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A series of the form

$$ \tag{1 }
\sum _ { n= 0} ^  \infty   a _ {n} \phi _ {n} ( x),\  x \in X,
$$

where  $  \{ \phi _ {n} \} $
is an [[Orthonormal system|orthonormal system]] of functions with respect to a measure  $  \mu $:

$$ 
\int\limits _ { X } \phi _ {i} ( x) \phi _ {j} ( x)  d \mu ( x)  = \ 
\left \{ 
\begin{array}{lll}
 0  & \textrm{ when }  &i \neq j,  \\
 1  & \textrm{ when }  &i = j.  \\
\end{array}

 \right .$$

Since the 18th century, certain special orthonormal systems and expansions of functions with respect to them have appeared in the research of L. Euler, D. Bernoulli, A. Legendre, P. Laplace, F. Bessel, and others into various questions of mathematics, astronomy, mechanics, and physics (planetary motion, oscillation of chords, membranes, etc.). The following have had a decisive influence on the creation of the theory of orthogonal series:

a) The research of J. Fourier (1807–1822) (the [[Fourier method|Fourier method]] for solving the boundary value problems of mathematical physics) and the related work of J. Sturm and J. Liouville (1837–1841);

b) The research of P.L. Chebyshev on interpolation and the problem of moments (mid-19th century), which led to his creation of the general theory of [[Orthogonal polynomials|orthogonal polynomials]];

c) The research of D. Hilbert (early 20th century) on integral equations (cf. [[Integral equation with symmetric kernel|Integral equation with symmetric kernel]]), in which he established, in particular, general theorems on the expansion of functions in a series with respect to an orthonormal system;

d) The creation by H. Lebesgue of measure theory and the [[Lebesgue integral|Lebesgue integral]], which were responsible for the theory of orthogonal series in its modern form.

The active development of the theory of orthogonal series in the 20th century has been enhanced by the use of orthonormal systems of functions and series with respect to them in the most varied areas of science (mathematical physics, computational mathematics, functional analysis, quantum mechanics, mathematical statistics, operational calculus, automatic regulation and control, various technical problems, etc.).

==Characteristic results and directions of research in the theory of orthogonal series.==

1) Let  $  X = [ a, b] $, 
let  $  d \mu ( x) = dx $
be Lebesgue measure and let  $  \{ \phi _ {n} \} $
be an orthonormal system. If  $  f \in L _ {2} ( a, b) $, 
then the numbers

$$ 
a _ {n} ( f  )  \equiv  ( f, \phi _ {n} )  \equiv  \int\limits _ { a } ^ { b }  f( x) \phi _ {n} ( x)  dx
$$

are called the Fourier coefficients, while the series (1) with  $  a _ {n} = a _ {n} ( f  ) $
is called the Fourier series of the function  $  f $
with respect to the system  $  \{ \phi _ {n} \} $.

The system  $  \{ \phi _ {n} \} $
is closed in the space  $  L _ {2} $
if for any function  $  f \in L _ {2} $
and any number  $  \epsilon > 0 $, 
a polynomial

$$ 
\Phi ( x)  =  \sum _ { n= 0} ^ { N }  c _ {n} \phi _ {n} ( x)
$$

can be found such that the norm  $  \| f- \Phi \| _ {2} < \epsilon $. 
The system  $  \{ \phi _ {n} \} $
is complete relative to  $  L _ {2} $
if it follows from the conditions  $  f \in L _ {2} $
and  $  a _ {n} ( f  ) = 0 $
for all  $  n \geq  0 $
that  $  f( x) = 0 $
almost-everywhere, i.e. that  $  f $
is the zero element of the space  $  L _ {2} $.

