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By Eells J. (ed.)

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Those notes grew out of lectures given via the writer on the Institut für Angewandte Mathematik, Heidelberg college, and on the Centre for Mathematical research, Australian nationwide Unviersity

A significant target used to be to provide the elemental principles of Geometric degree conception in a mode quite simply obtainable to analysts. i've got attempted to maintain the notes as short as attainable, topic to the constraint of masking the particularly vital and principal rules. There have after all been omissions; in an extended model of those notes (which i'm hoping to jot down within the close to future), issues which might evidently have a excessive precedence for inclusion are the idea of flat chains, extra functions of G. M. T. to geometric variational difficulties, P. D. E. features of the idea, and boundary regularity theory.

I am indebted to many mathematicians for worthwhile conversations pertaining to those notes. specifically C. Gerhardt for his invitation to lecture in this fabric at Heidelberg, ok. Ecker (who learn completely an prior draft of the 1st few chapters), R. Hardt for plenty of necessary conversations over a couple of years. such a lot in particular i need to thank J. Hutchinson for varied positive and enlightening conversations.

As a long way as content material of those notes is anxious, i've got drawn seriously from the normal references Federer [FH1] and Allard [AW1], even supposing the reader will see that the presentation and perspective usually differs from those references.

An define of the notes is as follows. bankruptcy 1 contains easy degree concept (from the Caratheodory standpoint of outer measure). lots of the effects are by means of now rather classical. For a extra wide therapy of a few of the themes lined, and for a few bibliographical comments, the reader is observed bankruptcy 2 of Federer's booklet [FH1], which used to be as a minimum the fundamental resource used for many of the fabric of bankruptcy 1.

Chapter 2 develops extra uncomplicated preliminaries from research. In getting ready the dialogue of the world and co-area formulae we came upon Hardt's Melbourne notes [HR1] really worthwhile. there's just a brief part on BV features, however it with ease suffices for all of the later purposes. We came across Giusti's Canberra notes [G] worthy in getting ready this fabric (especially) with regards to the later fabric on units of in the community finite perimeter).

Chapter three is the 1st really good bankruptcy, and offers a concise therapy of an important points of countably n-rectifiable units. There are even more common ends up in Federer's e-book [FH1], yet confidently the reader will locate the dialogue right here compatible for many functions, and a superb place to begin for any extensions which would sometimes be needed.

In Chapters four, five we advance the fundamental thought of rectifiable varifolds and end up Allard's regularity theorem. ([AW1]. ) Our therapy this is officially even more concrete than Allard's; in truth the total argument is given within the concrete environment of rectifiable varifolds, regarded as countably n-rectifiable units built with in the neighborhood Hn-integrable multiplicity functionality. optimistically this may make it more straightforward for the reader to determine the $64000 principles concerned with the regularity theorem (and within the initial concept regarding monotonicity formulae and so on. ).

Chapter 6 contians the fundamental thought of currents, together with integer multiplicity rectifiable currents, yet no longer together with a dialogue of flat chains. the fundamental references for this bankruptcy are the unique paper of Federer and Fleming [FF] and Federer's booklet [FH1], even supposing in a couple of respects our remedy is a bit assorted from those references.

In bankruptcy 7 there's a dialogue of the fundamental thought of minimizing currents. the concept 36. four, the evidence of that is kind of ordinary, doesn't appear to look in different places within the literature. within the final part we advance the regularity idea for condimension 1 minimizing currents. A function of this part is that we deal with the case while the currents in query are literally codimension 1 in a few gentle submanifold. (This was once after all in most cases recognized, yet doesn't explicitly look in different places within the literature. )

Finally in bankruptcy eight we describe Allard's concept of basic varifolds, which initially seemed in [AW1]. (Important facets of the speculation of varifolds had previous been built by way of Almgren [A3]. )

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Soit donc f(z)= A(z)eiAX” la fonction considérée précédemment, et soit $(z) une ondelette à support est maximale pour une certaine compact dans un intervalle I , telle que valeur = W O . 26) La transformée en ondelettes est donc localisée autour de a = wo/X et permet une mesure directe de X et A ( z ) ,à condition bien sûr que le reste R ( b , a ) soit négligeable, ce qui est le cas quand les variations de l’amplitude sont lentes par rapport aux oscillations provenant de l’exponentielle complexe.

Puis, nous nous focaliserons sur l’illustration des aspects “temps-échelle” et “temps-fréquence” des ondelettes. Nous utiliserons essentiellement des ondelettes “progressives”, c’est-à-dire ne possédant que des fréquences positives, qui sont mieux adapt& à l’étude des propriétés spectrales des signaux étudiés. I LE PLAN TEMPS-FRÉQUENCE L’une des premières utilisations de la transformée en ondelettes consiste à représenter graphiquement la transformée en ondelettes et à essayer de comprendre qualitativement sur la représentation graphique quelles sont les caractéristiques du signal étudié.

32 Analyse continue par ondelettes et par gaborettes Proposition 1 Soit T f la transformée en ondelettes de la fonction f E L’(El) par rapport à l’ondelette t,b E L’(El). 13) y)défini par le noyau reproduisant où 7 i + est le sous-espace de L2(lR;x lR, It7$. 14) U De plus, P$ est un projecteur orthogonal. Cette propriété de noyau reproduisant s’avère très intéressante en pratique, car elle conduit à des formules d’interpolation permettant la restitution (approchce) de la transformée continue à partir de versions échantillonnées.

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Complex Analysis by Eells J. (ed.)

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