Ce travail est dédié à l’étude des inégalités de type Landau-Kolmogorov en normes L2. Les mesures utilisées sont celles d’Hermite, de Laguerre-Sonin et de Jacobi. Ces inégalités sont obtenues en utilisant une méthode variationnelle. Elles font intervenir la norme d’un polynômes p et celles de ces dérivées. Dans un premier temps, on s'intéresse aux inégalités en une variable réelle qui font intervenir un nombre quelconque de normes. Les constantes correspondantes sont prises dans le domaine où une certaine forme bilinéaire est définie positive. Ensuite, on généralise ces résultats aux polynômes à plusieurs variables réelles en utilisant le produit tensoriel dans L2 et en faisant intervenir au plus les dérivées partielles secondes. Pour les mesures d'Hermite et de Laguerre-Sonin, ces inégalités sont étendues à toutes les fonctions d'un espace de Sobolev. Pour la mesure de Jacobi on donne des inégalités uniquement pour les polynômes d'un degré fixé par rapport à chaque variable. / This thesis is devoted to Landau-Kolmogorov type inequalities in L2 norm. The measures which are used, are the Hermite, the Laguerre-Sonin and the Jacobi ones. These inequalities are obtained by using a variational method and the involved the square norms of a polynomial p and some of its derivatives. Initially, we focused on inequalities in one real variable that involve any number of norms. The corresponding constants are taken in the domain where a certain biblinear form is positive definite. Then we generalize these results to polynomials in several real variables using the tensor product in L2 and involving at most the second partial derivatives. For the Hermite and Laguerrre-Sonin cases, these inequalities are extended to all functions of a Sobolev space. For the Jacobi case inequalities are given only for polynomials of degree fixed with respect to each variable.

The aim of this work was to model a four-port waveguide as a simple circuit,by first starting with a two-port then a three-port waveguide. Duethat the work was based on Nathan Marcuvitz book Waveguide handbook,an analytical solution for the circuit parameters was desired. In order toobtain an analytical solution three methods were studied; the Variationalmethod, the Integral equation method and the Static method. Out of thesethree methods the latter was chosen, because its strength of simplifying theboundary conditions. The goal to model a four-port and a three-portedwaveguide was too complicated. This led to that the goal was changed totrying to get a higher accuracy on the existing circuit model for a two-portwaveguide, by solving an extension to the circuit parameter. This was donebecause Marcuvitz only treated the first two modes correctly and it was notclear if the circuit model was stable for the higher orders of Taylor series. Inthe end a circuit model for a waveguide with an iris that treats the first 16modes correctly was solved. By looking at the dispersive properties of thecircuit a comparison with simulation software CST Microwave Studio couldbe done, which resulted in that the circuit model gave good results up to2b/ < 1. It was also showed that the accuracy was about the same as thecircuit model found in Waveguide handbook, but it can be mentioned thatthe accuracy is minimally better for the circuit model that was developed inthis work. Something that was discovered in this work is that the restrictionmentioned in Waveguide handbook for the case when the window is centeredis unreliable, it should be 2b/ < 1. It also appeared that the circuit modelremained stable for higher orders of the Taylor series, in this case up to the16:th order. / M°alet med detta arbete var att modellera en fyr-portars v°agledare somen simpel elektrisk krets, genom att f¨orst b¨orja med en tv°a-portars sedantre-portars -v°agledare. Detta arbete var baserat p°a Nathan Marcuvitz bokWaveguide handbook, d¨arav s¨oktes det en analytisk l¨osning f¨or kretsparametrarna.F¨or att kunna f°a en analytisk l¨osning, studerades tre metoder;Variationsmetoden, Integralsekvationsmetoden samt den Statiskametoden.Av dessa tre metoder valdes den sistn¨amnda, p°a grund av dess styrka medatt f¨orenkla randvillkoren. M°alet att modellera en fyr-portars samt en treportarsv°agledare var alldeles f¨or komplicerat. Detta ledde till att m°alet¨andrades till att f¨ors¨oka f°a en h¨ogre precision p°a den befintliga kretsmodellenf¨or en tv°a portars v°agledare, genom att l¨osa ut flera termer till kretsparametern.Detta gjordes d°a Marcuvitz endast hanterade de tv°a f¨orstamoderna korrekt, samt att det inte framgick ifall kretsmodellen ¨ar stabil f¨orh¨ogre ordningar av Taylor serier. I slut¨andan l¨ostes en kretsmodel f¨or env°agledare med en iris som hanterar de f¨orsta 16 moderna korrekt. Genomatt kolla p°a de dispersiva egenskaperna f¨or kretsen, kunde en j¨amf¨orelse medsimuleringsprogrammet CST Microwave Studio ske, d¨ar slutsatsen blev attkretsmodellen gav goda resultat upp till 2b/ < 1. Det visade ¨aven sig attprecisionen var ungef¨ar densamma som den kretsmodell som°aterfinns i Waveguidehandbook, men det kan n¨amnas att precisionen ¨ar minimalt b¨attref¨or den kretsmodell som togs fram i detta arbete. En sak som uppt¨acktes underdetta arbete var att restriktionen som n¨amns i Waveguide handbook f¨orfallet n¨ar gapet f¨or irisen ¨ar centrerad st¨ammer inte, den b¨or vara 2b/ < 1.Dessutom visade det sig att kretsmodellen fortfarande var stabil f¨or h¨ogreordningar av Taylorserier, i detta fall upp till den 16:e ordningen.