If for a function  $  f \in L _ {2} $
the equation

$$ \tag{2 }
\sum _ { n= 0} ^  \infty   a _ {n}  ^ {2} ( f  )  = \ 
\int\limits _ { a } ^ { b }  f ^ { 2 } ( x)  dx
$$

is fulfilled, then the function  $  f $
is said to satisfy the Lyapunov–Steklov closure condition (or Parseval identity). This condition is equivalent to convergence of the partial sums of the Fourier series of  $  f $
in the norm of  $  L _ {2} $
to  $  f $.

The definitions of closure and completeness, and conditions of closure, are given in the same way for more general spaces and measures.

One of the most important questions in the theory of orthogonal series is the question of the unique determination of a function by means of its Fourier coefficients. For the space  $  L _ {2} $
it is connected very closely to the fulfillment of (2) for all functions  $  f \in L _ {2} $.

Equation (2) was put forward in 1805 (though without proof) for the [[Trigonometric system|trigonometric system]] by M. Parseval, while in 1828, Bessel established that

$$ \tag{3 }
\sum _ { n= 0} ^  \infty   a _ {n}  ^ {2} ( f  )  \leq   \int\limits _ { a } ^ { b }  f ^ { 2 }
( x)  dx
$$

(the Bessel inequality). In 1896, A.M. Lyapunov proved (2) for Riemann-integrable functions and P. Fatou then proved it for the case  $  f \in L _ {2} $.

V.A. Steklov (1898–1904) put forward the question of the closure of general orthonormal systems, and solved it positively for many orthogonal systems (spherical functions, eigen functions of a Sturm–Liouville operator, systems of orthogonal Hermite polynomials, Laguerre polynomials, Lamé functions, and others).

Inequality (3) has proved to be true for arbitrary orthonormal systems and functions  $  f \in L _ {2} $.

In 1907, F. Riesz and E. Fischer proved that for any orthonormal system  $  \{ \phi _ {n} \} $
and for any sequence of numbers  $  \{ a _ {n} \} \in l _ {2} $
a function  $  f \in L _ {2} $
can be found for which  $  a _ {n} ( f  ) \equiv ( f, \phi _ {n} ) = a _ {n} $
and such that (2) is fulfilled. It follows from this theorem and the Bessel inequality that for any orthonormal system, completeness and closure are equivalent in  $  L _ {2} $; 
closure in the space  $  L _ {p} $
with  $  1 < p < \infty $
is equivalent to completeness in the space  $  L _ {p  ^  \prime   } $, 
where  $  1/p  ^  \prime  + 1 / p = 1 $ (S. Banach, 1931).

The Bessel inequality and the Riesz–Fischer theorem were extended by G.H. Hardy, J.E. Littlewood and R. Paley to the space  $  L _ {p} $. 
In fact, let  $  \{ \phi _ {n} \} $
be an orthonormal system,  $  | \phi _ {n} ( x) | \leq  M $, 
and let  $  1 < p < 2 $. 
Then:

a) if  $  f \in L _ {p} $,

$$ 
\sum _ { n= 1} ^  \infty   | ( f, \phi _ {n} ) |  ^ {p} n  ^ {p-2} \ 
\leq   A  \| f \| _ {p}  ^ {p} .
$$

b) If a sequence  $  \{ a _ {n} \} $
is given with

$$ 
I  \equiv  \sum _ { n= 1} ^  \infty   | a _ {n} | ^ {p  ^  \prime  } n ^ {p  ^  \prime  - 2 }  <  \infty ,
$$

then a function  $  f \in L _ {p  ^  \prime   } $
can be found for which  $  ( f, \phi _ {n} ) = a _ {n} $
and  $  \| f \| _ {p  ^  \prime   } ^ {p  ^  \prime  } \leq  AI $, 
where  $  A $
depends only on  $  p $
and  $  M $.

