An Approximation Theorem for Continuous Functions

Q: What is a delta sequence, and why does it matter for approximating continuous functions?

A delta sequence $\delta_k$ is a family of functions with three core properties: (1) $\delta_k\ge 0$, (2) $\int_{\mathbb{R}^n} \delta_k(y)\,dy = 1$ for every k, and (3) concentration near 0—given any $\varepsilon>0$, the integral of $\delta_k$ over the complement of the $\varepsilon$-ball around 0 tends to 0 as k $\to\infty$. Convolution $(\delta_k*f)(x)=\int \delta_k(y)f(x-y)dy$ then becomes an average of f near x; as $\delta_k$ concentrates, that average approaches f(x) for continuous f.

Q: How is the smooth bump function constructed so that it has compact support and integrates to 1?

The bump is built from an exponential term that depends on $\|x\|$ and uses a negative exponent of the form $1/(\|x\|^2-1)$. This makes the function smoothly decay to 0 outside the unit ball (where $\|x\|\ge 1$) while staying $C^\infty$ inside. A positive constant C is then chosen so the total integral over $\mathbb{R}^n$ equals 1, avoiding repeated explicit integral computations for each dimension.

Q: What scaling turns the bump into a delta sequence $\delta_k$?

The scaling shrinks the support and preserves normalization. The construction scales the input by k (so the effective radius becomes $1/k$) and scales the peak accordingly so that $\int \delta_k = 1$ remains true for every k. As k increases, the bump becomes taller and narrower, concentrating its mass near the origin.

Q: How does the proof split the convolution integral, and what kills the “far” part?

After rewriting $|f(x)-(\delta_k*f)(x)|$ as an integral of $|f(x)-f(x-y)|$ against $\delta_k(y)$, the integral is split into $\|y\|<\delta/2$ and $\|y\|\ge \delta/2$. The near part is bounded by $\varepsilon$ using uniform continuity. For the explicit bump-based delta sequence, the support of $\delta_k$ eventually lies entirely inside the $\delta/2$-ball, so for sufficiently large k the far part becomes exactly 0.

Q: What does the final convergence statement mean in practice?

The theorem concludes that $\sup_{x\in A}|(\delta_k*f)(x)-f(x)|\to 0$ as k $\to\infty$. In other words, the smooth functions $\delta_k*f$ approximate f uniformly on A, so the approximation error does not depend on the particular point x within the compact set.

TL;DR

Construct a $C^{\infty}$ compactly supported bump function using an exponential term that becomes 0 outside a ball, then normalize it so its integral equals 1.

Briefing Cornell Notes

Briefing

Continuous functions on n can be uniformly approximated on any compact set by smooth (-infinity) functions via convolution with a carefully constructed “delta sequence.” The key insight is that for a continuous function f and a compact set A, the quantity $sup_{x \in A} ∣ (δ_{k} * f) (x) - f (x) ∣$ can be forced below any prescribed $ε > 0$ by taking k large enough. That yields uniform convergence on A, which is exactly what makes the approximation theorem useful in analysis and PDEs.

The construction starts with a standard smooth “bump” function $E_{A}$ on $R^{n}$ that is $C^{\infty}$ and has compact support: it is positive on a bounded region and smoothly decays to 0 at the boundary. The bump is built from an exponential term that depends on $∥ x ∥$ : using a negative exponent of the form $1/ (∥ x ∥^{2} - 1)$ ensures the function becomes 0 outside the unit ball while remaining smooth inside. A normalization constant C is chosen so that the integral of $E_{A}$ over $R^{n}$ equals 1.

From this bump, a delta sequence $δ_{k}$ is defined by scaling both the height and the input: the support shrinks to a ball of radius $1/ k$ , while the normalization is preserved so that $\int_{R^{n}} δ_{k} = 1$ for every k. The sequence is characterized by three properties: (1) non-negativity, (2) total mass 1, and (3) concentration near the origin—outside any fixed $ε$ -ball around 0, the integral of $δ_{k}$ tends to 0 as k $\to \infty$ . This concentration property is what turns convolution into an “averaging” operator that becomes indistinguishable from pointwise evaluation for continuous functions.

With $δ_{k}$ in hand, the approximation uses convolution: $(δ_{k} * f) (x) = \int_{R^{n}} δ_{k} (y) f (x - y) d y$ . On a compact set A, continuity of f implies uniform continuity. That uniform continuity provides a global $δ$ such that whenever $∥ y ∥ < δ$ , the difference $∣ f (x) - f (x - y) ∣$ stays below $ε$ , uniformly for all $x \in A$ .

The proof then estimates $∣ f (x) - (δ_{k} * f) (x) ∣$ by inserting the integral formula and using the fact that $\int δ_{k} = 1$ to rewrite the difference as an integral of $f (x) - f (x - y)$ . The non-negativity of $δ_{k}$ allows the absolute value to be handled cleanly, and the concentration property splits the integral into two regions: where $∥ y ∥ < δ /2$ (controlled by uniform continuity) and where $∥ y ∥ \geq δ /2$ (controlled by the delta sequence’s vanishing outside small balls). For the explicit bump-based delta sequence, the support eventually lies entirely inside the chosen ball, making the “far” integral exactly 0 for sufficiently large k.

As a result, $sup_{x \in A} ∣ (δ_{k} * f) (x) - f (x) ∣ \to 0$ , giving uniform approximation of continuous functions on compact domains by $C^{\infty}$ functions. The theorem formalizes a practical principle: smooth mollifiers can replace rough functions without losing control on compact sets.

