G-Frames in Hilbert spaces are a redundant set of operators which yield a representation for each vector in the space. In this paper we investigate the connection between g-frames, g-orthonormal bases and g-Riesz bases. We show that a family of bounded operators is a g-Bessel sequences if and only if the Gram matrix associated to its denes a bounded operator.

Two Bessel sequences are orthogonal if the composition of the synthesis operator of one sequence with the analysis operator of the other sequence is the 0 operator. We characterize when two Bessel sequences are orthogonal when the Bessel sequences have the form of translates of a finite number of functions in L(R). The characterizations are applied to Bessel sequences which have an affine struc...

For a delta operator aD − bD we find the corresponding polynomial sequence of binomial type and relations with Fuss numbers. In the case of D − 1 2D 2 we show that the corresponding Bessel–Carlitz polynomials are moments of the convolution semigroup of inverse Gaussian distributions. We also find probability distributions νt, t > 0, for which {yn(t)}, the Bessel polynomials at t, is the moment ...

Generation of Bessel-like beam array from polymeric microstructures, which were created using four-beam interference lithography is demonstrated. Quality of the produced Bessel-like beams depends on the geometrical parameters of the manufactured axicon-like structures, and their shape can be efficiently controlled through the laser treatment parameters. The output beam characteristics indicate ...

We report a systematic treatment of the holographic generation of electron Bessel beams, with a view to applications in electron microscopy. We describe in detail the theory underlying hologram patterning, as well as the actual electron-optical configuration used experimentally. We show that by optimizing our nanofabrication recipe, electron Bessel beams can be generated with relative efficienc...

g-frames in hilbert spaces are a redundant set of operators which yield a repre-sentation for each vector in the space. in this paper we investigate the connection betweeng-frames, g-orthonormal bases and g-riesz bases. we show that a family of bounded opera-tors is a g-bessel sequences if and only if the gram matrix associated to its denes a boundedoperator.

in this paper we develop a natural generalization of schauder basis theory, we term operator-valued basis or simply ov-basis theory, using operator-algebraic methods. we prove several results for ov-basis concerning duality, orthogonality, biorthogonality and minimality. we prove that the operators of a dual ov-basis are continuous. we also dene the concepts of bessel, hilbert ov-basis and obt...

We study theoretically and numerically third-harmonic generation (THG) from model geometries (interfaces, slabs, periodic media) illuminated by Bessel beams produced by focusing an annular intensity profile. Bessel beams exhibit a phase and intensity distribution near focus different from Gaussian beams, resulting in distinct THG phase matching properties and coherent scattering directions. Exc...