The formulas below sometimes contain division by zero. We will present them formally for now and eliminate the singularities later as we go.

The main result here is the construction of corresponding parametric biorthogonal systems (see Section 2.3).

For $f\in L_2\left[\mathrm{0,1}\right]$, the following expansion is considered

$f\left(x\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{f}_k\cos \left(\pi kx\right),x\in \left[0,1\right],$(2.1)

where

$f_0={\displaystyle {\int }_0^1}f\left(t\right)\text{d}t,f_s=2{\displaystyle {\int }_0^1}f\left(t\right)\cos \left(\pi kt\right)\text{d}t,s=1,2,\cdots .$

Let’s denote for an integer $n\ge 0$

$\left(n_1\mathrm{,}{n}_2\right)=(\begin{array}{ll}\left(n/2\mathrm{,}\left(n-1\right)/2\right)\mathrm{,}\hfill & \text{if}n\text{is}\text{even}\hfill \\ \left(\left(n-1\right)/2\mathrm{,}\left(n-1\right)/2\right)\mathrm{,}\hfill & \text{if}n\text{is}\text{odd}\hfill \end{array}$(2.2)

The truncated Fourier cosine series can now be written as

$\begin{array}{c}{C}_n\left(x\right)={\displaystyle \underset{k=0}{\overset{n}{\sum }}}{f}_k\cos \left(\pi kx\right)\\ ={\displaystyle \underset{k=0}{\overset{n_1}{\sum }}}{f}_{2k}\cos \left(2\pi kx\right)+{\displaystyle \underset{k=0}{\overset{n_2}{\sum }}}{f}_{2k+1}\cos \left(\pi \left(2k+1\right)x\right),x\in \left[0,1\right],n\ge 1.\end{array}$(2.3)

Consider the vector-function

$c\left(\lambda \mathrm{,}x\right)=\left\{-\frac{\pi \csc \left(\frac{\pi \lambda }{2}\right)\cos \left(\pi \lambda \left(x-\frac{1}{2}\right)\right)}{4\lambda }\mathrm{,}-\frac{\pi \sec \left(\frac{\pi \lambda }{2}\right)\sin \left(\pi \lambda \left(x-\frac{1}{2}\right)\right)}{4\lambda }\right\}\mathrm{.}$ (2.4)

It is easy to verify that for a non-integer $\lambda \in ℂ$

$2{\displaystyle {\int }_0^1}c\left(\lambda ,x\right)\cos \left(\pi kx\right)\text{d}x=\left\{\frac{1+{\left(-1\right)}^k}{2\left(k^2-{\lambda }^2\right)},\frac{1+{\left(-1\right)}^k}{2\left(k^2-{\lambda }^2\right)}\right\},k=1,2\dots$(2.5)

From the Formula (2.1), it follows that the expansion (2.3) is the following representation space $L_2$ as an orthogonal sum of spaces ${\stackrel{1}{L}}_2$ and ${\stackrel{2}{L}}_2$.

$L_2\left(\mathrm{0,1}\right)={\stackrel{1}{L}}_2\left(\mathrm{0,1}\right)\oplus {\stackrel{2}{L}}_2\left(\mathrm{0,1}\right)\mathrm{.}$(2.6)

This partition is the starting point of our approach to trigonometric Fourier series.

Here

$f\left(x\right)={\displaystyle \underset{k=1}{\overset{\infty }{\sum }}}{f}_k\sin \left(\pi kx\right)\mathrm{,}x\in \left[\mathrm{0,1}\right]\mathrm{,}$(2.7)

where

$f\in L_2\left[\mathrm{0,1}\right]\mathrm{,}{f}_s=2{\displaystyle {\int }_0^1}f\left(t\right)\sin \left(\pi kt\right)\text{d}t\mathrm{,}s=\mathrm{1,2,}\cdots$

Now denote for an integer $n\ge 0$

$\left(n_1\mathrm{,}{n}_2\right)=(\begin{array}{ll}\left(n/2-\mathrm{1,}n/2-1\right)\mathrm{,}\hfill & \text{if}n\text{is}\text{even}\hfill \\ \left(\left(n-1\right)/2-\mathrm{1,}\left(n-1\right)/2\right)\mathrm{,}\hfill & \text{if}n\text{is}\text{odd}\hfill \end{array}$(2.8)

Here the truncated sine Fourier series has the form.

