## 2 Derivation of the Estimation Algorithm
### Intermediate Estimator  
The simplest, from the computational point of view, estimator \(\hat{f}(x;r)\) for the family \(f(x;r)\), \(r\in[R_{1},\,R_{2}]\), we obtain when we additionally assume that the observations of \(X(r)\)'s are made on an equidistant grid \(\rho_{1}<\rho_{2}<\ldots<\rho_{M}\) that splits \([R_{1},\,R_{2}]\) into non-overlapping intervals of the length \(\Delta_{r}>0\), which cover all \([R_{1},\,R_{2}]\) in such a way that \(R_{1}=\rho_{1}-\Delta_{r}/2\) and \(R_{2}=\rho_{M}+\Delta_{r}/2\). In this section, we tentatively impose the restriction that only repeated, but independent and identically distributed (i.i.d.) observations of \(X(\rho_{m})\), \(m=1,\,2,\,\ldots,\,M\) are available. In the asymptotic analysis at the end of this paper, we shall assume that \(M\) grows to infinity with the number of observations \(d\). Then, also positions of \(\rho_{m}\)'s and \(\Delta_{r}\) will be changing, but we do not display this fact in the notations, unless necessary.  
At each of \(\rho_{m}\), \(m=1\), \(2\),..., \(M\), \(n_{m}>0\) observations \((X_{l}(\rho_{m}),\rho_{m})\), \(l=1,2,\ldots,n_{m}\) are made, keeping the above-mentioned assumptions on mutual independence in force. Additionally, the mutual independence of the following lists of r.v.'s is postulated:  
\[\{X_{l}(\rho_{m}),\,l=1,2,\ldots,n_{m}\},\quad m=1,\,2,\,\ldots,\,M \tag{6}\]  
Then, one can estimate \(\alpha_{kj}\)'s by  
\[\hat{\alpha}_{kj}=\Delta_{r}\,\sum_{m=1}^{M}\hat{\alpha}_{k}(\rho_{m})T_{j}( \rho_{m}) \tag{7}\]  
where  
\[\hat{\alpha}_{k}(\rho_{m})=\frac{1}{n_{m}}\sum_{l=1}^{n_{m}}v_{k}(X_{l}(\rho_ {m})) \tag{8}\]  
Estimators (7) are justified by (4). Notice that in (7) the simplest quadrature formula is used for approximating the integral \(\int_{R_{1}}^{R_{2}}\). When an active experiment is applicable, then one can select \(\rho_{m}\)'s at nodes of more advanced quadratures with weights, in the same spirit as in [17; 18] for nonparametric regression estimators, where the bias reduction was proved. In turn, estimators (8) are motivated by replacing the expectations in (2) by the corresponding empirical means.  
Truncating series (5) at \(K\)-th and \(J\)-th terms and substituting \(\hat{\alpha}_{kj}\) instead of \(\alpha_{kj}\), we obtain the following estimator if the family \(f(x;r)\), \(r\in[R_{1},R_{2}]\)  
\[\hat{f}(x;r)=\sum_{k=1}^{K}\,\sum_{j=1}^{J}\hat{\alpha}_{kj}\,v_{k}(x)\,T_{j}( r). \tag{9}\]  
Later on, \(K\) and \(J\) will depend on the number of observations, but it is not displayed, unless necessary.  
Estimator (9) is quite general in the sense that one can select (and mix) different orthonormal bases \(v_{k}\)'s and \(T_{j}\)'s. In particular, the trigonometric system, Legendre polynomials, Haar system, and other orthogonal wavelets can be applied. The reduction of computational complexity of (9) is possible when, for a given orthogonal system, its discrete and fast counterpart exists. We illustrate this idea in the next section, by selecting \(v_{k}\)'s and \(T_{j}\)'s as the trigonometric bases, applying discrete Fourier transform (DFT) and the fast Fourier transform (FFT) as its fast implementation.