In  and some following publications, Tadmor and Gelb took up a well known property of conjugate Fourier series in 1-d, namely the property to detect jump discontinuities in given spectral data. In fact, this property of conjugate series is known for quite a long time. The research in papers around the year 1910 shows that there were also other means of detecting jumps observed and analysed. We review the classical results as well as the results of Gelb and Tadmor and demonstrate their discrete case using different estimates in all detail. It is worth noting that the techniques presented are not global but local techniques. Edges are a local phenomenon and can only be found appropriately by local means. Furthermore, applying a different approach in the proof of the main estimate leads to weaker preconditions in the discrete case. Finally an outlook to a two-dimensional approach based on the work of Móricz, in which jumps in the mixed second derivative of a 2-d function are detected, is made.
Cite this paper
P. Öffner, T. Sonar and M. Wirz, "Detecting Strength and Location of Jump Discontinuities in Numerical Data," Applied Mathematics
, Vol. 4 No. 12, 2013, pp. 1-14. doi: 10.4236/am.2013.412A001
 A. Gelb and E. Tadmor, “Detection of Edges in Spectral Data,” Applied and Computational Harmonic Analysis, Vol. 7, No. 1, 1999, pp. 101-135. http://dx.doi.org/10.1006/acha.1999.0262
 A. Gelb and E. Tadmor, “Detection of Edges in Spectral Data II. Nonlinear Enhancemen,” SIAM Journal on Numerical Analysis, Vol. 38, No. 4, 2000, pp. 1389-1408. http://dx.doi.org/10.1137/S0036142999359153
 A. Gelb and E. Tadmor, “Spectral Reconstruction of Piecewise Smooth Functions from Their Discrete Data,” ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 38, No. 2, 2002, pp. 155-175.
 E. Tadmor, “Filters, mollifiers and the computation of the Gibbs phenomenon,” Acta Numerica, Vol. 16, 2005, pp. 305-378. http://dx.doi.org/10.1017/S0962492906320016
 L. Fejér, “über die Bestimmung des Sprunges einer Funktion aus ihrer Fourierreihe,” Journal für die reine und angewandte Mathematik, Vol. 142, 1913, pp. 165188.
 L. Fejér, “über Konjugierte Trigonometrische Reihen,” Journal für die reine und angewandte Mathematik, Vol. 144, 1914, pp. 48-56.
 F. Lukács, “über die Bestimmung des Sprunges einer Funktion aus ihrer Fourierreihe,” Journal für die reine und angewandte Mathematik, Vol. 150, 1920, pp. 107112.
 A. Zygmund, “Trigonometric Series,” 2nd Edition, Cambridge University Press, 1959.
 F. Móricz, “Extension of a Theorem of Ferenc Lukács from Single to Double Conjugate Series,” Journal of Mathematical Analysis and Applications, Vol. 259, 2001, pp. 582-595. http://dx.doi.org/10.1006/jmaa.2001.7432
 M. Wirz, “Ein Spektrale-Differenzen-Verfahren mit modaler Filterung und zweidimensionaler Kantendetektierung mithilfe konjugierter Fourierreihen,” Dissertation, Cuvillier and TU Braunschweig, 2012.
 H. S. Carslaw, “An Introduction to the Theory of Fourier’s Series and Integrals,” Dover Publications, New York, 1950.
 L. V. Zhizhiashvili, “Trigonometric Fourier Series and Their Conjugates,” Kluwer Academic Publishers, New York, 1996.
 F. Móricz, “Approximation by Rectangular Partial Sums of Double Conjugate Fourier Series,” Journal of Approximation Theory, Vol. 103, 2000, pp. 130-150. http://dx.doi.org/10.1006/jath.1999.3422
 L. Zhizhiashvili and K. Sokol-Sokolowski, “On Trigonometric Series Conjugate to Fourier Series of Two Variables,” Fundamenta Mathematicae, Vol. 34, 1947, pp. 166-182.
 V. L. Shapiro, “Fourier Series in Several Variables,” Bulletin of the AMS, Vol. 70, No. 1, 1964, pp. 48-93. http://dx.doi.org/10.1090/S0002-9904-1964-11026-0
 J. M. Ash and L. Gluck, “Convergence and Divergence of Series Conjugate to a Convergent Multiple Fourier Series,” Trans-AMS, Vol. 207, 1975, pp. 127-142.
 á. Jenei, “Pointwise Convergence of Fourier and Conjugate Series of Periodic Functions in Two Variables,” Ph.D. Thesis, University of Szeged, Bolyai Institute, 2009.