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''of a complex sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s0911101.png" /> of numbers or functions (or of a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s0911102.png" /> with partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s0911103.png" />) to a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s0911104.png" />''
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Summability by a method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s0911105.png" /> (cf. [[Summation methods|Summation methods]]) such that for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s0911106.png" />:
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''of a complex sequence  $  \{ S _ {n} \} $
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of numbers or functions (or of a series  $  \sum _ {k=1}  ^  \infty  a _ {k} $
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with partial sums  $  S _ {n} $)
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to a number  $  S $''
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Summability by a method $  A = | a _ {nk} | $(
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cf. [[Summation methods|Summation methods]]) such that for a certain $  p > 0 $:
  
 
1) the sequence
 
1) the sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s0911107.png" /></td> </tr></table>
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$$
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\sigma _ {n}  = \sum_{k=1} ^  \infty  a _ {nk} | S _ {k} - S |  ^ {p}
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$$
  
converges for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s0911108.png" />, and for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s0911109.png" /> in the case of a sequence of functions;
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converges for every $  n > 1 $,  
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and for almost all $  x $
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in the case of a sequence of functions;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111010.png" />. By retaining 2) and replacing 1) by:
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2) $  \lim\limits _ {n \rightarrow \infty }  \sigma _ {n} = 0 $.  
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By retaining 2) and replacing 1) by:
  
1') for every monotone increasing sequence of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111011.png" />, the sequence
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1') for every monotone increasing sequence of indices $  \{ v _ {k} \} $,  
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the sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111012.png" /></td> </tr></table>
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$$
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\sigma _ {n}  = \sum_{k=1}^  \infty  a _ {nv _ {k}  } | S _ {v _ {k}  } - S |
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^ {p}
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$$
  
converges for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111013.png" />, and for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111014.png" /> in the case of a sequence of functions, one arrives at the concept of very strong summability.
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converges for every $  n > 1 $,  
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and for almost all $  x $
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in the case of a sequence of functions, one arrives at the concept of very strong summability.
  
The concept of strong summability was introduced in connection with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111015.png" />-summability of Fourier series (cf. [[Summation of Fourier series|Summation of Fourier series]]). The importance of this concept is well illustrated in the example of strong <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111016.png" />-summability. Strong <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111017.png" />-summability signifies that the partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111018.png" /> that spoil the convergence of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111019.png" /> are sufficiently scarcely positioned, i.e. have zero density. Unlike strong summability, very strong summability means that the convergence of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111020.png" /> is spoiled by very thin sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091110/s09111021.png" />.
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The concept of strong summability was introduced in connection with the $  ( C, 1) $-
 +
summability of Fourier series (cf. [[Summation of Fourier series|Summation of Fourier series]]). The importance of this concept is well illustrated in the example of strong $  ( C, 1) $-
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summability. Strong $  ( C, 1) $-
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summability signifies that the partial sums $  S _ {\nu _ {1}  } , S _ {\nu _ {2}  } \dots $
 +
that spoil the convergence of the sequence $  \{ S _ {n} \} $
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are sufficiently scarcely positioned, i.e. have zero density. Unlike strong summability, very strong summability means that the convergence of the sequence $  \{ S _ {n} \} $
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is spoiled by very thin sequences $  \{ S _ {\nu _ {m}  } \} $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  "Sur la série de Fourier d'une fonction à carré sommable"  ''C.R. Acad. Sci. Paris'' , '''156'''  (1913)  pp. 1307–1309</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Aleksich,  "Convergence problems of orthogonal series" , Pergamon  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Gen-Ichirô Sunouchi,  "Strong summability of Walsh–Fourier series"  ''Tôhoku Math. J.'' , '''16'''  (1964)  pp. 228–237</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Gen-Ichirô Sunouchi,  ''Acta Sci. Math.'' , '''27''' :  1–2  (1966)  pp. 71–76</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.A. Bolgov,  E.V. Efimov,  "On the rate of summability of orthogonal series"  ''Math. USSR Izv.'' , '''5''' :  6  (1071)  pp. 1399–1417  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35''' :  6  (1971)  pp. 1389–1408</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  Z. Zatewasser,  ''Studia Math.'' , '''6'''  (1936)  pp. 82–88</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  L. Leindler,  "Ueber die sehr starke Riesz-Summierbarkeit der Orthogonalreihen und Konvergenz lückenhafter Orthogonalreihen"  ''Acta Math. Acad. Sci. Hung.'' , '''13''' :  3–4  (1962)  pp. 401–414</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  "Sur la série de Fourier d'une fonction à carré sommable"  ''C.R. Acad. Sci. Paris'' , '''156'''  (1913)  pp. 1307–1309</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Aleksich,  "Convergence problems of orthogonal series" , Pergamon  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Gen-Ichirô Sunouchi,  "Strong summability of Walsh–Fourier series"  ''Tôhoku Math. J.'' , '''16'''  (1964)  pp. 228–237</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Gen-Ichirô Sunouchi,  ''Acta Sci. Math.'' , '''27''' :  1–2  (1966)  pp. 71–76</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.A. Bolgov,  E.V. Efimov,  "On the rate of summability of orthogonal series"  ''Math. USSR Izv.'' , '''5''' :  6  (1071)  pp. 1399–1417  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35''' :  6  (1971)  pp. 1389–1408</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  Z. Zatewasser,  ''Studia Math.'' , '''6'''  (1936)  pp. 82–88</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  L. Leindler,  "Ueber die sehr starke Riesz-Summierbarkeit der Orthogonalreihen und Konvergenz lückenhafter Orthogonalreihen"  ''Acta Math. Acad. Sci. Hung.'' , '''13''' :  3–4  (1962)  pp. 401–414</TD></TR>
 +
</table>