2) Another important problem in the theory of orthogonal series is the question of the expansion of a function, by means of simple functions, in a series converging to it in the norm of some space. A system of elements  $  \{ \phi _ {n} \} $
from a  $  B $-space  $  E $
is called a basis (an unconditional basis) if every element  $  f \in E $
can be uniquely represented in the form of a series

$$ \tag{4 }
f  =  \sum _ { n= 0} ^  \infty   a _ {n} \phi _ {n} ,\ \ 
a _ {n} = a _ {n} ( f  ),
$$

converging (unconditionally converging) to  $  f $
in the norm of the space  $  E $.

If  $  \{ \phi _ {n} \} $
is a basis in  $  E $, 
then the  $  a _ {n} ( f  ) $
are continuous linear functionals on  $  E $
and, if  $  E = L _ {p} ( 0, 1) $
with  $  1 < p < \infty $, 
take the form

$$ 
a _ {n} ( f  )  =  \int\limits _ { 0 } ^ { 1 }  f( t) \psi _ {n} ( t)  dt,
$$

where  $  \{ \psi _ {n} \} $
is a basis in  $  L _ {p  ^  \prime   } ( 0, 1) $
and  $  \{ \phi _ {n} , \psi _ {n} \} $
is a bi-orthonormal system (Banach). In particular, if  $  \phi _ {n} \equiv \psi _ {n} $, 
i.e. if  $  \{ \phi _ {n} \} $
is an orthonormal system, then an orthogonal basis in  $  L _ {p} $
is automatically a basis in all spaces  $  L _ {r} $, 
where  $  r $
is any number between  $  p $
and  $  p  ^  \prime  $.

Research into this problem has followed two directions:

$  \alpha $) 
for a given orthonormal system  $  \{ \phi _ {n} \} $, 
find the spaces in which  $  \{ \phi _ {n} \} $
is a basis;

$  \beta $) 
for a given space  $  E $, 
determine its bases or orthogonal bases.

In both cases, a mutual connection is sought between the properties of a function  $  f $
and its expansions.

As for the trigonometric system, it is not a basis in the space  $  C $
of continuous functions (P. du Bois-Reymond, 1876) but it is a basis in the space  $  L _ {p} $
with  $  1 < p < \infty $ (M. Riesz, 1927). The du Bois-Reymond result has been extended to all uniformly bounded orthonormal systems.

The orthonormal system of Legendre polynomials is a basis in the spaces  $  L _ {p} $
when  $  p \in ( 4/3, 4) $
and is not so in the other spaces  $  L _ {q} $ (1946–1952, H. Pollard, J. von Neumann and W. Rudin).

In 1910, an orthonormal system  $  \{ \chi _ {m} \} _ {m=0}  ^  \infty  $
was created such that every continuous function  $  f \in C ( 0, 1) $
can be uniquely expanded in a uniformly converging Fourier series with respect to this system (A. Haar). However, the [[Haar system|Haar system]]  $  \{ \chi _ {m} \} $
is not a basis in  $  C ( 0, 1) $, 
since the functions  $  \chi _ {m} $
are discontinuous when  $  m > 1 $. 
By integrating the system  $  \{ \chi _ {m} \} $, 
G. Faber (1910) established that the system

$$ 
\{ f _ {n} ( t) \}  \equiv  \left \{ 1, \int\limits _ { 0 } ^ { t }  \chi _ {m} ( x)  dx \right \}
$$

is a basis in  $  C ( 0, 1) $
and thereby the first basis in the space of continuous functions was found. Faber's result was rediscovered by J. Schauder (1927), who also determined a class of bases in  $  C ( 0, 1) $
of the type of the basis  $  \{ f _ {n} \} $; 
in honour of the latter, the term  "Schauder basisSchauder basis"  was introduced, although it would be more correct to call it a  "Faber–Schauder basisFaber–Schauder basis" .