$\begin{array}{c}{S}_n\left(x\right)={\displaystyle \underset{k=1}{\overset{n}{\sum }}}{f}_k\sin \left(\pi kx\right)\\ ={\displaystyle \underset{k=1}{\overset{n_1}{\sum }}}{f}_{2k}\sin \left(2\pi kx\right)+{\displaystyle \underset{k=0}{\overset{n_2}{\sum }}}{f}_{2k+1}\sin \left(\pi \left(2k+1\right)x\right),x\in \left[0,1\right],n\ge 1.\end{array}$

Consider the vector-function

$s\left(\lambda ,x\right)=\left\{-\frac{1}{4}\pi \csc \left(\frac{\pi \lambda }{2}\right)\sin \left(\pi \lambda \left(x-\frac{1}{2}\right)\right),\frac{1}{4}\pi \sec \left(\frac{\pi \lambda }{2}\right)\cos \left(\pi \lambda \left(x-\frac{1}{2}\right)\right)\right\}$ (2.9)

We see that again

$2{\displaystyle {\int }_0^1}s\left(\lambda ,x\right)\sin \left(\pi kx\right)\text{d}x=\left\{\frac{\left(1+{\left(-1\right)}^k\right)k}{2\left(k^2-{\lambda }^2\right)},\frac{\left(1+{\left(-1\right)}^k\right)k}{2\left(k^2-{\lambda }^2\right)}\right\},k=1,2,\cdots$(2.10)

Here, we also come to the representation of the form (2.6), in which “sin” appears instead of “cos”.

Let integers $m\ge 1$ are fixed and is a polynomial.

(2.11)

The coefficients $p_j\in ℂ\mathrm{,}j=\mathrm{0,}\cdots \mathrm{,}m-1$ are now considered arbitrary parameters.

In this section, we will construct biorthogonal systems that form the basis of our approach.

Consider the following two unbounded sequences for positive integer s and $r=n_1-m+\mathrm{1,}\cdots \mathrm{,}{n}_1$ in the first sequence and $r=n_2-m+\mathrm{1,}\cdots \mathrm{,}{n}_2$ in the second (see (2.2)).

(2.12)

It’s not hard to see that ${\stackrel{j}{\theta }}_{r\mathrm{,}r}=1$ and for $r\ne s$ ${\stackrel{j}{\theta }}_{r\mathrm{,}s}=0$, $j=\mathrm{1,2}$.

Remark 1. For $s=0$, instead of ${\stackrel{1}{\theta }}_{r\mathrm{,}s}$, it needs to take ${\stackrel{1}{\theta }}_{r\mathrm{,}s}\mathrm{<mo>\; /\; </mo>2}$. This circumstance is inherent only for this case, and the singularity in the previous formulas for the cosine series at $\lambda =0$ does not disappear (see Remark 4).

For a fixed r and $s\to \infty$ we have ${\stackrel{j}{\theta }}_{r\mathrm{,}s}=O\left(s^{-2}\right)\mathrm{,}j=\mathrm{1,2}$. Let’s look at the functions

$\begin{array}{l}{\stackrel{1}{\Theta }}_r\left(x\right)=\cos \left(2\pi rx\right)+{\displaystyle \underset{s=n_1-m}{\overset{\infty }{\sum }}}{\stackrel{1}{\theta }}_{r,s}\cos \left(2\pi sx\right),\\ {\stackrel{2}{\Theta }}_r\left(x\right)=\cos \left(\pi \left(2r+1\right)x\right)+{\displaystyle \underset{s=n_2-m}{\overset{\infty }{\sum }}}{\stackrel{2}{\theta }}_{r,s}\cos \left(\pi \left(2s+1\right)x\right).\end{array}$(2.13)

Here, we have two biotogonal and normalized systems, respectively, at ${\stackrel{1}{L}}_2$ and ${\stackrel{2}{L}}_2$ (see (2.6)).

