We find asymptotic formulas for the least upper bounds of the deviations of Fourier operators on classes of functions locally summable on the entire real axis and defined by ψ -integrals. On these classes, we also obtain asymptotic equalities for the upper bounds of functionals that characterize the simultaneous approximation of several functions.In this paper, we present results related to the approximation of the classes Ĉ ψ ᑨ of functions defined on the entire axis [nonperiodic analogs of the classes considered in [1] (see also the bibliography therein)] by using Fourier operators (see [2 -8]), i.e., entire functions of exponential type that coincide in the periodic case with partial Fourier sums of the corresponding orders for the functions approximated. Let ψ ψ ψ = ( ) 1 2 , be a pair of functions ψ 1 ( ) t and ψ 2 ( ) t such that ψ 1 ∈ᑛ and ψ 2 ∈ ′ ᑛ , where ᑛ [2, p. 193] is the set of all functions h continuous for t ≥ 0 and satisfying the conditions (i) h t ( ) ≥ 0 , h( ) 0 0 = , and h is increasing on 0 1 ; [ ], (ii) h is convex downwards on 1; ∞ [ ) and lim ( ) t h t →∞ = 0, and (iii) the derivative ′ = ′ + h t h t ( ) : ( ) 0 is a function of bounded variation on 0; ∞ [ ), i.e.,
The spectra of the Dirac quasi-electrons transmission through the Fibonacci quasi-periodical superlattice (SL) are calculated and analyzed in the continuum model with the help of the transfer matrix method. The onedimensional SL based on a monolayer graphene modulated by the Fermi velocity barriers is studied. A new quasi-periodical factor is proposed to be considered. We show that the Fibonacci quasi-periodic modulation in graphene superlattices with the velocity barriers can be effectively realized by virtue of a difference in the velocity barrier values (no additional factor is needed and we keep in mind that not each factor can provide the quasi-periodicity). This fact is true for a case of normal incidence of quasi-electrons on a lattice. In contrast to the case of other types of the graphene SL spectra studied reveal the remarkable property, namely the periodic character over all the energy scale and the transmission coefficient doesn't tend asymptotically to unity at rather large energies. Both the conductance (using the known Landauer-Buttiker formula) and the Fano factor for the structure considered are also calculated and analyzed. The dependence of spectra on the Fermi velocity magnitude and on the external electrostatic potential as well as on the SL geometrical parameters (width of barriers and quantum wells) is analyzed. Using the quasi-periodical SL one can control the transport properties of the graphene structures in a wide range. The obtained results can be used for applications in the graphene-based electronics.
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