Constructive approximations of spherical functions

Abstract

Approximation of functions on the sphere arises in almost all applications modeling data collected on the surface of the earth and for reconstruction of various processes in spherical coordinates. Constructive approximation of high dimensional spherical functions are useful for approximation of processes of several variables on a compact subset of a Euclidean space by mapping the data onto the unit sphere of a space having one higher dimension, avoiding the boundary effect of the set. Interpolation operators on the circle and periodic domains (based on a class of basis functions and data values) are essential for many high performance simulations. These operators are represented by analytical summation formulas that can be computed very efficiently using the fast Fourier transform. This work is concerned with construction of a similar class of interpolation and quasi-interpolation operators on the unit sphere in R q , for q = 3, 4, 5, · · · , using a new class of basis functions.

Development of Silicon Nitride‐Based Ceramic Radomes — A Review

“…27 Modern ground-based and ship-based radomes are manufactured using composite materials such as fiberglass, quartz, and aramid fibers held together with polyester, epoxy, and other resins. Materials for developing ceramic radomes include fused silica (SiO 2 ), 28 silicon nitride-barium aluminum silicate composites (Si 3 N 4 / BAS), 29,30 aluminum phosphate-mullite composites (AlPO 4 /3Al 2 O 3 .2SiO 2 ), 31 silica-aluminum nitride composites (SiO 2 -AlN), 32 silica fiber reinforced silicaboron nitride composites (SiO 2f /SiO 2 -BN), 33 aluminum borate whisker/aluminum phosphates composites (9Al₂O₃.2B₂O₃/AlPO 4 ), 34 liquid-phase silicon nitride, 35 Si 2 N 2 O composites, 36 reaction-bonded silicon nitride, 37 b-Si 4 Al 2 O 2 N 4 , 38 b-Si 4 Al 2 O 2 N 4 -SiO 2 , 39 b-SiAlON, 40,41 few of these materials with efficient properties with considerable importance are shown in Table I.…”

“…29 b-SiAlON material can be shaped into radome structures and can be sintered to full density without disturbing the structure, but fixing the best chemical composition based on the required properties and processing of these materials is quite difficult. 40,41 Si 3 N 4 ceramic can be shaped into radome structures using near-net shape forming techniques, and the major drawback of dense sintered silicon nitride materials is its poor sintering ability although it possesses exceptionally high flexural strength (600-1000 Mpa) 5 and it can withstand high temperatures and has well-acceptable dielectric properties. Low-density porous silicon nitride ceramic has been used as radome materials owing to its rather low dielectric constant and loss tangent, high chemical stability, high melting point and extremely low coefficient of thermal expansion.…”