$\left\{\cos \left(2\pi rx\right)\mathrm{,}{\stackrel{1}{\Theta }}_r\left(x\right)\right\}\mathrm{,}\left\{\cos \left(\pi \left(2r+1\right)x\right)\mathrm{,}{\stackrel{2}{\Theta }}_r\left(x\right)\right\}\mathrm{,}x\in \left[\mathrm{0,1}\right]\mathrm{.}$(2.14)

We emphasize that at this stage, this system contains arbitrary parameters $\left\{p_j\right\}$ (see (2.11)).

Similar to Formula (2.12) (see (2.8))

(2.15)

Similar to the previous case ${\stackrel{j}{\theta }}_{r\mathrm{,}r}=1$ and for $r\ne s$ ${\stackrel{j}{\theta }}_{r\mathrm{,}s}=0$, $j=\mathrm{1,2}$. For $s\to \infty$ we have $\stackrel{j}{\theta }=O\left(s^{-1}\right)$. Here, we will look at the functions.

$\begin{array}{l}{\stackrel{1}{\Theta }}_r\left(x\right)=\sin \left(2\pi rx\right)+{\displaystyle \underset{s=n_1-m}{\overset{\infty }{\sum }}}{\stackrel{1}{\theta }}_{r,s}\sin \left(2\pi sx\right),\\ {\stackrel{2}{\Theta }}_r\left(x\right)=\sin \left(\pi \left(2r+1\right)x\right)+{\displaystyle \underset{s=n_2-m}{\overset{\infty }{\sum }}}{\stackrel{2}{\theta }}_{r,s}\sin \left(\pi \left(2s+1\right)x\right)\end{array}$(2.16)

Here we have two biotogonal and normalized systems for sine, respectively, at ${\stackrel{1}{L}}_2$ and ${\stackrel{2}{L}}_2$.

$\left\{\sin \left(2\pi rx\right)\mathrm{,}{\stackrel{1}{\Theta }}_r\left(x\right)\right\}\mathrm{,}\left\{\sin \left(\pi \left(2r+1\right)x\right)\mathrm{,}{\stackrel{2}{\Theta }}_r\left(x\right)\right\}\mathrm{,}x\in \left[\mathrm{0,1}\right]\mathrm{.}$

When dealing with cosine series, our formula for summation takes the following form (see Section 2.3.1)

$\begin{array}{l}f\left(x\right)\simeq A_{n\mathrm{,}m}\left(x\right)\\ ={\displaystyle \underset{k=0}{\overset{n_1-m}{\sum }}}{f}_{2k}\cos \left(2\pi kx\right)+{\displaystyle \underset{k=0}{\overset{n_2-m}{\sum }}}{f}_{2k+1}\cos \left(\pi \left(2k+1\right)x\right)\\ +{\displaystyle \underset{r=n_1-m+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\Theta }}_r\left(x\right)+{\displaystyle \underset{r=n_2-m+1}{\overset{n_2}{\sum }}}{f}_{2r+1}{\stackrel{2}{\Theta }}_r\left(x\right),x\in \left[0,1\right],n\ge 1.\end{array}$(2.17)

It is easy to see that if we only know the Fourier coefficients $\left\{f_s\right\}\mathrm{,}s\le n$, the “error of summation” has the following form.

$\begin{array}{l}{\Vert f\left(x\right)-A_{n,m}\left(x\right)\Vert }^2\\ ={\displaystyle \underset{s=n_1+1}{\overset{\infty }{\sum }}}{\left|f_{2s}-{\displaystyle \underset{r=n_1-m+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\theta }}_{r,s}\right|}^2+{\displaystyle \underset{s=n_2+1}{\overset{\infty }{\sum }}}{\left|f_{2s+1}-{\displaystyle \underset{r=n_2-m+1}{\overset{n_2}{\sum }}}{f}_{2r+1}{\stackrel{2}{\theta }}_{r,s}\right|}^2\end{array}$(2.18)

Similarly, in the case of sine series (see Section 2.3.2)

