Function estimation of irregularly spaced data with long memory dependence

We examine the problem of estimating an underlying function from collected data. The methods considered include parametric regression, density estimation, kernel estimation, wavelet regression, and specific results from when our underlying function f(x) is a member of the Besov or the Triebel spaces. Then we consider the problem of long memory error in several settings, including data which is equally spaced, data which is unequally spaced, and data which is a member of the Holder class and several other spaces. Ultimately we focus on three different problems. The first involves using linear interpolation or local averaging to account for the problem of irregularly spaced data. The second involves using a function H to reorder the data in a more general space. The third involves solving the problem in the matrix setting and considers the use of penalty functions. This method leads to general equations which describe the Mean Square Error in terms of Oracle risk. All three of these problems attempt to bound the Mean Integrated Square Error when the data is subject to long memory error.