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Polynomial Approximation

by Heinrich Hartmann / August 11, 2025 / Stemwede

Polynomial approximation is a classical topic in analysis, with results spanning from the 19th century Weierstrass theorem to modern generalizations in Sobolev and smooth function spaces.

• Weierstrass (1885): Every continuous function on a closed interval can be uniformly approximated by polynomials [Weierstrass, 1885].
• Stone–Weierstrass theorem (1948): Extends Weierstrass to continuous functions on compact Hausdorff spaces [Stone, 1948].
• Nachbin’s theorem (1965): Characterizes polynomial density in \(C^k\) spaces on manifolds [Nachbin, 1965].
• Wannebo (2004): Discusses polynomial approximation in Sobolev spaces and the role of boundary regularity [Wannebo, 2004].

In this note, we give a number of polynomial density in a variety of topological vector spaces over a bounded domain \(\Omega \subset \IR^n\). We opt to give independent Fourier-analytic proofs that avoid reduction to any of the results cited above.

We also present a counterexample showing that regularity of \(\partial \Omega\) is essential for uniform \(C^k\) approximation, even in \(n=1,k=0\) case.

Theorem (Polynomial Density). Let \(\mathcal{P} = \IR[x^1,\dots,x^n]\) be the space of real polynomials on \(\IR^n\), and let \(\Omega \subset \IR^n\) be a bounded open domain with Lipschitz boundary. Then \(\mathcal{P}\) is dense in each of the following topological vector spaces:

1. \(L^2(\Omega)\) with inner product $$ \langle f,g \rangle_{L^2} = \int_\Omega f(x) g(x) \, dx $$

2. \(H^s(\Omega)\), \(s \in \IN\), with inner product $$ \langle f,g \rangle_{H^s} = \sum_{|\alpha| \le s} \int_\Omega \del^\alpha f(x) \del^\alpha g(x) \, dx $$

3. \(C^k(\Omega)\), \(k \in \IN\), with the compact–open \(C^k\) topology $$ \rho_{K,k}(f) = \max_{|\alpha| \le k} \ \sup_{x \in K} |\del^\alpha f(x)|, \quad K \Subset \Omega $$

4. \(C^k(\overline{\Omega})\) with the Banach norm $$ |f|{C^k(\overline{\Omega})} = \max |\del^\alpha f(x)| $$} \ \sup_{x \in \overline{\Omega}

5. \(C^\infty(\Omega) = \varprojlim_{k \in \IN} C^k(\Omega)\) with the inverse limit topology

6. \(C^\infty(\overline{\Omega}) = \varprojlim_{k \in \IN} C^k(\overline{\Omega})\) with the inverse limit topology

7. \(\mathcal{E}_0 = \varinjlim_{0 \in U \subset \IR^n} C^\infty(U)\), the space of smooth germs at \(0\), with the direct limit topology

Proof

1. \(L^2(\Omega)\)

Fourier–Paley–Wiener approach:

1. If \(a \in L^2(\Omega)\) is orthogonal to all \(p \in \mathcal{P}\), then \(\int_\Omega a(x) x^\alpha dx = 0\) for all multiindices \(\alpha\).
2. Extend \(a\) to \(A \in L^2(\IR^n)\) by zero outside \(\Omega\).
3. The Fourier transform \(\widehat{A}\) is entire (Paley–Wiener theorem, e.g. [Hörmander, Thm. 7.3.1]) since \(A\) has compact support.
4. Orthogonality to \(x^\alpha\) means \(\partial^\alpha \widehat{A}(0) = 0\) for all \(\alpha\).
5. An entire function with all derivatives zero at a point is identically zero; hence \(\widehat{A} \equiv 0\), \(A = 0\), \(a = 0\).

Thus \(\mathcal{P}^\perp = \{0\}\) and \(\overline{\mathcal{P}}^{L^2} = L^2(\Omega)\).

2. \(H^s(\Omega)\)

We avoid reduction to \(C^0\) by using Fourier truncation:

1. Let \(u \in H^s(\Omega)\), extend by a bounded linear extension \(E\) to \(H^s(\IR^n)\) (possible for Lipschitz \(\partial\Omega\); see [Adams–Fournier, Thm. 4.32]).
2. Let \(\chi_R(\xi)\) be the indicator of the ball \(B_R(0)\) in \(\IR^n\). Define \(u_R := \mathcal{F}^{-1}[\chi_R \widehat{Eu}]\).
3. Then \(u_R \in C^\infty(\IR^n)\) and \(u_R \to Eu\) in \(H^s\) as \(R \to \infty\).
4. Since \(u_R\) is real-analytic (compact Fourier support \(\implies\) entire), truncate its Taylor series to degree \(N\) to obtain a polynomial \(p_{R,N}\) approximating \(u_R\) in \(H^s\).
5. Choosing \(R, N\) large in succession gives \(p_{R,N} \to u\) in \(H^s(\Omega)\).