$\begin{array}{l}f\left(x\right)\simeq B_{n\mathrm{,}m}\left(x\right)\\ ={\displaystyle \underset{k=0}{\overset{n_1-m}{\sum }}}{f}_{2k}\sin \left(2\pi kx\right)+{\displaystyle \underset{k=0}{\overset{n_2-m}{\sum }}}{f}_{2k+1}\sin \left(\pi \left(2k+1\right)x\right)\\ +{\displaystyle \underset{r=n_1-m+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\Theta }}_r\left(x\right)+{\displaystyle \underset{r=n_2-m+1}{\overset{n_2}{\sum }}}{f}_{2r+1}{\stackrel{2}{\Theta }}_r\left(x\right),x\in \left[0,1\right],n\ge 1.\end{array}$(2.19)

$\begin{array}{l}{\Vert f\left(x\right)-B_{n,m}\left(x\right)\Vert }^2\\ ={\displaystyle \underset{s=n_1+1}{\overset{\infty }{\sum }}}{\left|f_{2s}-{\displaystyle \underset{r=n_1-m+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\theta }}_{r,s}\right|}^2+{\displaystyle \underset{s=n_2+1}{\overset{\infty }{\sum }}}{\left|f_{2s+1}-{\displaystyle \underset{r=n_2-m+1}{\overset{n_2}{\sum }}}{f}_{2r+1}{\stackrel{2}{\theta }}_{r,s}\right|}^2\end{array}$(2.20)

Let us consider the case of the cosine series. According to the orthogonal expansion (2.6) we have for $f\in {\stackrel{1}{L}}_2\left[\mathrm{0,1}\right]$, $f=\stackrel{1}{f}+\stackrel{2}{f}$, $\stackrel{1}{f}\in {\stackrel{1}{L}}_2$, $\stackrel{2}{f}\in {\stackrel{2}{L}}_2$.

On the other hand for sine series

$\stackrel{1}{f}\left(x\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{f}_{2k}\cos \left(2\pi kx\right),x\in \left[0,1\right]$

we have (see Section 2.3.3)

${\stackrel{1}{A}}_{n,m}\left(x\right)={\displaystyle \underset{k=0}{\overset{n_1-m}{\sum }}}{f}_{2k}\cos \left(2\pi kx\right)+{\displaystyle \underset{r=n_1-m+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\Theta }}_r\left(x\right),x\in \left[0,1\right].$

Therefore (see Section 2.3.1)

$\begin{array}{l}\stackrel{1}{f}\left(x\right)-{\stackrel{1}{A}}_{n,m}\left(x\right)\\ ={\displaystyle \underset{s=0}{\overset{n_1-m}{\sum }}}\left(f_{2s}-{\displaystyle \underset{r=n_1-m+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\theta }}_{r,s}\right)+{\displaystyle \underset{s=n+1}{\overset{\infty }{\sum }}}\left(f_{2s}-{\displaystyle \underset{r=n_1-m+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\theta }}_{r,s}\right),x\in \left[0,1\right].\end{array}$

From here

$\begin{array}{l}{\Vert \stackrel{1}{f}\left(x\right)-{\stackrel{1}{A}}_{n,m}\left(x\right)\Vert }^2\\ ={\displaystyle \underset{s=0}{\overset{n_1-m}{\sum }}}{\left|f_{2s}-{\displaystyle \underset{r=n_1-m+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\theta }}_{r,s}\right|}^2+{\displaystyle \underset{s=n+1}{\overset{\infty }{\sum }}}{\left|f_{2s}-{\displaystyle \underset{r=n_1-m+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\theta }}_{r,s}\right|}^2,x\in \left[0,1\right].\end{array}$(3.1)

The members of the sum at right are unknown to us. We minimize the sum on the left, at least partially (see the beginning of this Section).

