## Time-Dependent Cauchy-Navier Splines and their Application to Seismic Wave Front Propagation

• In this paper a known orthonormal system of time- and space-dependent functions, that were derived out of the Cauchy-Navier equation for elastodynamic phenomena, is used to construct reproducing kernel Hilbert spaces. After choosing one of the spaces the corresponding kernel is used to define a function system that serves as a basis for a spline space. We show that under certain conditions there exists a unique interpolating or approximating, respectively, spline in this space with respect to given samples of an unknown function. The name "spline" here refers to its property of minimising a norm among all interpolating functions. Moreover, a convergence theorem and an error estimate relative to the point grid density are derived. As numerical example we investigate the propagation of seismic waves.

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Verfasserangaben: Paula Kammann, Volker Michel urn:nbn:de:hbz:386-kluedo-14373 Schriften zur Funktionalanalysis und Geomathematik (26) Preprint Englisch 2006 2006 Technische Universität Kaiserslautern 23.08.2006 Cauchy-Navier-Gleichung; reproduzierender Kern Cauchy-Navier equation; reproducing kernel ; seismic wave ; sphere ; spline Approximation; Elastizität ; Seismische Welle ; Sphäre ; Spline ; Zeitabhängigkeit Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 41-XX APPROXIMATIONS AND EXPANSIONS (For all approximation theory in the complex domain, see 30E05 and 30E10; for all trigonometric approximation and interpolation, see 42A10 and 42A15; for numerical approximation, see 65Dxx) / 41Axx Approximations and expansions / 41A05 Interpolation [See also 42A15 and 65D05] 41-XX APPROXIMATIONS AND EXPANSIONS (For all approximation theory in the complex domain, see 30E05 and 30E10; for all trigonometric approximation and interpolation, see 42A10 and 42A15; for numerical approximation, see 65Dxx) / 41Axx Approximations and expansions / 41A15 Spline approximation 41-XX APPROXIMATIONS AND EXPANSIONS (For all approximation theory in the complex domain, see 30E05 and 30E10; for all trigonometric approximation and interpolation, see 42A10 and 42A15; for numerical approximation, see 65Dxx) / 41Axx Approximations and expansions / 41A52 Uniqueness of best approximation 65-XX NUMERICAL ANALYSIS / 65Dxx Numerical approximation and computational geometry (primarily algorithms) (For theory, see 41-XX and 68Uxx) / 65D07 Splines 86-XX GEOPHYSICS [See also 76U05, 76V05] / 86Axx Geophysics [See also 76U05, 76V05] / 86A17 Global dynamics, earthquake problems Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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