Bounded point derivations on $R^p(X)$ and approximate derivatives

  • Stephen Deterding

Abstract

It is shown that if a point $x_0$ admits a bounded point derivation on $R^p(X)$, the closure of rational function with poles off $X$ in the $L^p(dA)$ norm, for $p >2$, then there is an approximate derivative at $x_0$. A similar result is proven for higher-order bounded point derivations. This extends a result of Wang which was proven for $R(X)$, the uniform closure of rational functions with poles off $X$.

References

Bishop, E., A minimal boundary for function algebras, Pacific J. Math. 9 (1959), 629–642.

Brennan, J. E., Invariant subspaces and rational approximation, J. Functional Analysis 7 (1971), 285–310.

Browder, A., Point derivations on function algebras, J. Functional Analysis 1 (1967), 22–27.

Dolženko, E. P., Construction on a nowhere dense continuum of a nowhere differentiable function which can be expanded into a series of rational functions, Dokl. Akad. Nauk SSSR 125 (1959), 970–973.

Fernström, C. and Polking, J. C., Bounded point evaluations and approximation in $L^p$ by solutions of elliptic partial differential equations, J. Functional Analysis 28 (1978), no. 1, 1–20.

Hedberg, L. I., Bounded point evaluations and capacity, J. Functional Analysis 10 (1972), 269–280.

Hedberg, L. I., Non-linear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972), 299–319. https://doi.org/10.1007/BF01181619

Sinanjan, S. O., The uniqueness property of analytic functions on closed see without interior points, Sibirsk. Mat. Ž. 6 (1965), 1365–1381.

Wang, J. L. M., An approximate Taylor's theorem for $R(X)$, Math. Scand. 33 (1973), 343–358. https://doi.org/10.7146/math.scand.a-11496

Wilken, D. R., Bounded point derivations and representing measures on $R(X)$, Proc. Amer. Math. Soc. 24 (1970), 371–373. https://doi.org/10.2307/2036364
Published
2019-01-13
How to Cite
Deterding, S. (2019). Bounded point derivations on $R^p(X)$ and approximate derivatives. MATHEMATICA SCANDINAVICA, 124(1), 132-148. https://doi.org/10.7146/math.scand.a-109998
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Articles