Cornell Notes

A continuous function f on $R^{n}$ can be uniformly approximated on any compact set A by smooth functions of the form $δ_{k} * f$ . The method relies on building a “delta sequence” $δ_{k}$ : nonnegative, normalized so $\int δ_{k} = 1$ , and concentrating near 0 so that mass outside any $ε$ -ball goes to 0 as k $\to \infty$ . Convolution with $δ_{k}$ produces $C^{\infty}$ functions, and uniform continuity of f on compact A ensures that $f (x - y)$ stays close to f(x) when $∥ y ∥$ is small. Splitting the convolution integral into “near” and “far” regions shows the difference $sup_{x \in A} ∣ (δ_{k} * f) (x) - f (x) ∣$ can be made smaller than any $ε$ by choosing k large enough.

What is a delta sequence, and why does it matter for approximating continuous functions?

A delta sequence

δ_{k}

is a family of functions with three core properties: (1)

δ_{k} \geq 0

, (2)

\int_{R^{n}} δ_{k} (y) d y = 1

for every k, and (3) concentration near 0—given any

ε > 0

, the integral of

δ_{k}

over the complement of the

ε

-ball around 0 tends to 0 as k

\to \infty

. Convolution

(δ_{k} * f) (x) = \int δ_{k} (y) f (x - y) d y

then becomes an average of f near x; as

δ_{k}

concentrates, that average approaches f(x) for continuous f.

How is the smooth bump function constructed so that it has compact support and integrates to 1?

The bump is built from an exponential term that depends on

∥ x ∥

and uses a negative exponent of the form

1/ (∥ x ∥^{2} - 1)

. This makes the function smoothly decay to 0 outside the unit ball (where

∥ x ∥ \geq 1

) while staying

C^{\infty}

inside. A positive constant C is then chosen so the total integral over

R^{n}

equals 1, avoiding repeated explicit integral computations for each dimension.

What scaling turns the bump into a delta sequence $δ_{k}$ ?

The scaling shrinks the support and preserves normalization. The construction scales the input by k (so the effective radius becomes

1/ k

) and scales the peak accordingly so that

\int δ_{k} = 1

remains true for every k. As k increases, the bump becomes taller and narrower, concentrating its mass near the origin.

Why does uniform continuity on a compact set drive the main estimate?

On a compact set A, continuity of f implies uniform continuity. That means for any

ε > 0

, there exists a single

δ > 0

such that for all

x \in A

and all y with

∥ y ∥ < δ

, one has

∣ f (x) - f (x - y) ∣ < ε

. This uniform bound controls the convolution error on the “near” region where

δ_{k}

has its mass.

How does the proof split the convolution integral, and what kills the “far” part?

After rewriting

∣ f (x) - (δ_{k} * f) (x) ∣

as an integral of

∣ f (x) - f (x - y) ∣

against

δ_{k} (y)

, the integral is split into

∥ y ∥ < δ /2

and

∥ y ∥ \geq δ /2

. The near part is bounded by

ε

using uniform continuity. For the explicit bump-based delta sequence, the support of

δ_{k}

eventually lies entirely inside the

δ /2

-ball, so for sufficiently large k the far part becomes exactly 0.

What does the final convergence statement mean in practice?

The theorem concludes that

sup_{x \in A} ∣ (δ_{k} * f) (x) - f (x) ∣ \to 0

as k

\to \infty

. In other words, the smooth functions

δ_{k} * f

approximate f uniformly on A, so the approximation error does not depend on the particular point x within the compact set.

Review Questions

What three properties define a delta sequence, and how does each property enter the convolution error estimate?
Where exactly does uniform continuity of f on a compact set get used in the proof?
Why does choosing a delta sequence with compactly supported $δ_{k}$ make the “far” integral vanish for large k?

Key Points

1
Construct a $C^{\infty}$ compactly supported bump function using an exponential term that becomes 0 outside a ball, then normalize it so its integral equals 1.
2
Define $δ_{k}$ by scaling the bump so its support shrinks to radius $1/ k$ while keeping $\int δ_{k} = 1$ .
3
Use convolution $(δ_{k} * f) (x) = \int δ_{k} (y) f (x - y) d y$ to produce smooth approximants of f.
4
On a compact set A, continuity of f implies uniform continuity, giving a single $δ$ that controls $∣ f (x) - f (x - y) ∣$ for all $x \in A$ .
5
Rewrite $f (x) - (δ_{k} * f) (x)$ using $\int δ_{k} = 1$ so the error becomes an integral of $f (x) - f (x - y)$ .
6
Split the integral into near and far regions relative to $δ /2$ ; the near region is bounded by $ε$ , and the far region is controlled by the delta sequence’s concentration.
7
For the explicit compactly supported delta sequence, the far region integral becomes exactly 0 once k is large enough, yielding uniform convergence on A.

Highlights

A normalized, compactly supported smooth bump can be scaled into a delta sequence whose mass concentrates near 0 as k grows.

Uniform continuity on compact sets turns small ∥y∥ into a uniform bound on ∣f(x)−f(x−y)∣.

Splitting the convolution integral into near and far regions shows the approximation error can be forced below any ε.

With compact support, the “far” part of the error can vanish exactly for sufficiently large k, not just asymptotically. 

Topics

Approximation Theorem
Delta Sequence
Mollifiers
Uniform Convergence
Convolution