Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 077, 10 pages      arXiv:1508.06884
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Moments and Legendre-Fourier Series for Measures Supported on Curves

Jean B. Lasserre
LAAS-CNRS and Institute of Mathematics, University of Toulouse, 7 Avenue du Colonel Roche, BP 54 200, 31031 Toulouse Cédex 4, France

Received August 28, 2015, in final form September 26, 2015; Published online September 29, 2015

Some important problems (e.g., in optimal transport and optimal control) have a relaxed (or weak) formulation in a space of appropriate measures which is much easier to solve. However, an optimal solution $\mu$ of the latter solves the former if and only if the measure $\mu$ is supported on a ''trajectory'' $\{(t,x(t))\colon t\in [0,T]\}$ for some measurable function $x(t)$. We provide necessary and sufficient conditions on moments $(\gamma_{ij})$ of a measure $d\mu(x,t)$ on $[0,1]^2$ to ensure that $\mu$ is supported on a trajectory $\{(t,x(t))\colon t\in [0,1]\}$. Those conditions are stated in terms of Legendre-Fourier coefficients ${\mathbf f}_j=({\mathbf f}_j(i))$ associated with some functions $f_j\colon [0,1]\to {\mathbb R}$, $j=1,\ldots$, where each ${\mathbf f}_j$ is obtained from the moments $\gamma_{ji}$, $i=0,1,\ldots$, of $\mu$.

Key words: moment problem; Legendre polynomials; Legendre-Fourier series.

pdf (387 kb)   tex (16 kb)


  1. Beiglböck M., Griessler C., A land of monotone plenty, Technical report, Department of Mathematics, University of Vienna, 2015.
  2. Charina M., Lasserre J.B., Putinar M., Stöckler J., Structured function systems and applications, Oberwolfach Rep. 10 (2013), 579-655.
  3. Diaconis P., Freedman D., The Markov moment problem and de Finetti's theorem. I, Math. Z. 247 (2004), 183-199.
  4. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  5. Gottlieb D., Shu C.-W., On the Gibbs phenomenon. III. Recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function, SIAM J. Numer. Anal. 33 (1996), 280-290.
  6. Helton J.W., Lasserre J.B., Putinar M., Measures with zeros in the inverse of their moment matrix, Ann. Probab. 36 (2008), 1453-1471, math.PR/0702314.
  7. Lasserre J.B., Borel measures with a density on a compact semi-algebraic set, Arch. Math. (Basel) 101 (2013), 361-371, arXiv:1304.1716.
  8. Lasserre J.B., Henrion D., Prieur C., Trélat E., Nonlinear optimal control via occupation measures and LMI-relaxations, SIAM J. Control Optim. 47 (2008), 1643-1666, math.OC/0703377.
  9. McCann R.J., Guillen N., Five lectures on optimal transportation: geometry, regularity and applications, arXiv:1011.2911.
  10. Putinar M., Extremal solutions of the two-dimensional $L$-problem of moments, J. Funct. Anal. 136 (1996), 331-364.
  11. Putinar M., Extremal solutions of the two-dimensional $L$-problem of moments. II, J. Approx. Theory 92 (1998), 38-58, math.CA/9512222.
  12. Suetin P.K., On the representation of continuous and differentiable functions by Fourier series in Legendre polynomials, Sov. Math. Dokl. 158 (1964), 1408-1410.
  13. Villani C., Topics in optimal transportation, Graduate Studies in Mathematics, Vol. 58, Amer. Math. Soc., Providence, RI, 2003.
  14. Vinter R., Convex duality and nonlinear optimal control, SIAM J. Control Optim. 31 (1993), 518-538.
  15. Wang H., Xiang S., On the convergence rates of Legendre approximation, Math. Comp. 81 (2012), 861-877.

Previous article  Next article   Contents of Volume 11 (2015)