### Abstract

Let D be a bounded symmetric domain of tube type and Σ be the Shilov boundary of D. Denote by H_{2}(D) and A_{2}(D) the Hardy and Bergman spaces, respectively, of holomorphic functions on D; and let B(H_{2}(D)) and B(A_{2}(D)) denote the closed unit balls in these spaces. For an integer l≥0 we define the notion R^{l}f of the lth radial derivative of a holomorphic function f on D, and we prove the following results: Let 0<ρ<1. Denote by W the class of holomorphic functions f on D for which R^{l}f∈B(H_{2}(D)) and set X=C(ρΣ). Then we show that the linear and Gelfand N-widths of W in X coincide, and we compute the exact value. We do the same for the case in which W is the class of holomorphic functions f for which R^{l}f∈B(A_{2}(D)), and X=C(ρΣ). Next, let X=L^{p}(ρΣ) (respectively, L^{p}(ρD)) for 1≤p≤∞, and let W be a class of holomorphic functions f on D for which R^{l}f∈B(H_{p}(D)) (respectively, B(A_{p}(D))). We show that the Kolmogorov, linear, Gelfand, and Bernstein N-widths all coincide, we calculate the exact value, and we identify optimal subspaces or optimal linear operators. These results extend work of Yu. A. Farkov (1993, J. Approx. Theory75, 183-197) and K. Yu. Osipenko (1995, J. Approx. Theory82, 135-155), and initiate the study of N-widths of spaces of holomorphic functions on bounded symmetric domains.

Original language | English (US) |
---|---|

Pages (from-to) | 121-141 |

Number of pages | 21 |

Journal | Journal of Approximation Theory |

Volume | 104 |

Issue number | 1 |

DOIs | |

State | Published - May 1 2000 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Numerical Analysis
- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of Approximation Theory*,

*104*(1), 121-141. https://doi.org/10.1006/jath.1999.3445

}

*Journal of Approximation Theory*, vol. 104, no. 1, pp. 121-141. https://doi.org/10.1006/jath.1999.3445

**The N-Widths of Spaces of Holomorphic Functions on Bounded Symmetric Domains of Tube Type.** / Ding, Hongming; Gross, Kenneth I.; Richards, Donald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The N-Widths of Spaces of Holomorphic Functions on Bounded Symmetric Domains of Tube Type

AU - Ding, Hongming

AU - Gross, Kenneth I.

AU - Richards, Donald

PY - 2000/5/1

Y1 - 2000/5/1

N2 - Let D be a bounded symmetric domain of tube type and Σ be the Shilov boundary of D. Denote by H2(D) and A2(D) the Hardy and Bergman spaces, respectively, of holomorphic functions on D; and let B(H2(D)) and B(A2(D)) denote the closed unit balls in these spaces. For an integer l≥0 we define the notion Rlf of the lth radial derivative of a holomorphic function f on D, and we prove the following results: Let 0<ρ<1. Denote by W the class of holomorphic functions f on D for which Rlf∈B(H2(D)) and set X=C(ρΣ). Then we show that the linear and Gelfand N-widths of W in X coincide, and we compute the exact value. We do the same for the case in which W is the class of holomorphic functions f for which Rlf∈B(A2(D)), and X=C(ρΣ). Next, let X=Lp(ρΣ) (respectively, Lp(ρD)) for 1≤p≤∞, and let W be a class of holomorphic functions f on D for which Rlf∈B(Hp(D)) (respectively, B(Ap(D))). We show that the Kolmogorov, linear, Gelfand, and Bernstein N-widths all coincide, we calculate the exact value, and we identify optimal subspaces or optimal linear operators. These results extend work of Yu. A. Farkov (1993, J. Approx. Theory75, 183-197) and K. Yu. Osipenko (1995, J. Approx. Theory82, 135-155), and initiate the study of N-widths of spaces of holomorphic functions on bounded symmetric domains.

AB - Let D be a bounded symmetric domain of tube type and Σ be the Shilov boundary of D. Denote by H2(D) and A2(D) the Hardy and Bergman spaces, respectively, of holomorphic functions on D; and let B(H2(D)) and B(A2(D)) denote the closed unit balls in these spaces. For an integer l≥0 we define the notion Rlf of the lth radial derivative of a holomorphic function f on D, and we prove the following results: Let 0<ρ<1. Denote by W the class of holomorphic functions f on D for which Rlf∈B(H2(D)) and set X=C(ρΣ). Then we show that the linear and Gelfand N-widths of W in X coincide, and we compute the exact value. We do the same for the case in which W is the class of holomorphic functions f for which Rlf∈B(A2(D)), and X=C(ρΣ). Next, let X=Lp(ρΣ) (respectively, Lp(ρD)) for 1≤p≤∞, and let W be a class of holomorphic functions f on D for which Rlf∈B(Hp(D)) (respectively, B(Ap(D))). We show that the Kolmogorov, linear, Gelfand, and Bernstein N-widths all coincide, we calculate the exact value, and we identify optimal subspaces or optimal linear operators. These results extend work of Yu. A. Farkov (1993, J. Approx. Theory75, 183-197) and K. Yu. Osipenko (1995, J. Approx. Theory82, 135-155), and initiate the study of N-widths of spaces of holomorphic functions on bounded symmetric domains.

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U2 - 10.1006/jath.1999.3445

DO - 10.1006/jath.1999.3445

M3 - Article

AN - SCOPUS:0009581091

VL - 104

SP - 121

EP - 141

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

IS - 1

ER -