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Hyperbolic cross

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A summation domain of multiple Fourier series (cf. also Partial Fourier sum). Let 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=55490
This article was adapted from an original article by E.S. Belinsky (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article