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Difference between revisions of "Equiconvergent series"

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Convergent or divergent series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e0360001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e0360002.png" /> whose difference is a convergent series with zero sum: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e0360003.png" />. If their difference is only a convergent series, then the series are called equiconvergent in the wide sense.
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Convergent or divergent series $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ whose difference is a convergent series with zero sum: $\sum_{n=1}^\infty(a_n-b_n)=0$. If their difference is only a convergent series, then the series are called equiconvergent in the wide sense.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e0360004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e0360005.png" /> are functions, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e0360006.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e0360007.png" /> is any set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e0360008.png" /> is the set of real numbers, then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e0360009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e03600010.png" /> are called uniformly equiconvergent (uniformly equiconvergent in the wide sense) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e03600011.png" /> if their difference is a series that is uniformly convergent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e03600012.png" /> with sum zero (respectively, only uniformly convergent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e03600013.png" />).
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If $a_n=a_n(x)$ and $b_n=b_n(x)$ are functions, for example, $a_n,b_n\colon X\to\mathbf R$, where $X$ is any set and $\mathbf R$ is the set of real numbers, then the series $\sum_{n=1}^\infty a_n(x)$ and $\sum_{n=1}^\infty b_n(x)$ are called uniformly equiconvergent (uniformly equiconvergent in the wide sense) on $X$ if their difference is a series that is uniformly convergent on $X$ with sum zero (respectively, only uniformly convergent on $X$).
  
Example. If two integrable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e03600014.png" /> are equal on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e03600015.png" />, then their Fourier series are uniformly equiconvergent on every interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e03600016.png" /> interior to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e03600017.png" />, and the conjugate Fourier series are uniformly equiconvergent in the wide sense on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036000/e03600018.png" />.
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Example. If two integrable functions on $[-\pi,\pi]$ are equal on an interval $I\subset[-\pi,\pi]$, then their Fourier series are uniformly equiconvergent on every interval $I^*$ interior to $I$, and the conjugate Fourier series are uniformly equiconvergent in the wide sense on $I^*$.

Latest revision as of 18:36, 21 November 2018

Convergent or divergent series $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ whose difference is a convergent series with zero sum: $\sum_{n=1}^\infty(a_n-b_n)=0$. If their difference is only a convergent series, then the series are called equiconvergent in the wide sense.

If $a_n=a_n(x)$ and $b_n=b_n(x)$ are functions, for example, $a_n,b_n\colon X\to\mathbf R$, where $X$ is any set and $\mathbf R$ is the set of real numbers, then the series $\sum_{n=1}^\infty a_n(x)$ and $\sum_{n=1}^\infty b_n(x)$ are called uniformly equiconvergent (uniformly equiconvergent in the wide sense) on $X$ if their difference is a series that is uniformly convergent on $X$ with sum zero (respectively, only uniformly convergent on $X$).

Example. If two integrable functions on $[-\pi,\pi]$ are equal on an interval $I\subset[-\pi,\pi]$, then their Fourier series are uniformly equiconvergent on every interval $I^*$ interior to $I$, and the conjugate Fourier series are uniformly equiconvergent in the wide sense on $I^*$.

How to Cite This Entry:
Equiconvergent series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equiconvergent_series&oldid=18128
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article