# Difference between revisions of "Trigonometric series"

Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
Ulf Rehmann (talk | contribs) m (Undo revision 49037 by Ulf Rehmann (talk)) Tag: Undo |
||

Line 1: | Line 1: | ||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

A series in cosines and sines of multiple angles, that is, a series of the form | A series in cosines and sines of multiple angles, that is, a series of the form | ||

− | + | <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/t/t094/t094240/t0942401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table> | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

or, in complex form, | or, in complex form, | ||

− | + | <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/t/t094/t094240/t0942402.png" /></td> </tr></table> | |

− | |||

− | |||

− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t0942403.png" /> or, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t0942404.png" /> are called the coefficients of the trigonometric series. |

− | or, respectively, | ||

− | are called the coefficients of the trigonometric series. | ||

− | + | <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/t/t094/t094240/t0942405.png" /></td> </tr></table> | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | + | <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/t/t094/t094240/t0942406.png" /></td> </tr></table> | |

− | |||

− | + | <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/t/t094/t094240/t0942407.png" /></td> </tr></table> | |

− | |||

− | |||

− | |||

− | |||

− | |||

Line 67: | Line 23: | ||

− | The question introduced into the debates of the mathematicians in the 18th century was solved in 1807 by J. Fourier, who gave formulas for calculating the coefficients of the trigonometric series (1) that had to "represent" a function | + | The question introduced into the debates of the mathematicians in the 18th century was solved in 1807 by J. Fourier, who gave formulas for calculating the coefficients of the trigonometric series (1) that had to "represent" a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t0942408.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t0942409.png" />: |

− | on | ||

− | + | <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/t/t094/t094240/t09424010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table> | |

− | |||

− | and who applied them in the solution of problems of heat conduction. Formulas (2) have acquired the name Fourier formulas, although they were encountered earlier by A. Clairaut (1754) and Euler (1777) via term-by-term integration. The trigonometric series (1) whose coefficients are defined by (2) is called the Fourier series of | + | and who applied them in the solution of problems of heat conduction. Formulas (2) have acquired the name Fourier formulas, although they were encountered earlier by A. Clairaut (1754) and Euler (1777) via term-by-term integration. The trigonometric series (1) whose coefficients are defined by (2) is called the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424011.png" />, and the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424012.png" /> the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424013.png" />. |

− | and the numbers | ||

− | the Fourier coefficients of | ||

The character of the results obtained depends on how the "representation" of the function by a series is to be understood, and how the integral in (2) is to be understood. The modern form of the theory of trigonometric series was obtained after the appearance of the [[Lebesgue integral|Lebesgue integral]]. | |||

Line 83: | Line 35: | ||

− | For the study of the properties of an arbitrary series (1) with coefficients converging to zero, Riemann considered the continuous function | + | For the study of the properties of an arbitrary series (1) with coefficients converging to zero, Riemann considered the continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424014.png" /> that is the sum of the uniformly-convergent series |

− | that is the sum of the uniformly-convergent series | ||

− | + | <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/t/t094/t094240/t09424015.png" /></td> </tr></table> | |

− | |||

− | |||

− | |||

− | |||

− | + | obtained after twice term-by-term integration of the series (1). If the series (1) converges at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424016.png" /> to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424017.png" />, then the second [[Symmetric derivative|symmetric derivative]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424018.png" /> exists at this point and is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424019.png" />: | |

− | |||

− | |||

− | + | <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/t/t094/t094240/t09424020.png" /></td> </tr></table> | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

Since | Since | ||

− | + | <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/t/t094/t094240/t09424021.png" /></td> </tr></table> | |

− | + | <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/t/t094/t094240/t09424022.png" /></td> </tr></table> | |

− | |||

− | |||

− | + | this leads to the summation of (1) produced by the factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424023.png" />, called the Riemann summation method. By means of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424024.png" />, Riemann formulated the localization principle, according to which the behaviour of the series (1) at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424025.png" /> depends only on the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424026.png" /> in an arbitrarily small neighbourhood of this point. | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | this leads to the summation of (1) produced by the factors | ||

− | called the Riemann summation method. By means of the function | ||

− | Riemann formulated the localization principle, according to which the behaviour of the series (1) at a point | ||

