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A summation domain of multiple [[Fourier series|Fourier series]] (cf. also [[Partial Fourier sum|Partial Fourier sum]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h1301301.png" /> be an integrable periodic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h1301302.png" /> variables defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h1301303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h1301304.png" />. It has an expansion as a Fourier series, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h1301305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h1301306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h1301307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h1301308.png" />. Unlike in the one-dimensional case, there is no natural ordering of the Fourier coefficients, so the choice of the order of summation is of great importance.
 
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h1301309.png" /> with all coordinates positive, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h13013010.png" />. Consider the [[Differential operator|differential operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h13013011.png" /> with periodic boundary conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h13013012.png" />. Then the eigenvalues (cf. [[Eigen value|Eigen value]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h13013013.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h13013014.png" />, while the corresponding eigenfunctions are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h13013015.png" />. The partial sums of the Fourier series corresponding to the eigenfunctions with all eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h13013016.png" /> are called hyperbolic partial Fourier sums of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h13013017.png" /> (or hyperbolic crosses). This approach, in which the method of summation of the Fourier series is defined by the differential operator, is due to K. Babenko [[#References|[a1]]], who applied it to various problems in [[Approximation theory|approximation theory]] (e.g., Kolmogorov widths, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h13013019.png" />-entropy, etc.). Subsequently the hyperbolic cross itself became the object of study in connection with Lebesgue constants, the Bernshtein inequality, etc. Also, this approach initiated a detailed study and applications of spaces of functions with bounded mixed derivative (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130130/h13013020.png" />).
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A summation domain of multiple [[Fourier series|Fourier series]] (cf. also [[Partial Fourier sum|Partial Fourier sum]]). Let $f ( x )$ be an integrable periodic function of $n$ variables defined on $\mathbf{T} ^ { n }$, $\mathbf{T} = ( - \pi , \pi ]$. It has an expansion as a Fourier series, $\sum _ { \mathbf{k} } c_{ \mathbf{k} } e ^ { i \mathbf{kx} }$, $\mathbf{k} = ( k _ { 1 } , \dots , k _ { n } )$, ${\bf x} = ( x _ { 1 } , \ldots , x _ { n } )$, $\mathbf{k} \cdot \mathbf{x} = k _ { 1 } x _ { 1 } + \ldots + k _ { n } x _ { n }$. Unlike in the one-dimensional case, there is no natural ordering of the Fourier coefficients, so the choice of the order of summation is of great importance.
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Let $\mathbf{r} = ( r _ { 1 } , \dots , r _ { n } ) \in \mathbf{R} ^ { n }$ with all coordinates positive, $r_j > 0$. Consider the [[Differential operator|differential operator]] $D^{\mathbf{r}} = \partial^{r_1 + \dots + r_n} / \partial^{r_1} x_1 \dots \partial^{r_n} x_{n}$ with periodic boundary conditions on $\mathbf{T} ^ { n }$. Then the eigenvalues (cf. [[Eigen value|Eigen value]]) of $D ^ { \mathbf{r} }$ are $\lambda_{\mathbf{k}} = i^{r_1 + \dots + r_n} k_1^{r_1} \dots k_n^{r_n}$, while the corresponding eigenfunctions are $e ^ { i \mathbf k . \mathbf x }$. The partial sums of the Fourier series corresponding to the eigenfunctions with all eigenvalues $| \lambda _ { \mathbf{k} } | \leq N$ are called hyperbolic partial Fourier sums of order $N$ (or hyperbolic crosses). This approach, in which the method of summation of the Fourier series is defined by the differential operator, is due to K. Babenko [[#References|[a1]]], who applied it to various problems in [[Approximation theory|approximation theory]] (e.g., Kolmogorov widths, $\varepsilon$-entropy, etc.). Subsequently the hyperbolic cross itself became the object of study in connection with Lebesgue constants, the Bernshtein inequality, etc. Also, this approach initiated a detailed study and applications of spaces of functions with bounded mixed derivative (in $L _ { p }$).
  
