Naslov
Fractal Geometry of Oscillatory Integrals and Singularities of Differentiable Maps

Autori
Rolin, Jean-Philippe ; Vlah, Domagoj ; Županović, Vesna

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Skup
Fractals 2016: Summer School on Fractal Geometry and Complex Dimensions - In celebration of the 60th birthday of Michel Lapidus

Mjesto i datum
San Luis Obispo (CA), Sjedinjene Američke Države, 21.06.2016. - 29.06.2016

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
oscillatory integral; box dimension; Minkowski content; critical points; Newton diagram

Sažetak
It is well known that theory of singularities is closely related to the study of asymptotic of oscillatory integrals. We investigate the fractal properties of a geometrical representation of oscillatory integrals \[ I(\tau)=\int_{; ; \mathbb{; ; R}; ; ^n}; ; e^{; ; i\tau f(x)}; ; \phi(x) dx, \] for large values of a real parameter $\tau$, where $f$ is the analytic phase and $\phi$ is the smooth amplitude with compact support. We are motivated by a geometrical representation of Fresnel oscillatory integrals by a spiral called the clothoid, and the idea to produce a classification of singularities using the fractal dimension. We measure the oscillatority by the Minkowski dimension of the planar curve parametrized by the real part $X$ and imaginary part $Y$ of the integral $I$. Also, we measure the oscillatory dimension that is defined as the Minkowski dimension of the graph of the function $x(t) = X(1/t)$, near $t=0$, and analogously for $Y$. We provide explicit formulas connecting these Minkowski dimensions and associated Minkowski contents with asymptotics of the integral $I$ and the type of the critical point of the phase $f$. Used techniques include Newton diagrams and the resolution of singularities. Finally, it is possible to consider oscillatory integrals whose amplitude $\phi$ is a discontinuous function, which is related to more involved fractal geometry.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA