# bandwidth: Calculation of Bandwidth Parameter In SpherWave: Spherical Wavelets and SW-based Spatially Adaptive Methods

## Description

Given the number of observations at a level, this function calculates bandwidth parameter of spherical basis function (SBF).

## Usage

 1 bandwidth(n) 

## Arguments

 n the number of observations at a level

## Details

This function is used for obtaining the bandwidth of the coarsest network level L, h_L. From geometry, the surface area covered by surface mass distribution with variance σ^2 over unit sphere Ω is 2 π (1 - √{1 - σ^2}). Since the total surface area of the unit sphere is 4 π and the variance of SBF induced from Poisson kernel is σ^2=≤ft((1 - h^2)/(1 + h^2)\right)^2, the surface covered are is 2 π ≤ft(1 - √{1 - ((1 - h^2)/(1 + h^2))^2}\right). Under the assumption that the observations are distributed equally over the sphere, it can be easily known how many observation are needed in order to cover the whole sphere with fixed h, and how large the h is needed to cover the sphere when the number of observations are fixed as follows:

\# of observations = n = 2/(1 - √{1 - ((1 - h^2)/(1 + h^2))^2}) and h = √{(1 - a_n)/(1 + a_n)}

where a_n = √{1 - (1 - 2/n)^2}.

## Value

 h bandwidth parameter at a level

## References

Oh, H-S. (1999) Spherical wavelets and their statistical analysis with applications to meteorological data. Ph.D. Thesis, Department of Statistics, Texas A\&M University, College Station.

eta.comp
 1 bandwidth(20)