These bases are not orthogonal. The first orthonormal basis  $  \{ F _ {n} \} $
in  $  C ( 0, 1) $
was obtained by Ph. Franklin (1928), who orthogonalized, by the Schmidt method (cf. [[Orthogonalization|Orthogonalization]]), the Faber–Schauder system  $  \{ f _ {n} \} $
and obtained  $  \{ F _ {n} \} $. 
In this direction (orthogonalization and integration), a new class of bases has been introduced and studied. All orthonormal bases in  $  C ( 0, 1) $
are automatically bases in all spaces  $  L _ {p} $
with  $  1 \leq  p \leq  \infty $.

The Haar system  $  \{ \chi _ {m} \} $
is an unconditional basis in all spaces  $  L _ {p} $
with  $  1 < p < \infty $ (1931–1937, Paley, J. Marcinkiewicz). The same result also holds for the Franklin system  $  \{ F _ {n} \} $.

In the spaces  $  C $
and  $  L $
there are no unconditional bases, in general. Neither are there any normalized or uniformly bounded unconditional bases in the spaces  $  L _ {p} $
when  $  1 < p < \infty $
and  $  p \neq 2 $.

3) Much research has been devoted to the problem of the almost-everywhere convergence of trigonometric and orthogonal series.

In 1911, N.N. Luzin gave the first example of an almost-everywhere divergent trigonometric series whose coefficients tend to zero. A Fourier series of this type was constructed by A.N. Kolmogorov (1923). Luzin's result has been extended to arbitrary complete orthonormal systems, while Kolmogorov's result has been generalized to sets of positive measure for uniformly bounded orthonormal systems.

A non-negative sequence  $  \{ \omega ( n) \} $
with  $  \omega ( n _ {0} ) > 0 $
and  $  \omega ( n)\uparrow $
is called a Weyl multiplier for the almost-everywhere convergence of a series with respect to a system  $  \{ \phi _ {n} \} $
if every series (1) converges almost everywhere on  $  X \equiv [ 0, 1] $, 
when only

$$ 
\sum _ {n= n _ {0} } ^  \infty   a _ {n}  ^ {2} \omega ( n)  <  \infty .
$$

If  $  \omega ( n) \equiv 1 $
is a Weyl multiplier, then  $  \{ \phi _ {n} \} $
is called a system of almost-everywhere convergence. The sequence  $  \{ \omega ( n) \} $
is called an exact Weyl multiplier for the almost-everywhere convergence of a series (1) if  $  \{ \omega ( n) \} $
is a Weyl multiplier, while every  $  \tau ( n) = o( \omega ( n)) $, 
$  n \rightarrow \infty $, 
is not. Definitions of a Weyl multiplier for other forms of convergence and summability are given in the same way (convergence in measure, unconditional convergence almost-everywhere, and others).

Weyl multipliers have been found for some systems. In 1913, M. Plancherel proved that  $  \{  \mathop{\rm log}  ^ {3}  n \} $
is a Weyl multiplier for the almost-everywhere convergence of a series with respect to any orthonormal system  $  \{ \phi _ {n} \} $, 
while in 1922, D.E. Men'shov and H. Rademacher established that  $  \{  \mathop{\rm log}  ^ {2}  n \} $
can be taken as a Weyl multiplier. Most importantly, Men'shov proved that this result could not be improved upon in the whole class of orthonormal systems, i.e.  $  \{  \mathop{\rm log}  ^ {2}  n \} $
is an exact Weyl multiplier for certain orthonormal systems.

Necessary and sufficient conditions have subsequently been found for  $  \{ \omega ( n) \} $
to be a Weyl multiplier for almost-everywhere (in the mean, etc.) convergence or  $  ( C, 1) $-summability of orthogonal series. It has been demonstrated, for example, that the system  $  \{  \mathop{\rm sign}  \sin  \pi nx \} $
is not a system of almost-everywhere convergence. In 1975, the first complete orthonormal system  $  \{ \phi _ {n} \} $
of strong convergence i.e. the series (1) converges almost-everywhere on  $  X = [ 0, 1] $
if and only if  $  \{ a _ {n} \} \in l _ {2} $, 
was constructed.

