Zonal measure algebras on isotropy irreducible homogeneous spaces

AbstractThis paper analyzes the convolution algebra M(K\GK) of zonal measures on a Lie group G, with compact subgroup K, primarily for the case when M(K\GK) is commutative and GK is isotropy irreducible. A basic result for such (G, K) is that the convolution of dim GK continuous (on GK) zonal measures is absolutely continuous. Using this, the spectrum (maximal ideal space) of M(K\GK) is determined and shown to be in 1-1 correspondence with the bounded Borel spherical functions. Also, certain asymptotic results for the continuous spherical functions are derived. For the special case when G is compact, all the idempotents in M(K\GK) are determined

X-ray harmonic comb from relativistic electron spikes

X-ray devices are far superior to optical ones for providing nanometre spatial and attosecond temporal resolutions. Such resolution is indispensable in biology, medicine, physics, material sciences, and their applications. A bright ultrafast coherent X-ray source is highly desirable, for example, for the diffractive imaging of individual large molecules, viruses, or cells. Here we demonstrate experimentally a new compact X-ray source involving high-order harmonics produced by a relativistic-irradiance femtosecond laser in a gas target. In our first implementation using a 9 Terawatt laser, coherent soft X-rays are emitted with a comb-like spectrum reaching the 'water window' range. The generation mechanism is robust being based on phenomena inherent in relativistic laser plasmas: self-focusing, nonlinear wave generation accompanied by electron density singularities, and collective radiation by a compact electric charge. The formation of singularities (electron density spikes) is described by the elegant mathematical catastrophe theory, which explains sudden changes in various complex systems, from physics to social sciences. The new X-ray source has advantageous scalings, as the maximum harmonic order is proportional to the cube of the laser amplitude enhanced by relativistic self-focusing in plasma. This allows straightforward extension of the coherent X-ray generation to the keV and tens of keV spectral regions. The implemented X-ray source is remarkably easily accessible: the requirements for the laser can be met in a university-scale laboratory, the gas jet is a replenishable debris-free target, and the harmonics emanate directly from the gas jet without additional devices. Our results open the way to a compact coherent ultrashort brilliant X-ray source with single shot and high-repetition rate capabilities, suitable for numerous applications and diagnostics in many research fields