Uniform limits of sequences of polynomials and their derivatives
Authors:
Joseph A. Ball and Thomas R. Fanney
Journal:
Proc. Amer. Math. Soc. 114 (1992), 749755
MSC:
Primary 41A10; Secondary 40A30, 41A65, 47E05
MathSciNet review:
1091175
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Abstract: Let be a compact subset of the unit interval , and let denote the space of functions continuous on with the uniform norm. Consider the densely defined operator given by for all polynomials . Let represent the graph of , that is polynomials} considered as a submanifold of . Write the interior of the set as a countable union of disjoint open intervals and let be the union of the closure of these intervals. The main result is that the closure of is equal to the set of all functions such that is absolutely continuous on and . As a consequence, the operator is closable if and only if the set is the closure of its interior. On the other extreme, is dense in i.e. for any pair , there exists a sequence of polynomials so that and uniformly on , if and only if the interior of is empty.
 [B1]
Errett
Bishop, Approximation by a polynomial and its
derivatives on certain closed sets, Proc. Amer.
Math. Soc. 9
(1958), 946–953. MR 0109225
(22 #112), http://dx.doi.org/10.1090/S00029939195801092251
 [B2]
Errett
Bishop, Simultaneous approximation by a
polynomial and its derivatives, Proc. Amer.
Math. Soc. 10
(1959), 741–743. MR 0109226
(22 #113), http://dx.doi.org/10.1090/S00029939195901092264
 [BF]
Joseph
A. Ball and Thomas
R. Fanney, Closability of differential operators and real
subJordan operators, Topics in operator theory: Ernst D. Hellinger
memorial volume, Oper. Theory Adv. Appl., vol. 48, Birkhäuser,
Basel, 1990, pp. 93–156. MR 1207395
(94g:47038)
 [CV]
Charles
O. Christenson and William
L. Voxman, Aspects of topology, Marcel Dekker, Inc., New
YorkBasel, 1977. Pure and applied Mathematics, Vol. 39. MR 0487938
(58 #7521)
 [F]
T. R. Fanney, Closability of differential operators and subjordan operators, Ph.D. dissertation, Virginia Polytechnic Institute and State University, 1989.
 [FR]
U.
Fixman and L.
A. Rubel, Limits of sequences of rational functions and their
derivatives, Houston J. Math. 5 (1979), no. 1,
75–87. MR
533641 (83h:30044)
 [R]
Walter
Rudin, Functional analysis, McGrawHill Book Co., New
YorkDüsseldorfJohannesburg, 1973. McGrawHill Series in Higher
Mathematics. MR
0365062 (51 #1315)
 [B1]
 E. Bishop, Approximation by a polynomial and its derivatives on certain closed sets, Proc. Amer. Math. Soc. 9 (1958), 946953. MR 0109225 (22:112)
 [B2]
 , Simultaneous approximation by a polynomial and its derivatives, Proc. Amer. Math. Soc. 10 (1959), 741743. MR 0109226 (22:113)
 [BF]
 J. A. Ball and T. R. Fanney, Closability of differential operators and real subjordan operators, Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, (L. de Branges, I. Gohberg, and J. Rounyak, eds.), Operator Theory: Advances and Applications, vol. 48, BirkhäuserVerlag, Basel, 1990, pp. 93156. MR 1207395 (94g:47038)
 [CV]
 C. O. Christenson and W. L. Voxman, Aspects of topology, Marcel Dekker, New York, 1977. MR 0487938 (58:7521)
 [F]
 T. R. Fanney, Closability of differential operators and subjordan operators, Ph.D. dissertation, Virginia Polytechnic Institute and State University, 1989.
 [FR]
 U. Fixman and L. A. Rubel, Limits of sequences of rational functions and their derivatives, Houston J. Math. 5 (1979), 7587. MR 533641 (83h:30044)
 [R]
 W. Rudin, Functional analysis, McGrawHill, New York, 1973. MR 0365062 (51:1315)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199210911758
PII:
S 00029939(1992)10911758
Article copyright:
© Copyright 1992
American Mathematical Society