− | depends only on the behaviour of | ||

− | in an arbitrarily small neighbourhood of this point. | ||

− | One of the central problems in the theory of general trigonometric series is that of "representing" an arbitrary function by a trigonometric series. By strengthening results of N.N. Luzin (1915) on the representation of functions by trigonometric series that are summable almost-everywhere by the methods of Abel–Poisson and Riemann, D.E. Men'shov proved (1940) the following theorem, relating to the most important case when the representation of | + | One of the central problems in the theory of general trigonometric series is that of "representing" an arbitrary function by a trigonometric series. By strengthening results of N.N. Luzin (1915) on the representation of functions by trigonometric series that are summable almost-everywhere by the methods of Abel–Poisson and Riemann, D.E. Men'shov proved (1940) the following theorem, relating to the most important case when the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424027.png" /> is understood as convergence of the trigonometric series to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424028.png" /> almost everywhere. There exists for any measurable, almost-everywhere finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424029.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424030.png" /> a trigonometric series converging to it almost-everywhere (Men'shov's theorem). It should be noted that even if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424031.png" /> is integrable, one cannot in general choose the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424032.png" /> for this series, since there exist Fourier series that are everywhere divergent. |

− | is understood as convergence of the trigonometric series to | ||

− | almost everywhere. There exists for any measurable, almost-everywhere finite | ||

− | periodic function | ||

− | a trigonometric series converging to it almost-everywhere (Men'shov's theorem). It should be noted that even if the function | ||

− | is integrable, one cannot in general choose the Fourier series of | ||

− | for this series, since there exist Fourier series that are everywhere divergent. | ||

− | Men'shov's theorem can be strengthened as follows: If a | + | Men'shov's theorem can be strengthened as follows: If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424033.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424034.png" /> is measurable and finite almost everywhere, then there exists a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424035.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424036.png" /> almost everywhere and the differentiated Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424037.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424038.png" /> almost everywhere (N.K. Bari, 1952). |

− | periodic function | ||

− | is measurable and finite almost everywhere, then there exists a continuous function | ||

− | such that | ||

− | almost everywhere and the differentiated Fourier series of | ||

− | converges to | ||

− | almost everywhere (N.K. Bari, 1952). | ||

− | It is not known (1984) whether the condition that | + | It is not known (1984) whether the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424039.png" /> be finite almost-everywhere can be dropped in Men'shov's theorem. In particular, it is not known (1984) whether there is a trigonometric series converging almost-everywhere to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424040.png" />. |

− | be finite almost-everywhere can be dropped in Men'shov's theorem. In particular, it is not known (1984) whether there is a trigonometric series converging almost-everywhere to | ||

− | Therefore the problem on the representation of functions that can take infinite values on a set of positive measure has been considered for the case when convergence almost-everywhere is replaced by a weaker condition, namely, convergence in measure. Convergence in measure to functions that can take infinite values is defined as follows: The sequence | + | Therefore the problem on the representation of functions that can take infinite values on a set of positive measure has been considered for the case when convergence almost-everywhere is replaced by a weaker condition, namely, convergence in measure. Convergence in measure to functions that can take infinite values is defined as follows: The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424041.png" /> of partial sums of a trigonometric series converges in measure to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424042.png" /> if |

− | of partial sums of a trigonometric series converges in measure to a function | ||

− | if | ||

− | + | <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/t/t094/t094240/t09424043.png" /></td> </tr></table> | |

− | |||

− | |||

− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424044.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424045.png" /> almost everywhere, while the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424046.png" /> converges in measure to zero. In this formulation, the question of the representation of functions has been completely solved: there exists for any measurable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424047.png" />-periodic function a trigonometric series converging to it in measure (Men'shov, 1948). |

− | converges to | ||

− | almost everywhere, while the sequence | ||

− | converges in measure to zero. In this formulation, the question of the representation of functions has been completely solved: there exists for any measurable | ||

− | periodic function a trigonometric series converging to it in measure (Men'shov, 1948). | ||

− | The following terminology has been established. A set | + | The following terminology has been established. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424048.png" /> is called a set of uniqueness (cf. [[Uniqueness set|Uniqueness set]]), or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424050.png" />-set, if the convergence everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424051.png" /> of a trigonometric series to zero except, perhaps, at points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424052.png" />, implies that all coefficients of the series are zero. Otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424053.png" /> is called a set of multiplicity, or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424055.png" />-set. As G. Cantor showed (1872), the empty, as well as any finite, set is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424056.png" />-set. Any countable set is also a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424057.png" />-set (Young, 1909). On the other hand, every set of positive measure is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424058.png" />-set. |