3. \(C^k(\Omega)\) (compact–open)

Pick \(s > k + n/2\). Sobolev embedding \(H^s(\Omega) \hookrightarrow C^k(\Omega)\) is continuous on compacts ([Adams–Fournier, Thm. 4.12]). Approximate \(f\) in \(H^s\) by polynomials and use the embedding to get convergence in the \(C^k\) seminorm on each \(K \Subset \Omega\).

4. \(C^k(\overline{\Omega})\)

If \(\partial\Omega\) is \(C^\infty\) (or \(C^{k,1}\) for finite \(k\)), there is a bounded extension operator \(E: C^k(\overline{\Omega}) \to C^k(\IR^n)\). Approximate \(Ef\) uniformly in \(C^k\) on a large compact by polynomials (via \(H^s\) embedding as above), then restrict to \(\overline{\Omega}\).

5. \(C^\infty(\Omega)\)

Inverse limit \(\varprojlim_k C^k(\Omega)\). Density holds in each \(C^k(\Omega)\) by Step 3; density passes to the inverse limit by definition of the topology.

6. \(C^\infty(\overline{\Omega})\)

Analogous to Step 5, using Step 4 at each finite \(k\).

7. \(\mathcal{E}_0\) (smooth germs)

Direct limit \(\varinjlim_{0 \in U} C^\infty(U)\). Given \(f \in C^\infty(U)\), pick a ball \(B_r(0) \subset U\). Approximate in \(C^\infty(B_r(0))\) by polynomials, then view the result as a germ. Density follows from Step 5.

Counterexample (Role of Boundary Regularity). Let \(\Omega\) be the complement of the Cantor set on \([0,1]\). There is a \(C^\infty\) (locally constant) function \(f\) with \(\sup_{x\in\Omega} |p(x)-f(x)| > 1/2\) for all polynomials \(p \in \IR[x]\).

Construction

\(\Omega\) is a countable union of disjoint open intervals constructed as follows:

• Let \(A_0=\{[0,1]\}\), \(B_0=\varnothing\).
• For a closed interval \(I=[a,b]\) let \((l,m,r) = \text{Split}(I)\) be the split operator that divides an interval into closed intervals \(l,r\) and an open middle \(m\).
• Given \(A_k\), set
  • \(A_{k+1} := \{l,r \mid l,\_,r = \text{Split}(I), I \in A_k\}\)
  • \(B_{k+1} := B_k \cup \{m \mid \_,m,\_ = \text{Split}(I), I \in A_k\}\)

Now \(B = \bigcup_k B_k\) is a set of disjoint open intervals. Let \(\Omega = \bigcup_{m \in B} m\), and for \(x \in \Omega\) let \(g(x) := \min\{k \mid x \in m \text{ for some } m \in B_k\}\).

Now \(f(x):=(-1)^{g(x)}\) on \(\Omega\) is a locally constant function that alternates sign on successive gaps in every level.

Let \(p\) be a polynomial. If \(|p(x)-f(x)| < 1/2\), then \(p\) must change sign infinitely often in \([0,1]\). Since polynomials have finitely many zeros, this is impossible.

References

• Weierstrass, K. (1885). Zur Theorie der eindeutigen analytischen Funktionen. Mathematische Werke, 2.
• Stone, M.H. (1948). The generalized Weierstrass approximation theorem. Mathematics Magazine, 21(5), 237–254.
• Nachbin, L. (1965). Sur les algèbres denses de fonctions différentiables sur une variété. Comptes Rendus de l'Académie des Sciences de Paris, 260, 1658–1660.
• Wannebo, A. (2004). Polynomial approximation in Sobolev spaces. Journal of Approximation Theory, 128(1), 1–10.
• Hörmander, L. (1983). The Analysis of Linear Partial Differential Operators I. Springer.
• Adams, R.A., Fournier, J.J.F. (2003). Sobolev Spaces. 2nd ed., Academic Press.