Here the coefficients [a.sub.k,r] and [b.sub.k,r] are independent random variables with N(0, h/2[r.sup.2][[pi].sup.2]) distributed and we could derive them from the Fourier integrals
Ho, Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces, Ann.
In , we find that [X.sub.[alpha]] is related to the mapping properties of the fractional integral operators, the convolution operators and the Fourier integral operators in r.i.q.B.f.s.
The topics include local inverse scattering problems as a tool of perturbation analysis for resonance systems, Fourier integrals
and a new representation of Maslov's canonical operator near caustics, the central limit theorem for linear eigenvalue statistics of the sum of independent random matrices of rank one, a homogenized model of oscillations of an elastic medium with small caverns filled with viscous incompressible fluid, and recovering a potential of the Sturm-Liouville problem from finite sets of spectral data.
where we use the connection between the continuum fractional derivatives of the order a and the correspondent Fourier integrals transforms
In the expression of the Fourier integral for lattice fields, the momentum integration with respect to wave-vector components [k.sub.[mu]]([mu] = 1, 2, 3, 4) is restricted by the Brillouin zone [k.sub.[mu]] [member of] [-[pi]/[a.sub.[mu]][pi]/[a.sub.[pi]]], where [a.sub.[mu]] are the lattice constants.
These solutions are based on the field representation in the form of Fourier integrals and series.
These conclusions hold also true for the representation of fields outside the slot plane in the form of Fourier integrals (4).
He proceeds from the elementary theory of Fourier series and Fourier integrals
to abstract harmonic analysis on locally compact abelian groups.
To obtain the bandregion of [q.sub.r[OMEGA]]([beta], [alpha]; x), the Fourier integrals
in both the a and [beta] directions must be evaluated.
We start with approximation by functions of exponential type which are defined as Fourier integrals
WIENER, The Fourier Integral
and Certain of Its Applications, Cambridge University Press, Cambridge, 1933.