− | is called a set of uniqueness (cf. [[Uniqueness set|Uniqueness set]]), or a | ||

− | set, if the convergence everywhere on | ||

− | of a trigonometric series to zero except, perhaps, at points of | ||

− | implies that all coefficients of the series are zero. Otherwise | ||

− | is called a set of multiplicity, or an | ||

− | set. As G. Cantor showed (1872), the empty, as well as any finite, set is a | ||

− | set. Any countable set is also a | ||

− | set (Young, 1909). On the other hand, every set of positive measure is an | ||

− | set. | ||

− | The existence of | + | The existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424059.png" />-sets of measure zero was established by Men'shov (1916), who constructed the first example of a [[Perfect set|perfect set]] possessing these properties. This result is of prime importance in the uniqueness problem. It follows from the existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424060.png" />-sets of measure zero that in the representation of functions by trigonometric series converging almost everywhere, these series are automatically non-uniquely determined. |

− | sets of measure zero was established by Men'shov (1916), who constructed the first example of a [[Perfect set|perfect set]] possessing these properties. This result is of prime importance in the uniqueness problem. It follows from the existence of | ||

− | sets of measure zero that in the representation of functions by trigonometric series converging almost everywhere, these series are automatically non-uniquely determined. | ||

− | Perfect sets can also be | + | Perfect sets can also be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424061.png" />-sets (Bari; A. Rajchman, 1921). In the uniqueness problem, an important role is played by very precise characteristics of sets of measure zero. The general question on the classification of sets of measure zero into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424062.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424063.png" />-sets remains open (1984). It has not even been solved for perfect sets. |

− | sets (Bari; A. Rajchman, 1921). In the uniqueness problem, an important role is played by very precise characteristics of sets of measure zero. The general question on the classification of sets of measure zero into | ||

− | and | ||

− | sets remains open (1984). It has not even been solved for perfect sets. | ||

− | The following problem touches on the uniqueness problem. If a trigonometric series converges to a function | + | The following problem touches on the uniqueness problem. If a trigonometric series converges to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424064.png" />, then does this series have to be the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424065.png" />? P. du Bois-Reymond (1877) gave an affirmative answer to this question when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424066.png" /> is Riemann integrable and the series converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424067.png" /> at all points. Results of Ch.J. de la Vallée-Poussin (1912) imply that the answer is also affirmative in case the series is convergent to a finite sum everywhere except on a countable set of points. |

− | then does this series have to be the Fourier series of | ||

− | P. du Bois-Reymond (1877) gave an affirmative answer to this question when | ||

− | is Riemann integrable and the series converges to | ||

− | at all points. Results of Ch.J. de la Vallée-Poussin (1912) imply that the answer is also affirmative in case the series is convergent to a finite sum everywhere except on a countable set of points. | ||

− | If a trigonometric series converges absolutely at some point | + | If a trigonometric series converges absolutely at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424068.png" />, then the points of convergence of the series, as well as the points of its absolute convergence, are distributed symmetrically about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424069.png" /> (P. Fatou, 1906). |

− | then the points of convergence of the series, as well as the points of its absolute convergence, are distributed symmetrically about | ||

− | P. Fatou, 1906). | ||

− | According to the [[Denjoy–Luzin theorem|Denjoy–Luzin theorem]], absolute convergence on a set of positive measure of a trigonometric series (1) implies that the series | + | According to the [[Denjoy–Luzin theorem|Denjoy–Luzin theorem]], absolute convergence on a set of positive measure of a trigonometric series (1) implies that the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424070.png" /> is convergent, and hence that (1) is absolutely convergent for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424071.png" />. This property is also possessed by sets of the second category and certain sets of measure zero. |

− | is convergent, and hence that (1) is absolutely convergent for all | ||

− | This property is also possessed by sets of the second category and certain sets of measure zero. | ||

Line 206: | Line 83: | ||

====References==== | ====References==== | ||

<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N.N. Luzin, "The integral and trigonometric series" , Moscow-Leningrad (1951) (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" H. Weber (ed.) , ''B. Riemann's Gesammelte Mathematische Werke'' , Dover, reprint (1953) pp. 227–265 ((Original: Göttinger Akad. Abh. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424072.png" /> (1868)))</TD></TR></table> | |||

