Weierstrass Approximation Theorem¶

Theorem¶

For a continuous function $f\colon [a, b] \to \mathbb{R}$ and any $\varepsilon > 0\text{,}$ there exists a polynomial $P\colon [a, b]\to\mathbb{R}$ such that $\|P-f\|_\infty < \varepsilon\text{.}$

Interpretation¶

Any continuous function on a closed bounded interval can be approximated as closely as desired by a polynomial, uniformly on the whole interval: for each tolerance $\varepsilon$ there is a polynomial whose graph stays within $\varepsilon$ of the graph of $f$ at every point of $[a,b]$.

The figure approximates $f(x)=|x|$ on $[-1,1]$, which has a corner at $0$ and is not differentiable there, by even polynomials $p_2$, $p_4$ and $p_8$. Their largest errors $\|f-p_n\|_\infty$ in the uniform norm are about $0.19$, $0.12$ and $0.07$, and in each case the largest error sits at the corner. The errors shrink as the degree grows, which is what the theorem promises.

Both hypotheses are needed. A polynomial is continuous, and a uniform limit of continuous functions is continuous, so a function with a jump cannot be approximated uniformly by polynomials. The interval must be closed and bounded as well: every polynomial is bounded on $(0,1]$, so none stays within $\varepsilon$ of the unbounded function $1/x$ there, and on $\mathbb{R}$ even $\mathrm{e}^x$ cannot be approximated, as in the Examples.

The approximating polynomials are not partial sums of one power series. The sum of a power series is differentiable inside its interval of convergence, so $|x|$ is not such a sum around $0$, yet it is a uniform limit of polynomials; in the figure even the constant term changes from one $p_n$ to the next. The theorem guarantees existence only; Bernstein polynomials, used in the interactive scene, are one explicit construction. It differs from the Taylor polynomials met later, which are built from derivatives at a single point.

Key Results¶

• Equivalently, every continuous function $f$ on $[a,b]$ is a uniform limit of polynomials: there are polynomials $P_n$ with $\|P_n-f\|_\infty\to0$ in the uniform norm.

Examples¶

• On $[0,1]$, the polynomials $\displaystyle P_n(x)=\sum_{k=0}^{n}x^k/k!$ approximate $f(x)=\mathrm{e}^x$ uniformly: $\displaystyle \mathrm{e}^x-P_n(x)=\sum_{k>n}x^k/k!$ is largest at $x=1$$1$, so $\displaystyle \|f-P_n\|_\infty=\sum_{k>n}1/k!\to0$$0$. For $n=3$ the error is $\mathrm{e}-\tfrac{8}{3}\approx0.052$$0.052$.
• Continuous functions, e.g., $f(x)=\mathrm{e}^x$${\mathrm{e}}^x$, over the entirety of $\mathbb{R}$ aren't always uniformly approximable by polynomials.

Prerequisites¶

• Continuity
• Polynomial
• Uniform Norm

Related Concepts¶

• Power Series
• Taylor Polynomial

Sources¶

• Math III: Analysis · Lecture 2

Symbols: epsilon, infinity, real numbers, sum

Last updated: 20 October 2026 03:02