In 1927 it was established that the sequence  $  \omega ( n) = \tau ( n)  \mathop{\rm log}  ^ {2}  n $
is a Weyl multiplier for almost-everywhere unconditional convergence of any orthogonal series if

$$ 
\sum _ {n = n _ {0} } ^  \infty   
\frac{1}{n \tau ( n)  \mathop{\rm log}  n }
  <  \infty .
$$

This result can not be strengthened.

In 1960 it was demonstrated that the Haar system  $  \{ \chi _ {n} \} $
is not a system of almost-everywhere unconditional convergence. It was also demonstrated, on the basis of this result, that many systems (bases in  $  L _ {2} $, 
complete orthonormal systems, etc.) are not systems of almost-everywhere unconditional convergence. For the system  $  \{ \chi _ {n} \} $, 
a sequence  $  \{ \omega ( n) \} $
is a Weyl multiplier for almost-everywhere unconditional convergence only if

$$ 
\sum _ {n = n _ {0} } ^  \infty   
\frac{1}{n \omega ( n) }
  <  \infty .
$$

For this reason, not every complete orthonormal system has an exact Weyl multiplier for almost-everywhere unconditional convergence.

A great deal of research has been carried out into the problem of the representation of functions by series converging almost-everywhere, in measure or in other ways. So, in 1957 it was established that for any complete orthonormal system  $  \{ \phi _ {n} \} $
with  $  X = [ 0, 1] $
and any measurable function  $  f $
there is a series of the form (1) which converges in measure to  $  f( x) $ (in the case of the trigonometric system this assertion was obtained in 1947 by Men'shov). This result becomes invalid, even in the case of bounded measurable functions, if instead of convergence in measure, almost-everywhere convergence is considered.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Luzin,   "The integral and trigonometric series" , Moscow-Leningrad  (1915)  (In Russian)  (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Banach,   "Théorie des opérations linéaires" , Chelsea, reprint  (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Ya.L. Geronimus,   "Orthogonal polynomials" , Consultants Bureau  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kaczmarz,   H. Steinhaus,   "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Jackson,   "Fourier series and orthogonal polynomials" , ''Carus Math. Monogr.'' , '''6''' , Math. Assoc. Amer.  (1971)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G. Szegö,   "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G. Alexits,   "Konvergenzprobleme der Orthogonalreihen" , Ungar. Akad. Wissenschaft.  (1961)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  F.G. Tricomi,   "Vorlesungen über Orthogonalreihen" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.M. Olevskii,   "Fourier series with respect to general orthogonal systems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  D.E. Men'shov,   P.L. Ul'yanov,   "On the metric theory of functions at Moscow University over 50 years"  ''Vestnik Moskov. Univ. Ser. 1. Mat. Mekh.'' , '''5'''  (1967)  pp. 24–36  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  A.A. Talalyan,   "The representation of measurable functions by series"  ''Russian Math. Surveys'' , '''15''' :  5  (1960)  pp. 75–136  ''Uspekhi Mat. Nauk'' , '''15''' :  5  (1960)  pp. 77–141</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  P.L. Ul'yanov,   "Solved and unsolved problems in the theory of trigonometric and orthogonal series"  ''Russian Math. Surveys'' , '''19''' :  1  (1964)  pp. 1–62  ''Uspekhi Mat. Nauk'' , '''19''' :  1  (1964)  pp. 3–69</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  ''J. Soviet Math.'' , '''1''' :  6  (1973)  ''Itogi Nauk. Mat. Anal. 1970''  (1971)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  N. Bourbaki,   "Eléments d'histoire des mathématiques" , Hermann  (1974)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  A.B. Paplauskas,   "Trigonometric series from Euler to Lebesgue" , Moscow  (1966)  (In Russian)</TD></TR></table>

====Comments====
For (Weyl) multipliers see also [[Multiplier theory|Multiplier theory]].