+ | |||

+ | |||

====Comments==== | ====Comments==== | ||

Line 212: | Line 91: | ||

The problems related to Men'shov's theorem cited above that were unsolved in 1984 have now been solved. In [[#References|[a1]]], S.V. Konyagin proved the necessity part of the following brilliant theorem. (The sufficiency part had already been proved by D.E. Men'shov (see [[#References|[a2]]] or [[#References|[1]]], vol. II, p. 437).) | |||

− | Men'shov–Konyagin theorem. Let | + | Men'shov–Konyagin theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424074.png" /> be measurable functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424075.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424076.png" />. In order that a trigonometric series (1) having partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424077.png" /> exists for which |

− | and | ||

− | be measurable functions from | ||

− | into | ||

− | In order that a trigonometric series (1) having partial sums | ||

− | exists for which | ||

− | + | <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/t/t094/t094240/t09424078.png" /></td> </tr></table> | |

− | |||

− | |||

− | |||

− | for almost-all | + | for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424079.png" />, it is necessary and sufficient that a measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424080.png" /> exists such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424081.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424082.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424084.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424085.png" />. |

− | it is necessary and sufficient that a measurable set | ||

− | exists such that | ||

− | for almost-all | ||

− | while | ||

− | and | ||

− | for almost-all | ||

− | In particular, no trigonometric series has sum | + | In particular, no trigonometric series has sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094240/t09424086.png" /> at every point of a set of positive measure. |

− | at every point of a set of positive measure. | ||

====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.V. Konyagin, "Limits of indeterminacy of trigonometric series" ''Math. Notes'' , '''44''' (1988) pp. 910–920 ''Mat. Zametki'' , '''44''' : 6 (1988) pp. 770–784</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D.E. Men'shov, "On limits of indeterminacy of Fourier series" ''Mat. Sb.'' , '''30''' : 3 (1952) pp. 601–650 (In Russian)</TD></TR></table> |

## Revision as of 14:53, 7 June 2020

A series in cosines and sines of multiple angles, that is, a series of the form

(1) |

or, in complex form,

where or, respectively, are called the coefficients of the trigonometric series.

However, the significance of the arguments was even greater. In fact, out of these discussions various questions arose connected with many fundamentally important concepts and ideas of mathematical analysis in general, such as the "representation" of functions by Taylor series and the analytic continuation of functions, the use of divergent series, interchange of limits, infinite systems of equations, interpolation of functions by polynomials, etc.

The question introduced into the debates of the mathematicians in the 18th century was solved in 1807 by J. Fourier, who gave formulas for calculating the coefficients of the trigonometric series (1) that had to "represent" a function on :

(2) |

and who applied them in the solution of problems of heat conduction. Formulas (2) have acquired the name Fourier formulas, although they were encountered earlier by A. Clairaut (1754) and Euler (1777) via term-by-term integration. The trigonometric series (1) whose coefficients are defined by (2) is called the Fourier series of , and the numbers the Fourier coefficients of .

The character of the results obtained depends on how the "representation" of the function by a series is to be understood, and how the integral in (2) is to be understood. The modern form of the theory of trigonometric series was obtained after the appearance of the Lebesgue integral.

The theory of trigonometric series can conditionally be divided into two main branches: the theory of Fourier series, in which it is supposed that the series (1) is the Fourier series of some function, and the theory of general trigonometric series, where this hypothesis is not made. The main results in the theory of general trigonometric series are given below (here the measure of sets and measurability of functions are to be understood in the sense of Lebesgue).

For the study of the properties of an arbitrary series (1) with coefficients converging to zero, Riemann considered the continuous function that is the sum of the uniformly-convergent series

obtained after twice term-by-term integration of the series (1). If the series (1) converges at some point to the number , then the second symmetric derivative of exists at this point and is equal to :

Since

this leads to the summation of (1) produced by the factors , called the Riemann summation method. By means of the function , Riemann formulated the localization principle, according to which the behaviour of the series (1) at a point depends only on the behaviour of in an arbitrarily small neighbourhood of this point.

One of the central problems in the theory of general trigonometric series is that of "representing" an arbitrary function by a trigonometric series. By strengthening results of N.N. Luzin (1915) on the representation of functions by trigonometric series that are summable almost-everywhere by the methods of Abel–Poisson and Riemann, D.E. Men'shov proved (1940) the following theorem, relating to the most important case when the representation of is understood as convergence of the trigonometric series to almost everywhere. There exists for any measurable, almost-everywhere finite -periodic function a trigonometric series converging to it almost-everywhere (Men'shov's theorem). It should be noted that even if the function is integrable, one cannot in general choose the Fourier series of for this series, since there exist Fourier series that are everywhere divergent.

