## 5 Two-Parameter Family of Densities with Different Degrees of Smoothness
### Analysis of Convergence Rate for Expansions of Test Functions  
Consider the function  
\[y_{\
u}^{-}(x)=\left\{ \begin{array}{ll}x^{\
u}&\text{for }x\in[-1,0),\\ 0&\text{for }x\in[0,1],\end{array} \right.\]  
where \(\
u\) is a parameter (natural number). Its expansion coefficients \(Y_{\
u,i}^{-}\) with respect to Legendre polynomials (6) are expressed through the previously found values \(Q_{\
u,i}^{-}\):  
\[y_{\
u}^{-}(x)=\sum_{i=0}^{\infty}Y_{\
u,i}^{-}\hat{P}_{i}(x),\ \ \ Y_{\
u,i}^{-}=\sqrt{\frac{2i+1}{2}}\,Q_{\
u,i}^{-}.\]  
Further, we derive the recurrence relation for \(Q_{\
u,i}^{-}\), different from relation (38). We multiply the left-hand and right-hand sides of relation (1) by \(x^{\
u}\) and integrate over the interval \([-1,0]\):  
\[(i+1)\int_{-1}^{0}x^{\
u}P_{i+1}(x)dx=(2i+1)\int_{-1}^{0}x^{\
u+1}P_{i}(x)dx-i \int_{-1}^{0}x^{\
u}P_{i-1}(x)dx.\]  
Next, we use the rule of integration by parts:  
\[\int_{-1}^{0}x^{\
u+1}P_{i}(x)dx=x^{\
u+1}\Theta_{i}(x)\bigg{|}_{-1}^{0}-(\
u +1)\int_{-1}^{0}x^{\
u}\Theta_{i}(x)dx,\ \ \ \Theta_{i}(x)=\int_{-1}^{x}P_{i}(\theta)d\theta,\]  
and consequently, taking into account equality (17), we obtain  
\[\int_{-1}^{0}x^{\
u+1}P_{i}(x)dx=-\frac{\
u+1}{2i+1}\int_{-1}^{0}x^{\
u}\big{(} P_{i+1}(x)-P_{i-1}(x)\big{)}dx;\]  
therefore,  
\[(i+1)\int_{-1}^{0}x^{\
u}P_{i+1}(x)dx=-(\
u+1)\int_{-1}^{0}x^{\
u}\big{(}P_{i +1}(x)-P_{i-1}(x)\big{)}dx-i\int_{-1}^{0}x^{\
u}P_{i-1}(x)dx,\]  
or  
\[(i+1)Q_{\
u,i+1}^{-}=-(\
u+1)(Q_{\
u,i+1}^{-}-Q_{\
u,i-1}^{-})-iQ_{\
u,i-1}^{-},\]  
i.e.,  
\[Q_{\
u,i+1}^{-}=\frac{\
u-i+1}{\
u+i+2}\,Q_{\
u,i-1}^{-}\ \ \text{and}\ \ \ Y_{\
u,i+1}^{-}=\sqrt{\frac{2i+3}{2i-1}}\,\frac{\
u-i+1}{\
u+i+2}\,Y_{\
u,i-1} ^{-}.\]The series formed by the squared expansion coefficients \(Y_{\
u,i}^{-}\) converges since \(y_{\
u}^{-}\in L_{2}(\Omega)\). It can be represented as a sum of two series:  
\[\sum_{i=0}^{\infty}(Y_{\
u,i}^{-})^{2}=\sum_{j=0}^{\infty}(Y_{\
u,2j}^{-})^{2}+ \sum_{j=0}^{\infty}(Y_{\
u,2j+1}^{-})^{2}.\]  
The Raabe-Duhamel test [24] implies that the first series (similarly for the second one) converges in the same way as the series  
\[\sum_{j=1}^{\infty}\frac{1}{j^{2\
u+2}},\]  
since the sequence  
\[R_{j}=j\bigg{(}\frac{(Y_{\
u,2j}^{-})^{2}}{(Y_{\
u,2j+2}^{-})^{2}}-1\bigg{)}=j \bigg{(}\frac{4j-1}{4j+3}\bigg{[}\frac{\
u+2j+2}{\
u-2j+1}\bigg{]}^{2}-1\bigg{)}\]  
has the limit \(\lim\limits_{j\to\infty}R_{j}=2\
u+2\), but the convergence of this series is equivalent to the convergence of the integral  
\[(i+1)\int_{-1}^{0}x^{\
u}P_{i+1}(x)dx=(2i+1)\int_{-1}^{0}x^{\
u+1}P_{i}(x)dx-i \int_{-1}^{0}x^{\
u}P_{i-1}(x)dx.\]0  
which takes place under the condition \(2\
u+1>0\), or \(\
u>-1/2\). As a result, using [PERSON]'s identity, we find  
\[(i+1)\int_{-1}^{0}x^{\
u}P_{i+1}(x)dx=(2i+1)\int_{-1}^{0}x^{\
u+1}P_{i}(x)dx-i \int_{-1}^{0}x^{\
u}P_{i-1}(x)dx.\]1  
where  
\[(i+1)\int_{-1}^{0}x^{\
u}P_{i+1}(x)dx=(2i+1)\int_{-1}^{0}x^{\
u+1}P_{i}(x)dx-i \int_{-1}^{0}x^{\
u}P_{i-1}(x)dx.\]2  
and this corresponds to estimate (9) with the limit value \(s=\
u+1/2\) (see Remark 1). The obtained result can be transferred to the function  
\[y_{\
u}^{+}(x)=\left\{ \begin{array}{ll}0&\text{for $x\in[-1,0)$},\\ x^{\
u}&\text{for $x\in[0,1]$},\end{array} \right.\]  
and its expansion coefficients \(Y_{\
u,i}^{+}\) with respect to Legendre polynomials (6). The easiest way to prove this is the use of the equality \(|Y_{\
u,i}^{-}|=|Y_{\
u,i}^{+}|\), which follows from the relation between expansion coefficients of an arbitrary function from \(L_{2}(\Omega)\) and its reflection [25]. Therefore, the same result holds for the function \(g_{\
u_{1},\
u_{2}}\). This result can be extended to the function \(f_{\
u_{1},\
u_{2}}\) due to its degree of smoothness is greater by one than the degree of smoothness of \(g_{\
u_{1},\
u_{2}}\). Thus,  
\[e_{g}^{\langle n\rangle}\approx\frac{C^{g}}{n^{\min\{\
u_{1},\
u_{2}\}+1/2}}, \ \ e_{f}^{\langle n\rangle}\approx\frac{C^{f}}{n^{\min\{\
u_{1},\
u_{2}\}+3/2}}, \tag{40}\]  
where \(C^{g},C^{f}>0\) are some constants.