View source for Closed system of elements (functions)

A system of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c0226001.png" /> of some normed linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c0226002.png" /> such that any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c0226003.png" /> can be approximated with arbitrary accuracy (in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c0226004.png" />) by a finite linear combination of elements of the system; in other words: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c0226005.png" /> there exist numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c0226006.png" /> such that

<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/c/c022/c022600/c0226007.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

For example, the system of powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c0226008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c0226009.png" /> is closed in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260010.png" /> of functions summable in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260011.png" />-th degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260012.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260013.png" />-summable) on a finite interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260014.png" /> with (integral) weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260015.png" />. The inequality (1) then reads

<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/c/c022/c022600/c02260016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260017.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260018.png" />. In the most familiar cases, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260019.png" /> is an orthonormal sequence of elements in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260020.png" />. The closure condition (1) is then equivalent to the condition: For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260021.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/c/c022/c022600/c02260022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260023.png" /> are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260024.png" /> relative to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260025.png" />. In the case of a trigonometric system of functions, condition (3) is known as the Parseval identity; explicitly, it takes 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/c/c022/c022600/c02260026.png" /></td> </tr></table>

This equality was studied by M. Parseval (1806), Ch.J. de la Vallée-Poussin (1890) and A. Hurwitz (1901–1903); a rigorous proof of its validity (for bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260027.png" />) was given by A.M. Lyapunov (1896).

The general case of the closure condition (3) was first examined in detail by V.A. Steklov (1898), in connection with the solution of certain problems in mathematical physics. Introducing the term  "closed" , Steklov studied the relevant aspects of various specific systems of orthogonal functions — in particular, the fundamental solutions of the Sturm–Liouville equation (see [[Sturm–Liouville problem|Sturm–Liouville problem]]). Equality (3) is therefore often called the Parseval–Steklov closure condition.

The concept of closure is widely applied in the theory of [[Orthogonal polynomials|orthogonal polynomials]]. If the orthogonality interval is finite, a system of orthogonal polynomials is closed for any weight function. For an infinite orthogonality interval, Steklov established various conditions on the weight function that are sufficient to ensure the closure of the corresponding system of orthogonal polynomials; in particular, he proved that the systems of Hermite and Laguerre polynomials are closed. One of Steklov's sufficient conditions for an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260028.png" /> is the existence of a sequence of positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260029.png" /> such that

<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/c/c022/c022600/c02260030.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/c/c022/c022600/c02260031.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260032.png" /> is a differential weight (function) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260033.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260034.png" />). A sufficient condition for the entire real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260035.png" /> is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260036.png" /> be almost-everywhere positive and satisfy an inequality 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/c/c022/c022600/c02260037.png" /></td> </tr></table>

In a certain sense this condition is  "nearly"  necessary; indeed, as Steklov proved, a system of polynomials is non-closed if the differential weight function is 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/c/c022/c022600/c02260038.png" /></td> </tr></table>

A complete solution to the problem of closure conditions for a system of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260039.png" /> for an infinite interval was given by M. Riesz (1922). He proved that a system of polynomials is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260040.png" /> if and only if either the corresponding [[Moment problem|moment problem]] is well-defined or, if it is not, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260041.png" /> is a so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260043.png" />-extremal solution to the moment problem.

A concept intimately connected with that of closure is that of completeness of a system of functions: A system of functions is called complete if, whenever a bounded linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260044.png" /> vanishes on all elements of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260045.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260046.png" />. In a Hilbert space, completeness and closure are equivalent. Completeness of a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260047.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260048.png" /> is equivalent to closure of the same system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260052.png" />.

Analogues of the inequalities (1) or (2) in a complex domain have been studied in detail for systems of polynomials and for more general systems of functions (see [[Orthogonal polynomials on a complex domain|Orthogonal polynomials on a complex domain]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,   H. Steinhaus,   "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Ya.L. Geronimus,   "Theory of orthogonal polynomials. A survey of the achievements in Soviet mathematics" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Szegö,   "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  V.A. Steklov,   ''Zap. Akad. Nauk Ser. Fiz.-Mat.'' , '''30''' :  4  (1911)  pp. 1–86</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  V.A. Steklov,   ''Zap. Akad. Nauk Ser. Fiz.-Mat.'' , '''33''' :  8  (1914)  pp. 1–59</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.A. Steklov,   "Fundamental problems of mathematical physics" , '''1–2''' , St. Petersburg  (1922–1923)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M. Riesz,   "Sur le problème des moments et le théorème de Parseval correspondant"  ''Acta Szeged Sect. Math.'' , '''1'''  (1923)  pp. 209–225  ''Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephine, Sect. Sci. Math.'' , '''1'''  (1922–1923)  pp. 209–225</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E. Hewitt,   "Remark on orthonormal sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022600/c02260053.png" />"  ''Amer. Math. Monthly'' , '''61'''  (1954)  pp. 249–250</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Ya.L. Geronimus,   "Polynomials orthogonal on circle and interval" , Pergamon  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Alexits,   "Convergence problems of orthogonal series" , Pergamon  (1961)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Freud,   "Orthogonal polynomials" , Pergamon  (1971)  (Translated from German)</TD></TR></table>