 
Many of these and related classes, as well as various problems in approximation theory, are considered in [[#References|[a2]]]. This method of summation has also been applied to other series expansions, e.g., multiple wavelets systems.
 
Many of these and related classes, as well as various problems in approximation theory, are considered in [[#References|[a2]]]. This method of summation has also been applied to other series expansions, e.g., multiple wavelets systems.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Babenko,  "Approximation of periodic functions of many variables by trigonometric polynomials"  ''Soviet Math.'' , '''1'''  (1960)  pp. 513–516  ''Dokl. Akad. Nauk. SSSR'' , '''132'''  (1960)  pp. 247–250</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V. Temlyakov,  "Approximation of periodic functions" , Nova Sci.  (1993)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  K. Babenko,  "Approximation of periodic functions of many variables by trigonometric polynomials"  ''Soviet Math.'' , '''1'''  (1960)  pp. 513–516  ''Dokl. Akad. Nauk. SSSR'' , '''132'''  (1960)  pp. 247–250</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  V. Temlyakov,  "Approximation of periodic functions" , Nova Sci.  (1993)</td></tr></table>

Latest revision as of 01:04, 15 February 2024

A summation domain of multiple Fourier series (cf. also Partial Fourier sum). Let $f ( x )$ be an integrable periodic function of $n$ variables defined on $\mathbf{T} ^ { n }$, $\mathbf{T} = ( - \pi , \pi ]$. It has an expansion as a Fourier series, $\sum _ { \mathbf{k} } c_{ \mathbf{k} } e ^ { i \mathbf{kx} }$, $\mathbf{k} = ( k _ { 1 } , \dots , k _ { n } )$, ${\bf x} = ( x _ { 1 } , \ldots , x _ { n } )$, $\mathbf{k} \cdot \mathbf{x} = k _ { 1 } x _ { 1 } + \ldots + k _ { n } x _ { n }$. Unlike in the one-dimensional case, there is no natural ordering of the Fourier coefficients, so the choice of the order of summation is of great importance.

Let $\mathbf{r} = ( r _ { 1 } , \dots , r _ { n } ) \in \mathbf{R} ^ { n }$ with all coordinates positive, $r_j > 0$. Consider the differential operator $D^{\mathbf{r}} = \partial^{r_1 + \dots + r_n} / \partial^{r_1} x_1 \dots \partial^{r_n} x_{n}$ with periodic boundary conditions on $\mathbf{T} ^ { n }$. Then the eigenvalues (cf. Eigen value) of $D ^ { \mathbf{r} }$ are $\lambda_{\mathbf{k}} = i^{r_1 + \dots + r_n} k_1^{r_1} \dots k_n^{r_n}$, while the corresponding eigenfunctions are $e ^ { i \mathbf k . \mathbf x }$. The partial sums of the Fourier series corresponding to the eigenfunctions with all eigenvalues $| \lambda _ { \mathbf{k} } | \leq N$ are called hyperbolic partial Fourier sums of order $N$ (or hyperbolic crosses). This approach, in which the method of summation of the Fourier series is defined by the differential operator, is due to K. Babenko [a1], who applied it to various problems in approximation theory (e.g., Kolmogorov widths, $\varepsilon$-entropy, etc.). Subsequently the hyperbolic cross itself became the object of study in connection with Lebesgue constants, the Bernshtein inequality, etc. Also, this approach initiated a detailed study and applications of spaces of functions with bounded mixed derivative (in $L _ { p }$).

Many of these and related classes, as well as various problems in approximation theory, are considered in [a2]. This method of summation has also been applied to other series expansions, e.g., multiple wavelets systems.

References

[a1] K. Babenko, "Approximation of periodic functions of many variables by trigonometric polynomials" Soviet Math. , 1 (1960) pp. 513–516 Dokl. Akad. Nauk. SSSR , 132 (1960) pp. 247–250
[a2] V. Temlyakov, "Approximation of periodic functions" , Nova Sci. (1993)
How to Cite This Entry:
Hyperbolic cross. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_cross&oldid=15654
This article was adapted from an original article by E.S. Belinsky (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article