TY - JOUR
T1 - Integrability, mean convergence, and parseval's formula for double trigonometric series
AU - Chen, Chang Pao
AU - Lin, Chin Cheng
PY - 1998/6
Y1 - 1998/6
N2 - Consider the double trigonometric series whose coefficients satisfy conditions of bounded variation of order (p, 0), (0, p), and (p, p) with the weight (|j|̄ |k|̄)p-1 for some p > 1. The following properties concerning the rectangular partial sums of this series are obtained: (a) regular convergence; (b) uniform convergence; (c) weighted Lr-integrability and weighted Lr-convergence; and (d) Parseval's formula. Our results generalize Bary [1, p. 656], Boas [2, 3], Chen [6, 7], Kolmogorov [9], Marzug [10], Móricz [11, 12, 13, 14], Ul'janov [15], Young [16], and Zygmund [17, p. 4].
AB - Consider the double trigonometric series whose coefficients satisfy conditions of bounded variation of order (p, 0), (0, p), and (p, p) with the weight (|j|̄ |k|̄)p-1 for some p > 1. The following properties concerning the rectangular partial sums of this series are obtained: (a) regular convergence; (b) uniform convergence; (c) weighted Lr-integrability and weighted Lr-convergence; and (d) Parseval's formula. Our results generalize Bary [1, p. 656], Boas [2, 3], Chen [6, 7], Kolmogorov [9], Marzug [10], Móricz [11, 12, 13, 14], Ul'janov [15], Young [16], and Zygmund [17, p. 4].
KW - Conditions of bounded variation
KW - Double trigonometric series
KW - Parseval's formula
KW - Rectangular partial sums
KW - Regular convergence
KW - Uniform convergence
KW - Weighted L-convergence
KW - Weighted L-integrability
UR - http://www.scopus.com/inward/record.url?scp=0346876144&partnerID=8YFLogxK
U2 - 10.11650/twjm/1500406932
DO - 10.11650/twjm/1500406932
M3 - 期刊論文
AN - SCOPUS:0346876144
SN - 1027-5487
VL - 2
SP - 191
EP - 212
JO - Taiwanese Journal of Mathematics
JF - Taiwanese Journal of Mathematics
IS - 2
ER -