View source for Partial Fourier sum

A partial sum of the [[Fourier series|Fourier series]] of a given [[Function|function]].

In the classical one-dimensional case where a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p1200701.png" /> is integrable on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p1200702.png" /> and

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is its trigonometric Fourier series, the partial Fourier sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p1200704.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p1200705.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p1200706.png" /> is the [[Trigonometric polynomial|trigonometric polynomial]]

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With the use of the sequence of partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p1200708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p1200709.png" />, the notion of convergence of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007010.png" /> is introduced and its sum at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007011.png" /> is defined as follows:

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At every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007013.png" />, the Dirichlet formula

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is true; here,

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is the [[Dirichlet kernel|Dirichlet kernel]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007016.png" />. This formula plays a key role in many problems in the theory of [[Fourier series|Fourier series]].

If a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007017.png" /> is given in complex form, i.e., 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/p/p120/p120070/p12007018.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/p/p120/p120070/p12007019.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007020.png" /> is the set of all integers, then

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In the multi-dimensional case, a notion of partial sum can be introduced in numerous different ways, none of which can be regarded as preferable.

One of the possible general approaches is as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007022.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007023.png" />-dimensional Euclidean space of points (vectors) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007024.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007025.png" /> be the integer lattice in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007026.png" />, i.e., the set of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007027.png" /> with integer coordinates. For vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007028.png" />, let

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Further, let

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let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007031.png" /> be a function that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007032.png" />-periodic in each variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007033.png" /> and integrable over a cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007034.png" />, and let

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be its Fourier series.

Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007037.png" /> be a family of bounded domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007038.png" /> that depend on a real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007039.png" /> and are such that any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007040.png" /> belongs to all domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007041.png" /> for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007042.png" />. In this case, the expression

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is called a partial Fourier sum of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007044.png" /> corresponding to the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007045.png" />, and the expression

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is called the Dirichlet kernel corresponding to this domain. It is clear that, for any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007047.png" />, the following formula holds:

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This definition allows one to consider the problem of the convergence (or summability) of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007049.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007050.png" />. By virtue of the boundedness of the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007051.png" /> the expression for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007052.png" /> is always a [[Trigonometric polynomial|trigonometric polynomial]].

The cases where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007053.png" />-dimensional spheres or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007054.png" />-dimensional intervals centred at the origin are taken as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007055.png" /> are most often encountered and are well studied. The expressions

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are called spherical partial sums, and the expressions

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007058.png" /> is an arbitrary vector from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007059.png" /> with positive coordinates, are called rectangular partial sums. In recent years, in connection with problems in the approximation of functions from Sobolev spaces, partial Fourier sums constructed by  "hyperbolic crosses" , namely, expressions 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/p/p120/p120070/p12007060.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/p/p120/p120070/p12007061.png" /></td> </tr></table>

have been extensively used. For Fourier series in general orthonormal systems of functions, partial Fourier series are constructed analogously. (Cf. also [[Orthonormal system|Orthonormal system]].)

Various properties of partial Fourier sums and their applications to the theory of approximation and other fields of science can be found in, e.g., [[#References|[a1]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a7]]]. [[#References|[a6]]], [[#References|[a2]]],

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bary,   "Treatise on trigonometric series" , '''1; 2''' , Pergamon  (1964) {{ZBL|0129.28002}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Zygmund,   "Trigonometrical series" , '''1; 2''' , Cambridge Univ. Press  (1959)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Edwards,   "Fourier series: A modern introduction" , '''1; 2''' , Springer  (1979)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''1; 2''' , Springer  (1963/70)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Rudin,   "Fourier analysis on groups" , Interscience  (1962)</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Szegő,   "Orthogonal polynomials" , Amer. Math. Soc.  (1959)</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Stepanets,   "Classification and approximation of periodic functions" , Kluwer Acad. Publ.  (1995)</TD></TR>
</table>