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SPHA: SubPeriodic Harmonic Analyis


SubPeriodic Harmonic Analysis concerns trigonometric approximation on subintervals of the period. It is connected to the theory of Fourier Extensions (Boyd, Huybrechs, Adcock, et al.), to the recent theory of nonperiodic trigonometric approximations (Tal-Ezer), and to multivariate approximation and cubature on regions defined by circular arcs, such as planar circular sections and surface/solid sections of sphere, cylinder and torus (for example sectors, lenses, lunes, quadrangles, collars, slices, caps).

There are several potential multivariate applications, for example to numerical cubature on polygonal elements with curved sides for the discretization of PDEs, to computational adaptive optics (ray tracing and optical system design with vignetted annular pupils, e.g. for the the LSST: Large-Synoptic Survey Telescope camera), to regional scale modelling/simulation in spherical and toroidal geometry (in particular, geomathematical applications such as local geomagnetic field modelling).

Software

Posters

Papers

  1. Near-optimal polynomial interpolation on spherical triangles
    draft - A. Sommariva and M. Vianello
  2. Numerical hyperinterpolation over spherical triangles
    draft - A. Sommariva and M. Vianello
  3. Near-algebraic Tchakaloff-like quadrature on spherical triangles
    preprint - A. Sommariva and M. Vianello
    Appl. Math. Lett., published online 7 April 2021
  4. Algebraic cubature on polygonal elements with a circular edge
    draft - E. Artioli, A. Sommariva and M. Vianello
    Comput. Math. Appl. 79 (2020), 2057--2066
  5. Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization
    preprint - M. Vianello
    Optim. Lett. 12 (2018), 1659--1667
  6. Discrete norming inequalities on sections of sphere, ball and torus
    preprint - A. Sommariva and M. Vianello
    J. Inequal. Spec. Funct. 9-4 (2018), 113--121
  7. Stability inequalities for Lebesgue constants via Markov-like inequalities
    preprint - F. Piazzon and M. Vianello
    Dolomites Res. Notes Approx. DRNA 11 (2018), 1--9
  8. Subperiodic Trigonometric Hyperinterpolation
    preprint - G. Da Fies, A. Sommariva and M. Vianello
    in: "Contemporary Computational Mathematics - a celebration of the 80th birthday of Ian Sloan" (invited paper)
    J. Dick, F.Y. Kuo, H. Wozniakowski Eds., Springer, 2018, pp. 283--304
  9. Numerical quadrature on the intersection of planar disks
    preprint - A. Sommariva and M. Vianello
    FILOMAT 31:13 (2017), 4105--4115
  10. Subperiodic trigonometric subsampling: a numerical approach
    preprint - A. Sommariva and M. Vianello
    Appl. Anal. Discrete Math. 11 (2017), 470--483
  11. Polynomial approximation and quadrature on geographic rectangles
    preprint - M. Gentile, A. Sommariva and M. Vianello
    Appl. Math. Comput. 297 (2017), 159--179
  12. Numerical hyperinterpolation over nonstandard planar regions
    preprint - A. Sommariva and M. Vianello
    Math. Comput. Simulation, published online 30 July 2016
  13. Jacobi norming meshes
    preprint - F. Piazzon and M. Vianello
    Math. Inequal. Appl. 19 (2016), 1089--1095
  14. Polynomial fitting and interpolation on circular sections
    preprint - A. Sommariva and M. Vianello
    Appl. Math. Comput. 258 (2015), 410--424
  15. Product Gaussian quadrature on circular lunes
    preprint - G. Da Fies and M. Vianello
    Numer. Math. Theory Methods Appl. 7 (2014), 251--264
  16. Norming meshes by Bernstein-like inequalities
    preprint - M. Vianello
    Math. Inequal. Appl. 17 (2014), 929--936
  17. Algebraic cubature by linear blending of elliptical arcs
    preprint - G. Da Fies, A. Sommariva and M. Vianello
    Appl. Numer. Math. 74 (2013), 49--61
  18. Polynomial approximation on pyramids, cones and solids of rotation
    S. De Marchi and M. Vianello
    Dolomites Res. Notes Approx. DRNA 6 (2013), 20--26
  19. On the Lebesgue constant of subperiodic trigonometric interpolation
    preprint - G. Da Fies and M. Vianello
    J. Approx. Theory 167 (2013), 59--64
  20. Algebraic cubature on planar lenses and bubbles
    G. Da Fies and M. Vianello
    Dolomites Res. Notes Approx. DRNA 5 (2012), 7--12
  21. Trigonometric Gaussian quadrature on subintervals of the period
    preprint - G. Da Fies and M. Vianello
    Electron. Trans. Numer. Anal. 39 (2012), 102--112
  22. Subperiodic trigonometric interpolation and quadrature
    preprint - L. Bos and M. Vianello
    Appl. Math. Comput. 218 (2012), 10630--10638