100 Years of Fourier Series and Spherical Harmonics in Convexity

2004 ◽  
pp. 97-118
Author(s):  
H. Groemer
Keyword(s):  
Fourier Series ◽  
Spherical Harmonics

scholarly journals Weakly Orthogonal Spherical Harmonics in a Non-Polar Spherical Coordinates and its Application to Functions on Cubed-Sphere

2012 ◽  
Vol 17 (2) ◽  
pp. 200
Author(s):  
M.A.A.M. Faham ◽  
H.M. Nasir
Keyword(s):  
Fourier Series ◽  
Coordinate System ◽  
Spherical Harmonics ◽  
Spherical Coordinate System ◽  
Spherical Functions ◽  
Spherical Coordinates ◽  
Spherical Coordinate ◽  
Linear Relations ◽  
Inner Products ◽  
Cubed Sphere

In a recent paper (Nasir, 2007), a set of weakly orthogonal and completely orthogonal spherical harmonics in a non-polar spherical coordinate system based on a cubed-sphere was constructed. In this work, we explore some linear relations between these two sets of spherical harmonics. Moreover, a power representation for the set of weakly orthogonal spherical harmonics corresponding to a mode is presented. We also determine the norm of the orthogonal spherical harmonics and hence the inner products for the weakly orthogonal spherical harmonics. As an immediate application of these properties, we present a Fourier series formulation of spherical functions defined on the cubed-sphere.  

scholarly journals A Dynamical Core with Double Fourier Series: Comparison with the Spherical Harmonics Method

2006 ◽  
Vol 134 (4) ◽  
pp. 1299-1315 ◽  
Author(s):  
Hyeong-Bin Cheong
Keyword(s):  
Fourier Series ◽  
Helmholtz Equation ◽  
Gravity Wave ◽  
Spherical Harmonics ◽  
Surface Pressure ◽  
Double Fourier Series ◽  
Implicit Time ◽  
Dynamical Core ◽  
Fourier Filtering

Abstract A dynamical core of a general circulation model with the spectral method using double Fourier series (DFS) as basis functions is presented. The model uses the hydrostatic balance approximation and sigma coordinate system in the vertical direction and includes no topography. The model atmosphere is divided into 25 layers with equal sigma depths. Prognostic equations for the vorticity, divergence, temperature, and logarithmic surface pressure are solved by the DFS spectral-transform method with the Fourier filtering at middle and high latitudes. A semi-implicit time-stepping procedure, which deals with the eigendecomposition and inversion of the 3D Helmholtz equation associated with the gravity wave terms, is incorporated for the gravity wave–related terms. The DFS model is tested in terms of the solution of the 3D Helmholtz equation, balanced initial state, developing baroclinic waves, and short- and long-term Held–Suarez–Williamson simulations for T42, T62, T84, and T106 resolutions. It is found that the DFS model is stable and accurate and produces almost the same results as the spherical harmonics method (SHM). The normalized difference (i.e., L2 norm error) measured from the results of highest-resolution SHM-T106 showed a desirable convergence of the DFS solution with the resolution. The convergence property, however, varies with the test case and prognostic variables. The total mass (or global integrated surface pressure) is conserved to a good approximation in the long-term simulations. Computing on the high-performance computer NEC SX-5 (parallel-vector architecture) indicated that DFS is more efficient than the SHM and the efficiency increases with the resolution, for example, by factors of 2.09 and 7.68 for T212 and T1022, respectively.

scholarly journals Approximations in $$L^1$$ with convergent Fourier series

2021 ◽  
Author(s):  
Zhirayr Avetisyan ◽  
Martin Grigoryan ◽  
Michael Ruzhansky
Keyword(s):  
Fourier Series ◽  
Homogeneous Space ◽  
Spherical Harmonics ◽  
Measurable Function ◽  
Measure Space ◽  
Orthonormal Basis ◽  
Measurable Subset ◽  
Diffuse Measure ◽  
Bounded Functions ◽  
Hausdorff Group

AbstractFor a separable finite diffuse measure space $${\mathcal {M}}$$ M and an orthonormal basis $$\{\varphi _n\}$$ { φ n } of $$L^2({\mathcal {M}})$$ L 2 ( M ) consisting of bounded functions $$\varphi _n\in L^\infty ({\mathcal {M}})$$ φ n ∈ L ∞ ( M ) , we find a measurable subset $$E\subset {\mathcal {M}}$$ E ⊂ M of arbitrarily small complement $$|{\mathcal {M}}{\setminus } E|<\epsilon $$ | M \ E | < ϵ , such that every measurable function $$f\in L^1({\mathcal {M}})$$ f ∈ L 1 ( M ) has an approximant $$g\in L^1({\mathcal {M}})$$ g ∈ L 1 ( M ) with $$g=f$$ g = f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of $${\mathcal {M}}=G/H$$ M = G / H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

ON THE RATE OF CONVERGENCE OF THE SPHERICAL-HARMONICS METHOD FOR A SANDWICH REACTOR OF SMALL LATTICE PITCH

1958 ◽  
Vol 36 (6) ◽  
pp. 784-800 ◽  
Author(s):  
B. Davison
Keyword(s):  
Exact Solution ◽  
Spatial Distribution ◽  
Fourier Series ◽  
Neutron Flux ◽  
Rate Of Convergence ◽  
Spherical Harmonics ◽  
Thermal Neutron Flux ◽  
Arbitrary Order ◽  
Order Of Approximation ◽  
Small Lattice

There are some cases where the spherical-harmonics-method calculations can be carried out purely algebraically, without specifying numerically the order of approximation used. Such is the problem of determining the spatial distribution of the thermal neutron flux in an infinite sandwich reactor of a small lattice pitch. The spherical-harmonics-method solution of this problem, in an arbitrary order of approximation, is compared with the exact solution. It is shown that if both are expanded in the Fourier series in terms of the optical depth, the nth term of the Fourier series for the spherical-harmonics-method solution differs from the corresponding term for the exact solution by the factor[Formula: see text]where N is the order of approximation used in the spherical-harmonics method and ε is one-half of the lattice pitch on the optical scale.This result should provide some guidance in assessing the rate of convergence of the spherical-harmonics method also for the more complex (and realistic) cases.

scholarly journals REPRESENTATION OF GRADIENTS OF A SCALAR FIELD ON THE SPHERE USING A 2D FOURIER EXPRESSION

2015 ◽  
Vol XL-1-W5 ◽  
pp. 689-694 ◽  
Author(s):  
M. A. Sharifi ◽  
K. Ghobadi-Far
Keyword(s):  
Fourier Series ◽  
Scalar Field ◽  
Spherical Harmonics ◽  
Legendre Function ◽  
Series Representation ◽  
Spectral Domain ◽  
Third Order ◽  
Numerical Examples ◽  
Local Frame ◽  
Fixed Frame

Representation of data on the sphere is conventionally done using spherical harmonics. Making use of the Fourier series of the Legendre function in the SH representation results in a 2D Fourier expression. So far the 2D Fourier series representation on the sphere has been confined to a scalar field like geopotential or relief data. We show that if one views the 2D Fourier formulation as a representation in a rotated frame, instead of the original Earth-fixed frame, one can easily generalize the representation to any gradient of the scalar field. Indeed, the gradient and the scalar field itself are simply linked in the spectral domain using spectral transfers. We provide the spectral transfers of the first-, second- and third-order gradients of a scalar field in a local frame. Using three numerical examples based on gravity and geometrical quantities, we show the applicability of the presented formulation.

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