Math 2.0

statement theorem

Weierstrass approximation theorem

Statement

Let a<ba < b and let f ⁣:[a,b]→Rf \colon [a, b] \to \mathbb{R} be continuous. Then for every ε>0\varepsilon > 0 there is a polynomial pp with ∣f(x)−p(x)∣<ε|f(x) - p(x)| < \varepsilon for all x∈[a,b]x \in [a, b].

Remarks

Equivalently, there is a sequence of polynomials that converges uniformly to ff on [a,b][a, b]. One proof uses the Bernstein polynomials on [0,1][0, 1], together with uniform continuity of ff. The Stone–Weierstrass theorem generalizes the result to compact spaces. Mathlib states the theorem as polynomialFunctions_closure_eq_top.

History

Karl Weierstrass proved the theorem in 1885.

Explanations

L1 human

Every continuous function on a closed bounded interval is uniformly close to a polynomial. So polynomials can imitate any continuous graph as closely as we like.

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To read this node, read these 22 nodes in this order. The foundations come first, and the list ends with this node.

Bernstein polynomial, definition, no evidence levelBernstein polynomialdefinitionLimit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementPointwise convergence, definition, no evidence levelPointwise convergencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementUniform continuity, definition, no evidence levelUniform continuitydefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementUniform convergence, definition, no evidence levelUniform convergencedefinitionUniform continuity on a set, definition, no evidence levelUniform continuity on…definitionContinuous functions on a closed bounded interval are uniformly continuous, statement, no evidence levelContinuous functions …statementWeierstrass approximation theorem, statement, no evidence levelWeierstrass approxima…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Bernstein polynomial definition
  2. Limit point of a set definition
  3. Sequence definition
  4. Limit of a function definition
  5. Real sequence definition
  6. Subsequence definition
  7. Continuity at a point definition
  8. Bounded sequence definition
  9. Limit of a sequence definition
  10. Monotone sequence definition
  11. Continuous function definition
  12. Epsilon-delta definition of continuity statement
  13. Convergent sequence definition
  14. Monotone convergence theorem for sequences statement
  15. Pointwise convergence definition
  16. Every sequence has a monotone subsequence statement
  17. Uniform continuity definition
  18. Bolzano–Weierstrass theorem statement
  19. Uniform convergence definition
  20. Uniform continuity on a set definition
  21. Continuous functions on a closed bounded interval are uniformly continuous statement
  22. Weierstrass approximation theorem statement

Sources

  1. Web https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem, Weierstrass approximation theorem

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