R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg, Entropy dissipation and long-range interactions, Archive for Rational Mechanics and, Analysis, vol.152, pp.327-355, 2000.

R. Alexandre and M. Elsafadi, LITTLEWOOD???PALEY THEORY AND REGULARITY ISSUES IN BOLTZMANN HOMOGENEOUS EQUATIONS I: NON-CUTOFF CASE AND MAXWELLIAN MOLECULES, Mathematical Models and Methods in Applied Sciences, vol.15, issue.06, pp.907-920, 2005.
DOI : 10.1142/S0218202505000613

R. Alexandre and L. He, Integral estimates for a linear singular operator linked with Boltzmann operators part ii: High singularities 1 ? ? < 2, Kinetic and Related Models, vol.1, pp.491-513, 2008.

R. Alexandre, Y. Morimoto, S. Ukai, C. Xu, and T. Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation, Archive for Rational Mechanics and Analysis, pp.39-123, 2010.

L. Arkeryd, On the Boltzmann equation, Archive for Rational Mechanics and Analysis, pp.1-16, 1972.

L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Archive for Rational Mechanics and Analysis, pp.11-21, 1981.

A. V. Bobylev, Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwellian gas, translated from Teoreticheskaya i Matematicheskaya Fizika 60, 280?310, Theoretical and Mathematical Physics, pp.820-841, 1984.

L. Boltzmann and W. Studien-Über-das-wärmegleichgewicht-unter-gasmolekülen, Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften II, Abtheilung, vol.66, pp.275-370, 1872.

L. Desvillettes, About the regularizing properties of the non-cut-off Kac equation, Communications in Mathematical Physics, vol.5, issue.2, pp.417-440, 1995.
DOI : 10.1007/BF02101556

L. Desvillettes, Regularization properties of the 2-dimensional non radially symmetric non cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules, Transport Theory and Statistical Physics, vol.34, issue.3, pp.341-357, 1997.
DOI : 10.1080/03605309408821082

L. Desvillettes, About the use of Fourier transform for the Boltzmann equation, pp.1-99, 2003.

L. Desvillettes, G. Furioli, and E. Terraneo, Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules, Transactions of the American Mathematical Society, vol.361, issue.04, pp.1731-1747, 2009.
DOI : 10.1090/S0002-9947-08-04574-1

L. Desvillettes and B. Wennberg, Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff, Communications in Partial Differential Equations, vol.45, issue.1-2, pp.133-155, 2004.
DOI : 10.1080/03605309408821082

R. A. Devore and G. G. Lorentz, Constructive Approximation, Grundlehren der mathematischen Wissenschaften 303, 1993.

W. Gautschi, On inverses of Vandermonde and confluent Vandermonde matrices, Numerische Mathematik, vol.63, issue.1, pp.117-123, 1962.
DOI : 10.1007/BF01386302

L. Glangetas and M. Najeme, Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation, Kinetic and Related Models, vol.6, issue.2, pp.407-427, 2013.
DOI : 10.3934/krm.2013.6.407
URL : https://hal.archives-ouvertes.fr/hal-00701857

T. Goudon, On boltzmann equations and fokker???planck asymptotics: Influence of grazing collisions, Journal of Statistical Physics, vol.24, issue.1/2, pp.751-776, 1997.
DOI : 10.1007/BF02765543

S. Karlin, Interpolation properties of generalized perfect splines and the solutions of certain extremal problems. i, Transactions of the, pp.25-66, 1975.

N. Lekrine and C. Xu, Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation, Kinetic and Related Models, pp.647-666, 2009.

N. Lerner, Y. Morimoto, K. Pravda-starov, and C. Xu, Gelfand???Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff, Journal of Differential Equations, vol.256, issue.2, pp.797-831, 2014.
DOI : 10.1016/j.jde.2013.10.001
URL : https://hal.archives-ouvertes.fr/hal-01116715

C. D. Levermore and M. Oliver, Analyticity of Solutions for a Generalized Euler Equation, Journal of Differential Equations, vol.133, issue.2, pp.321-339, 1997.
DOI : 10.1006/jdeq.1996.3200

S. Lin, Gevrey Regularity for the Noncutoff Nonlinear Homogeneous Boltzmann Equation with Strong Singularity, Article ID 584169, 2014.
DOI : 10.1215/21562261-1625154

P. Lions, Compactness in Boltzmann???s equation via Fourier integral operators and applications. I, Journal of Mathematics of Kyoto University, vol.34, issue.2, pp.391-427, 1994.
DOI : 10.1215/kjm/1250519017

S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, pp.467-501, 1999.

Y. Morimoto and S. Ukai, Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff, Journal of Pseudo-Differential Operators and Applications, vol.143, issue.1, pp.139-159, 2010.
DOI : 10.1007/s11868-010-0008-z

Y. Morimoto, S. Ukai, C. Xu, and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete and Continuous Dynamical Systems, pp.187-212, 2009.

Y. Morimoto and C. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations, Journal of Differential Equations, vol.247, issue.2, pp.596-617, 2009.
DOI : 10.1016/j.jde.2009.01.028
URL : https://hal.archives-ouvertes.fr/hal-00368263

C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous boltzmann equation with cut-off, Archive for Rational Mechanics and Analysis, pp.169-212, 2004.

A. Pinkus, Some extremal properties of perfect splines and the pointwise Landau problem on the finite interval, Journal of Approximation Theory, vol.23, issue.1, pp.37-64, 1978.
DOI : 10.1016/0021-9045(78)90077-1

A. Shadrin, The Landau-Kolmogorov inequality revisited, Discrete and Continuous Dynamical Systems, pp.1183-1210, 2014.

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, Journal of Statistical Physics, vol.94, issue.3/4, pp.619-637, 1999.
DOI : 10.1023/A:1004589506756

S. Ukai, Local solutions in gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan Journal of Applied Mathematics, vol.84, issue.1, pp.141-156, 1984.
DOI : 10.1007/BF03167864

C. Villani, On a New Class of Weak Solutions to the Spatially Homogeneous Boltzmann and Landau Equations, Archive for Rational Mechanics and Analysis, vol.187, issue.Ser.2, pp.273-307, 1998.
DOI : 10.1007/s002050050106

C. Villani, A Review of Mathematical Topics in Collisional Kinetic Theory, Handbook of Mathematical Fluid Dynamics, pp.71-305, 2002.
DOI : 10.1016/S1874-5792(02)80004-0

Z. Yin and T. Zhang, Gevrey regularity for solutions of the non-cutoff Boltzmann equation: The spatially inhomogeneous case, Nonlinear Analysis: Real World Applications, vol.15, pp.246-261, 2014.

T. Zhang and Z. Yin, Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff, Journal of Differential Equations, vol.253, issue.4, pp.1172-1190, 2012.
DOI : 10.1016/j.jde.2012.04.023

J. Université, C. Cnrs, C. , and C. , 83957 La Garde, France E-mail: barbarou@univ-tln.fr Dirk Hundertmark Karlsruhe Institute of Technology, mail: tobias.ried@kit.edu Semjon Vugalter Karlsruhe Institute of Technology, p.76131