Latest revision as of 12:35, 6 January 2024


of a complex sequence $ \{ S _ {n} \} $ of numbers or functions (or of a series $ \sum _ {k=1} ^ \infty a _ {k} $ with partial sums $ S _ {n} $) to a number $ S $

Summability by a method $ A = | a _ {nk} | $( cf. Summation methods) such that for a certain $ p > 0 $:

1) the sequence

$$ \sigma _ {n} = \sum_{k=1} ^ \infty a _ {nk} | S _ {k} - S | ^ {p} $$

converges for every $ n > 1 $, and for almost all $ x $ in the case of a sequence of functions;

2) $ \lim\limits _ {n \rightarrow \infty } \sigma _ {n} = 0 $. By retaining 2) and replacing 1) by:

1') for every monotone increasing sequence of indices $ \{ v _ {k} \} $, the sequence

$$ \sigma _ {n} = \sum_{k=1}^ \infty a _ {nv _ {k} } | S _ {v _ {k} } - S | ^ {p} $$

converges for every $ n > 1 $, and for almost all $ x $ in the case of a sequence of functions, one arrives at the concept of very strong summability.

The concept of strong summability was introduced in connection with the $ ( C, 1) $- summability of Fourier series (cf. Summation of Fourier series). The importance of this concept is well illustrated in the example of strong $ ( C, 1) $- summability. Strong $ ( C, 1) $- summability signifies that the partial sums $ S _ {\nu _ {1} } , S _ {\nu _ {2} } \dots $ that spoil the convergence of the sequence $ \{ S _ {n} \} $ are sufficiently scarcely positioned, i.e. have zero density. Unlike strong summability, very strong summability means that the convergence of the sequence $ \{ S _ {n} \} $ is spoiled by very thin sequences $ \{ S _ {\nu _ {m} } \} $.

References

[1] G.H. Hardy, J.E. Littlewood, "Sur la série de Fourier d'une fonction à carré sommable" C.R. Acad. Sci. Paris , 156 (1913) pp. 1307–1309
[2] G. Aleksich, "Convergence problems of orthogonal series" , Pergamon (1961) (Translated from Russian)
[3] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)
[4] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[5] Gen-Ichirô Sunouchi, "Strong summability of Walsh–Fourier series" Tôhoku Math. J. , 16 (1964) pp. 228–237
[6] Gen-Ichirô Sunouchi, Acta Sci. Math. , 27 : 1–2 (1966) pp. 71–76
[7] V.A. Bolgov, E.V. Efimov, "On the rate of summability of orthogonal series" Math. USSR Izv. , 5 : 6 (1071) pp. 1399–1417 Izv. Akad. Nauk SSSR Ser. Mat. , 35 : 6 (1971) pp. 1389–1408
[8] Z. Zatewasser, Studia Math. , 6 (1936) pp. 82–88
[9] L. Leindler, "Ueber die sehr starke Riesz-Summierbarkeit der Orthogonalreihen und Konvergenz lückenhafter Orthogonalreihen" Acta Math. Acad. Sci. Hung. , 13 : 3–4 (1962) pp. 401–414
How to Cite This Entry:
Summability, strong. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Summability,_strong&oldid=13277
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article