Absolute Convergence of Fourier Integrals, Summability of Fourier Series, and Polynomial Approximation of Functions on the Torus

Abstract

The Mega Ampère Spherical Tokamak (MAST) programme is strongly focused on addressing key physics issues in preparation for operation of ITER as well as providing solutions for DEMO design choices. In this regard, MAST has provided key results in understanding and optimizing H-mode confinement, operating with smaller edge localized modes (ELMs), predicting and handling plasma exhaust and tailoring auxiliary current drive. In all cases, the high-resolution diagnostic capability on MAST is complemented by sophisticated numerical modelling to facilitate a deeper understanding. Mitigation of ELMs with resonant magnetic perturbations (RMPs) with toroidal mode number n RMP = 2, 3, 4, 6 has been demonstrated: at high and low collisionality; for the first ELM following the transition to high confinement operation; during the current ramp-up; and with rotating n RMP = 3 RMPs. n RMP = 4, 6 fields cause less rotation braking whilst the power to access H-mode is less with n RMP = 4 than n RMP = 3, 6. Refuelling with gas or pellets gives plasmas with mitigated ELMs and reduced peak heat flux at the same time as achieving good confinement. A synergy exists between pellet fuelling and RMPs, since mitigated ELMs remove fewer particles. Inter-ELM instabilities observed with Doppler backscattering are consistent with gyrokinetic simulations of micro-tearing modes in the pedestal. Meanwhile, ELM precursors have been strikingly observed with beam emission spectroscopy (BES) measurements. A scan in beta at the L-H transition shows that pedestal height scales strongly with core pressure. Gyro-Bohm normalized turbulent ion heat flux (as estimated from the BES data) is observed to decrease with increasing tilt of the turbulent eddies. Fast ion redistribution by energetic particle modes depends on density, and access to a quiescent domain with 'classical' fast ion transport is found above a critical density. Highly efficient electron Bernstein wave current drive (1 A W −1 ) has been achieved in solenoid-free start-up. A new proton detector has characterized escaping fusion products. Langmuir probes and a high-speed camera suggest filaments play a role in particle transport in the private flux region whilst coherence imaging has measured scrape-off layer (SOL) flows. BOUT++ simulations show that fluxes due to filaments are strongly dependent on resistivity and magnetic geometry of the SOL, with higher radial fluxes at higher resistivity. Finally, MAST Upgrade is due to begin operation in 2016 to support ITER preparation and importantly to operate with a Super-X divertor to test extended leg concepts for particle and power exhaust.

On Approximation Methods Generated by Bochner-Riesz Kernels

“…In other words, equivalence (1.5) is valid for all admissable pairs (1/p, α) with 1 ≤ p ≤ +∞ and α > 0 if the method S (α) n converges in L p . This statement is essentially stronger than the result on equivalence (1.5) obtained in [21] and its consequences (1.6) and (1.8), because the condition α > (d − 1)/2 is required no longer.…”

“…Analysing Theorems 2 and 7 we can see that in contrast to the result on the equivalence (1.5) in [21] the condition α > (d − 1)/2 which was essentially used in the proof turns out to be not necessary. Just the convergence property of the BochnerRiesz means is required.…”

“…The number (d − 1)/2 is known as the critical index for the convergence of Riesz means (see [18, p. 255]). Moreover, in this case their approximation error in the norm of the space C of continuous 2π -periodic functions (as usual, it corresponds to p = +∞) turns out to be equivalent to the modulus of smoothness at the point 1/n which is given by (1.4) where μ(D) is the measure of the d-dimensional unit ball D [21]. Using standard arguments of duality and interpolation, one can easily transfer this result to the case 1 ≤ p < +∞.…”

An Interplay of Multidimensional Variations in Fourier Analysis

“…Already this variation has numerous applications; besides those in [1] see also [21,Sects. 3.3.9-3.3.10] and [19].…”

Modal approach in structural-dynamic equations

“…The method of expansion in eigenfunctions of a linear homogeneous operator with constant coefficients has no such restrictions, which form the system of orthogonal functions [4], satisfying the conditions of theorem Steklov [4,5,6], i.e. it is possible to build an absolutely convergent generalized Fourier series [7]. As such a system, one can effectively use the solution of the problem of free vibrations of linearly elastic bodies with homogeneous kinematic conditions [8,9].…”