Men'shov's theorem can be strengthened as follows: If a -periodic function is measurable and finite almost everywhere, then there exists a continuous function such that almost everywhere and the differentiated Fourier series of converges to almost everywhere (N.K. Bari, 1952).

It is not known (1984) whether the condition that be finite almost-everywhere can be dropped in Men'shov's theorem. In particular, it is not known (1984) whether there is a trigonometric series converging almost-everywhere to .

Therefore the problem on the representation of functions that can take infinite values on a set of positive measure has been considered for the case when convergence almost-everywhere is replaced by a weaker condition, namely, convergence in measure. Convergence in measure to functions that can take infinite values is defined as follows: The sequence of partial sums of a trigonometric series converges in measure to a function if

where converges to almost everywhere, while the sequence converges in measure to zero. In this formulation, the question of the representation of functions has been completely solved: there exists for any measurable -periodic function a trigonometric series converging to it in measure (Men'shov, 1948).

The following terminology has been established. A set is called a set of uniqueness (cf. Uniqueness set), or a -set, if the convergence everywhere on of a trigonometric series to zero except, perhaps, at points of , implies that all coefficients of the series are zero. Otherwise is called a set of multiplicity, or an -set. As G. Cantor showed (1872), the empty, as well as any finite, set is a -set. Any countable set is also a -set (Young, 1909). On the other hand, every set of positive measure is an -set.

The existence of -sets of measure zero was established by Men'shov (1916), who constructed the first example of a perfect set possessing these properties. This result is of prime importance in the uniqueness problem. It follows from the existence of -sets of measure zero that in the representation of functions by trigonometric series converging almost everywhere, these series are automatically non-uniquely determined.

Perfect sets can also be -sets (Bari; A. Rajchman, 1921). In the uniqueness problem, an important role is played by very precise characteristics of sets of measure zero. The general question on the classification of sets of measure zero into - and -sets remains open (1984). It has not even been solved for perfect sets.

The following problem touches on the uniqueness problem. If a trigonometric series converges to a function , then does this series have to be the Fourier series of ? P. du Bois-Reymond (1877) gave an affirmative answer to this question when is Riemann integrable and the series converges to at all points. Results of Ch.J. de la Vallée-Poussin (1912) imply that the answer is also affirmative in case the series is convergent to a finite sum everywhere except on a countable set of points.

If a trigonometric series converges absolutely at some point , then the points of convergence of the series, as well as the points of its absolute convergence, are distributed symmetrically about (P. Fatou, 1906).

According to the Denjoy–Luzin theorem, absolute convergence on a set of positive measure of a trigonometric series (1) implies that the series is convergent, and hence that (1) is absolutely convergent for all . This property is also possessed by sets of the second category and certain sets of measure zero.

#### References

[1] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |

[2] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |

[3] | N.N. Luzin, "The integral and trigonometric series" , Moscow-Leningrad (1951) (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212) |

[4] | B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" H. Weber (ed.) , B. Riemann's Gesammelte Mathematische Werke , Dover, reprint (1953) pp. 227–265 ((Original: Göttinger Akad. Abh. (1868))) |

#### Comments

For the convergence of Fourier series see also Carleson theorem.

The problems related to Men'shov's theorem cited above that were unsolved in 1984 have now been solved. In [a1], S.V. Konyagin proved the necessity part of the following brilliant theorem. (The sufficiency part had already been proved by D.E. Men'shov (see [a2] or [1], vol. II, p. 437).)

Men'shov–Konyagin theorem. Let and be measurable functions from into . In order that a trigonometric series (1) having partial sums exists for which

for almost-all , it is necessary and sufficient that a measurable set exists such that for almost-all , while and for almost-all .

In particular, no trigonometric series has sum at every point of a set of positive measure.

#### References

[a1] | S.V. Konyagin, "Limits of indeterminacy of trigonometric series" Math. Notes , 44 (1988) pp. 910–920 Mat. Zametki , 44 : 6 (1988) pp. 770–784 |

[a2] | D.E. Men'shov, "On limits of indeterminacy of Fourier series" Mat. Sb. , 30 : 3 (1952) pp. 601–650 (In Russian) |

**How to Cite This Entry:**

Trigonometric series.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Trigonometric_series&oldid=49480