In the framework of Sobolev (Bessel potential) spaces Hn(R,R or C), we consider the nonlinear Nemytskij operator sending a function x ∈ R 7→ f(x) into a composite function x ∈ R 7→ G(f(x), x). Assuming sufficient smoothness for G, we give a ”tame” bound on the Hn norm of this composite function in terms of a linear function of the Hn norm of f , with a coefficient depending on G and on the Ha n...

In the framework of Sobolev (Bessel potential) spaces Hn(R,R or C), we consider the nonlinear Nemytskij operator sending a function x ∈ R 7→ f(x) into a composite function x ∈ R 7→ G(f(x), x). Assuming sufficient smoothness for G, we give a ”tame” bound on the Hn norm of this composite function in terms of a linear function of the Hn norm of f , with a coefficient depending on G and on the Ha n...

In this paper, we give some conditions under which the finite sum of continuous $g$-frames is again a continuous $g$-frame. We give necessary and sufficient conditions for the continuous $g$-frames $Lambda=left{Lambda_w in Bleft(H,K_wright): win Omegaright}$ and $Gamma=left{Gamma_w in Bleft(H,K_wright): win Omegaright}$ and operators $U$ and $V$ on $H$ such that $Lambda U+Gamma V={Lambda_w U+Ga...

In the framework of Sobolev (Bessel potential) spaces Hn(R,R or C), we consider the nonlinear Nemytskij operator sending a function x ∈ R 7→ f(x) into a composite function x ∈ R 7→ G(f(x), x). Assuming sufficient smoothness for G, we give a ”tame” bound on the Hn norm of this composite function in terms of a linear function of the Hn norm of f , with a coefficient depending on G and on the Ha n...

We report the creation of white-light, achromatic Bessel beams using a spatial light modulator and a prism to compensate for the dispersion. Unlike the Bessel beam created by a refractive axicon, this achromatic beam has a radial wavevector and hence an intensity cross-section which is independent of wavelength. The technique also lends itself to the generation of higher order Bessel beams with...

Recently Pogány and Süli [Proc. Amer. Math. Soc. 137(7) (2009), 2363–2368] derived a closed-form integral expression for Neumann series of Bessel functions of the first kind Jν . In this paper our aim is to establish analogous integral representations for the Neumann-type series of modified Bessel functions of the first kind Iν and for Bessel functions of the second kind Yν , Kν , and to give l...

We establish a simple relative trace formula for GSp(4) and inner forms with respect to Bessel subgroups to obtain a certain Bessel identity. From such an identity, one can hope to prove a formula relating central values of degree four spinor L-functions to squares of Bessel periods as conjectured by Böcherer. Under some local assumptions, we obtain nonvanishing results, i.e., a global Gross–Pr...

A theory of the recently discussed and demonstrated novel phenomenon of self-phase-matching [B.Glushko, B. Kryzhanovsky, and D. Sarkisyan, Phys. Rev. Lett. 71, 243 (1993)j is given by representing the conical beam, produced in the experiments, by a Bessel beam and as a superposition of Bessel beams. The predictions of the theory with the superposition of Bessel beams are in conformity with the ...

We prove a limit relation for the Dunkl–Bessel function of type BN with multiplicity parameters k1 on the roots ±ei and k2 on ±ei ± ej where k1 tends to infinity and the arguments are suitably scaled. It gives a good approximation in terms of the Dunkltype Bessel function of type AN−1 with multiplicity k2. For certain values of k2 an improved estimate is obtained from a corresponding limit rela...