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 St P.

N1 - Funding Information:
Let D be a bounded symmetric domain of tube type and 7 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<\<1. Denote by W the class of holomorphic functions f on D for which Rlf#B(H2(D)) and set X=C(\7). 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(\7). Next, let X=Lp(\7) (respectively, Lp(\D)) 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. Theory 75, 183 197) and K. Yu. Osipenko (1995, J. Approx. Theory 82, 135 155), and initiate the study of N-widths of spaces of holomorphic functions on bounded symmetric domains. ν 2000 Academic Press 1Research supported by National Science Foundation Grant DMS-9500907. 2Research supported by National Science Foundation Grant DMS-9501191. 3Research supported by National Science Foundation Grant DMS-9703705.

PY - 2000/5

Y1 - 2000/5

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.

UR - http://www.scopus.com/inward/record.url?scp=0009581091&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0009581091&partnerID=8YFLogxK

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 -