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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Mixed and directional derivatives
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by W. Chen and Z. Ditzian PDF
Proc. Amer. Math. Soc. 108 (1990), 177-185 Request permission

Abstract:

The estimate \[ \left \| {\frac {{{\partial ^k}f\left ( x \right )}}{{\partial {\xi _1} \cdots \partial {\xi _k}}}} \right \| \leq {\sup _\xi }\left \| {\frac {{{\partial ^k}f\left ( x \right )}}{{\partial {\xi ^k}}}} \right \|\] is proved for various spaces of functions over domains in ${R^d}$ , where $\partial f / \partial {\xi _i}$ is the directional derivative of $f$ in the ${\xi _i}$, direction and ${\xi _1}, \ldots ,{\xi _k}$ are any $k$ directions.
References
  • Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
  • Z. Ditzian, Multivariate Landau-Kolmogorov-type inequality, Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 2, 335–350. MR 974990, DOI 10.1017/S0305004100067839
  • Robert C. Sharpley, Cone conditions and the modulus of continuity, Second Edmonton conference on approximation theory (Edmonton, Alta., 1982) CMS Conf. Proc., vol. 3, Amer. Math. Soc., Providence, RI, 1983, pp. 341–351. MR 729339
  • Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 108 (1990), 177-185
  • MSC: Primary 26D10; Secondary 41A44
  • DOI: https://doi.org/10.1090/S0002-9939-1990-0994773-0
  • MathSciNet review: 994773