATION THEOREM
ANTON R. SCHEP
At age 70 Weierstrass published the proof of his well-known Approximation
Theorem. In this note we will present a self-contained version, which is essentially
his proof. For a bounded uniformly continuous function f : R → R define for h > 0
Z ∞
u−x 2
1
f (u)e−( h ) du.
Sh f (x) = √
h π −∞
Theorem 1. Let f : R → R be a bounded uniformly continuous function. Then
Sh f converges uniformly to f as h ↓ 0.
Proof. Let > 0. then there exists δ > 0 such that |f (x)−f (y)| < 2 for all x, y ∈ R
R∞
√
2
with |x − y| < δ. Assume |f (x)| ≤ M on R. Using that −∞ e−v dv = π, one
also verifies easily that
Z ∞
u−x 2
1
√
e−( h ) du = 1.
h π −∞
This implies that we can write
Z ∞
u−x 2
1
f (x) = √
f (x)e−( h ) du.
h π −∞
Now let h0 > 0 such that h0 <
√
δ π
2M ,
Z ∞
then
u−x 2
1
|f (u) − f (x)|e−( h ) du
|Sh f (x) − f (x)| ≤ √
h π −∞
Z
u−x 2
1
≤ + √
|f (u) − f (x)|e−( h ) du
2 h π |x−u|≥δ
Z
u−x 2
2M
≤ + √
e−( h ) du
2 h π |x−u|≥δ
Z
Z
2
2
2M
2M h
= + √
e−v dv ≤ √
|v|e−v dv
2
π |v|≥ hδ
δ π |v|≥ hδ
Z ∞
2
4M h
2hM
≤ + √
ve−v dv = + √ < 2
2
δ π 0
δ π
for all 0 < h ≤ h0 and all x ∈ R.
Theorem 2 (Weierstrass Approximation Theorem). Let f : [a, b] → R be a continuous function. Then f is on [a, b] a uniform limit of polynomials.
Proof. We begin by extending f to a bounded uniformly continuous function on R
by defining f (x) = f (a)(x − a + 1) on [a − 1, a), f (x) = −f (b)(x − b − 1) on (b, b + 1],
and f (x) = 0 on R\[a−1, b+1]. In particular there exists R > 0 such that f (x) = 0
Date: May 1, 2007.
1
2
ANTON R. SCHEP
for |x| > R. Let > 0 and M such that |f (x)| ≤ M for all x. Then by the above
theorem there exists h0 > 0 such that for all x ∈ R we have |Sh0 f (x) − f (x)| < 2 .
Since f (u) = 0 for |u| > R, we can write
Z R
u−x 2
1
f (u)e−( h0 ) du.
Sh0 f (x) = √
h0 π −R
2
2R
−v
converges uniformly, so there exists N such
On [ −2R
h0 , h0 ] the power series of e
that
2k N
1
1 X (−1)k u − x
−( u−x
)2
h
0
− √
√ e
<
h0 π
4RM
k!
h0
h0 π
k=0
for all |x| ≤ R and all |u| ≤ R, since in that case |u − x| ≤ 2R. This implies that
2k Z R
N
X
1
(−1)k u − x
f (u)
du <
Sh0 f (x) − √
2
k!
h0
h0 π −R
k=0
for all |x| ≤ R. If we put
P (x) =
1
√
h0 π
Z
R
f (u)
−R
2k
N
X
(−1)k u − x
du,
k!
h0
k=0
then P (x) is a polynomial in x of degree at most 2N such that |Sh0 f (x)−P (x)| < 2
for all |x| ≤ R. This implies that |f (x) − P (x)| < for all x ∈ [a, b].
Remark. The function Sh f is the convolution of f with a Gaussian heat kernel.
These heat kernels form an approximate identity. The following figure shows the
kernel for the values h = 1, h = 41 and h = 81 .
5
h=1/8
2.5
h=1/4
h=1
-5
-4
-3
-2
-1
0
1
2
3
4
References
[1] K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen
einer reellen Veränderlichen, Verl. d. Kgl. Akad. d. Wiss. Berlin 2(1885) 633–639.