Spectacularly large coefficients in Müntz's theorem
Müntz’s theorem asserts, for example, that the linear span of the even powers 1,x2,x4,… is dense in C([0,1]). We show that the associated expansions are so inefficient as to have no conceivable relevance to any actual computation. For example, approximating f(x)=x to accuracy ε=10−6 in this basis...
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| Format: | Journal article |
| Language: | English |
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Springer Nature
2023
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| _version_ | 1826309616355508224 |
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| author | Trefethen, LN |
| author_facet | Trefethen, LN |
| author_sort | Trefethen, LN |
| collection | OXFORD |
| description | Müntz’s theorem asserts, for example, that the linear span of the even powers 1,x2,x4,… is dense in C([0,1]). We show that the associated expansions are so inefficient as to have no conceivable relevance to any actual computation. For example, approximating f(x)=x to accuracy ε=10−6 in this basis requires powers larger than x280,000 and coefficients larger than 10107,000. We present a theorem establishing exponential growth of coefficients with respect to 1/ε
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| first_indexed | 2024-03-07T07:38:26Z |
| format | Journal article |
| id | oxford-uuid:bed5ef9b-6c60-447d-bbe3-c679e747a4e8 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:38:26Z |
| publishDate | 2023 |
| publisher | Springer Nature |
| record_format | dspace |
| spelling | oxford-uuid:bed5ef9b-6c60-447d-bbe3-c679e747a4e82023-03-28T10:47:06ZSpectacularly large coefficients in Müntz's theoremJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:bed5ef9b-6c60-447d-bbe3-c679e747a4e8EnglishSymplectic ElementsSpringer Nature2023Trefethen, LNMüntz’s theorem asserts, for example, that the linear span of the even powers 1,x2,x4,… is dense in C([0,1]). We show that the associated expansions are so inefficient as to have no conceivable relevance to any actual computation. For example, approximating f(x)=x to accuracy ε=10−6 in this basis requires powers larger than x280,000 and coefficients larger than 10107,000. We present a theorem establishing exponential growth of coefficients with respect to 1/ε . |
| spellingShingle | Trefethen, LN Spectacularly large coefficients in Müntz's theorem |
| title | Spectacularly large coefficients in Müntz's theorem |
| title_full | Spectacularly large coefficients in Müntz's theorem |
| title_fullStr | Spectacularly large coefficients in Müntz's theorem |
| title_full_unstemmed | Spectacularly large coefficients in Müntz's theorem |
| title_short | Spectacularly large coefficients in Müntz's theorem |
| title_sort | spectacularly large coefficients in muntz s theorem |
| work_keys_str_mv | AT trefethenln spectacularlylargecoefficientsinmuntzstheorem |