Who proved the Weierstrass approximation theorem?

It was proved by the German mathematician Karl Weierstrass in 1885.

Does the theorem hold for functions of several variables?

The classical theorem is for one variable. However, the Stone–Weierstrass theorem, a major generalization, applies to continuous functions on compact Hausdorff spaces, which includes multivariable cases.

Is the approximation in the theorem pointwise or uniform?

It is a uniform approximation. The polynomial approximates the function closely over the entire interval simultaneously, not just at individual points.

What is a common application of this theorem?

It provides the theoretical foundation for many polynomial-based numerical methods and is a key step in proving other important results in analysis, like the completeness of certain function spaces.

Is “weierstrass approximation theorem” formal?

Weierstrass approximation theorem is usually formal, technical in register.

How do you pronounce “weierstrass approximation theorem”?

Weierstrass approximation theorem: in British English it is pronounced /ˈvaɪəʃtrɑːs əˌprɒksɪˈmeɪʃən ˌθɪərəm/, and in American English it is pronounced /ˈvaɪərstrɑːs əˌprɑːksɪˈmeɪʃən ˌθɪrəm/. Tap the audio buttons above to hear it.

weierstrass approximation theorem: meaning, definition, pronunciation and examples

Specialized

UK/ˈvaɪəʃtrɑːs əˌprɒksɪˈmeɪʃən ˌθɪərəm/US/ˈvaɪərstrɑːs əˌprɑːksɪˈmeɪʃən ˌθɪrəm/

Formal, Technical

My Flashcards

Quick answer

What does “weierstrass approximation theorem” mean?

A fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated as closely as desired by a polynomial function.

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Pronunciation

Definition

Meaning and Definition

A fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated as closely as desired by a polynomial function.

A cornerstone theorem of approximation theory named after Karl Weierstrass, establishing the density of polynomials in the space of continuous functions with the supremum norm (or uniform norm). It guarantees the existence of polynomial approximations for modelling continuous phenomena.

Dialectal Variation

British vs American Usage

Differences

No significant differences in meaning. Spelling of related terms may differ (e.g., 'modelled' vs. 'modeled'). Pronunciation of the name 'Weierstrass' may have slight regional variation.

Connotations

Identical in both regions. It is a standard term in advanced calculus and functional analysis curricula.

Frequency

Used with equal, highly specialized frequency in the global mathematics community.

Grammar

How to Use “weierstrass approximation theorem” in a Sentence

The Weierstrass approximation theorem guarantees [that-clause]According to the Weierstrass approximation theorem, [sentence]One can use the Weierstrass approximation theorem to [infinitive phrase]

Vocabulary

Collocations

strong

prove the Weierstrass approximation theoremapply the Weierstrass approximation theoremgeneralization of the Weierstrass approximation theoremby the Weierstrass approximation theoremcorollary of the Weierstrass approximation theorem

medium

state the Weierstrass approximation theoremuse the Weierstrass approximation theoremconstructive proof of the Weierstrass approximation theoremdensity implied by the Weierstrass approximation theorem

weak

discuss the Weierstrass approximation theoremteach the Weierstrass approximation theoremclassical Weierstrass approximation theorempower of the Weierstrass approximation theorem

Examples

Examples of “weierstrass approximation theorem” in a Sentence

adjective

British English

The proof uses a Weierstrass-approximation-theorem-type argument.

American English

The argument is of a Weierstrass-approximation-theorem nature.

Usage

Meaning in Context

Business

Virtually never used.

Academic

Exclusively used in advanced mathematics, particularly in real analysis, functional analysis, and numerical analysis courses and research.

Everyday

Not used.

Technical

Used in theoretical discussions of function approximation, numerical methods, and proofs about the completeness of function spaces.

Vocabulary

Synonyms of “weierstrass approximation theorem”

Strong

Polynomial approximation theorem (in the context of real analysis)Classical approximation theorem

Neutral

Weierstrass theorem (in this specific context)

Weak

Density theorem for polynomialsStone–Weierstrass theorem (as a generalization)

Vocabulary

Antonyms of “weierstrass approximation theorem”

Counterexample to uniform convergenceNon-approximable function

Watch out

Common Mistakes When Using “weierstrass approximation theorem”

Misspelling: 'Weirstrass', 'Weierstrauss'.
Confusing it with the Bolzano–Weierstrass theorem (which is about sequences).
Incorrectly stating it applies to functions that are differentiable (only continuity is required).

FAQ

Frequently Asked Questions

: It was proved by the German mathematician Karl Weierstrass in 1885.
: The classical theorem is for one variable. However, the Stone–Weierstrass theorem, a major generalization, applies to continuous functions on compact Hausdorff spaces, which includes multivariable cases.
: It is a uniform approximation. The polynomial approximates the function closely over the entire interval simultaneously, not just at individual points.
: It provides the theoretical foundation for many polynomial-based numerical methods and is a key step in proving other important results in analysis, like the completeness of certain function spaces.
: A fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated as closely as desired by a polynomial function.
: Weierstrass approximation theorem is usually formal, technical in register.
: Weierstrass approximation theorem: in British English it is pronounced /ˈvaɪəʃtrɑːs əˌprɒksɪˈmeɪʃən ˌθɪərəm/, and in American English it is pronounced /ˈvaɪərstrɑːs əˌprɑːksɪˈmeɪʃən ˌθɪrəm/. Tap the audio buttons above to hear it.

Learning

Memory Aids

Mnemonic

Think: 'WeierSTRASS can make any smooth line pass through a polynomial path arbitrarily closely.'

Conceptual Metaphor

A powerful net (polynomials) that can catch any continuous fish (function) in a finite pond (closed interval).

Practice

Quiz

Fill in the gap

The guarantees that a continuous function on a closed interval can be uniformly approximated by polynomials.

Multiple Choice

What is the primary requirement on the function for the classical Weierstrass approximation theorem to apply?