Let’s choose a set of m even different coefficients ${\Sigma }_m\subseteq \left\{\mathrm{0,2,}\cdots \mathrm{,2}\left(n_1-m\right)\right\}$. This is only possible at $m\le n_1/2$. Our goal is to minimize the sum

${\displaystyle \underset{s\in {\Sigma }_m}{\sum }}{\left|f_{2s}-{\displaystyle \underset{r=n_1-m+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\theta }}_{r,s}\right|}^2\to \min .$

in the norm of ${\stackrel{1}{L}}_2\left[\mathrm{0,1}\right]$ concerning coefficients $\left\{p_j\right\}$ of the polynomial (see (2.11)). More details, the following amount is minimized

(3.2)

This leads (see numerators of this amount) to the pseudo-solution of the following linear system

(3.3)

In the ${\stackrel{2}{L}}_2$ we have in corresponding way

${\displaystyle \underset{s\in {\Sigma }_m}{\sum }}{\left|f_{2s+1}-{\displaystyle \underset{r=n_1-m+1}{\overset{n_1}{\sum }}}{f}_{2r+1}{\stackrel{2}{\theta }}_{r,s}\right|}^2\to \min .$

in the norm of ${\stackrel{2}{L}}_2\left[\mathrm{0,1}\right]$ concerning coefficients $\left\{p_j\right\}$ of the polynomial . Here, the corresponding linear system has the form

(3.4)

Note that here ${\Sigma }_m$ consists of m different odd coefficients: ${\Sigma }_m\subseteq \left\{\mathrm{1,3,}\cdots \mathrm{,2}\left(n_1-m\right)+1\right\}$, $m\le n_2/2$.

Similar to the previous case, here we get two linear systems

(3.5)

where ${\Sigma }_m\subseteq \left\{\mathrm{2,4,}\cdots \mathrm{,2}\left(n_1-m\right)\right\}$, $m\le n_1/2$.

(3.6)

where ${\Sigma }_m\subseteq \left\{\mathrm{1,3,}\cdots \mathrm{,2}\left(n_2-m\right)+1\right\}$, $m\le n_2/2$.

Remark 2. The resulting four equations are essential in our approach (see the beginning of Section 3). Let’s explain some details.

To minimize the norms mentioned above, linear systems (3.3) - (3.6) must be solved using the method of least squares. When a discrepancy occurs, we record them. Let’s consider the case of a cosine series. We denote the zeros of the

polynomial in (3.3) by $\left\{{\stackrel{1}{\mu }}_j={\left({\stackrel{1}{\lambda }}_{2j}\right)}^2\right\}$ and the zeros of the polynomial in (3.4) by $\left\{{\stackrel{1}{\mu }}_j={\left({\stackrel{2}{\lambda }}_{2j+1}\right)}^2\right\}$. Note that these polynomials are different. As a result, we have

$\begin{array}{l}{\stackrel{1}{\theta }}_{r,s}=\frac{4r^2-{\left({\stackrel{1}{\lambda }}_{2r}\right)}^2}{4s^2-{\left({\stackrel{1}{\lambda }}_{2r}\right)}^2}{\displaystyle \underset{\begin{array}{c}p=n_1-m+1\\ p\ne r\end{array}}{\overset{n_1}{\prod }}}\frac{s^2-p^2}{r^2-p^2}\frac{4r^2-{\left({\stackrel{1}{\lambda }}_{2p}\right)}^2}{4s^2-{\left({\stackrel{1}{\lambda }}_{2p}\right)}^2},\\ {\stackrel{2}{\theta }}_{r,s}=\frac{{\left(2r+1\right)}^2-{\left({\stackrel{2}{\lambda }}_{2r+1}\right)}^2}{{\left(2s+1\right)}^2-{\left({\stackrel{2}{\lambda }}_{2r+1}\right)}^2}{\displaystyle \underset{\begin{array}{c}p=n_2-m+1\\ p\ne r\end{array}}{\overset{n_2}{\prod }}}\frac{{\left(2s+1\right)}^2-{\left(2p+1\right)}^2}{{\left(2r+1\right)}^2-{\left(2p+1\right)}^2}\frac{{\left(2r+1\right)}^2-{\left({\stackrel{2}{\lambda }}_{2p+1}\right)}^2}{{\left(2s+1\right)}^2-{\left({\stackrel{2}{\lambda }}_{2p+1}\right)}^2}.\end{array}$ (3.7)

Remark 3. The notation used here for the zeros of polynomials is convenient because, for example, we see that if ${\stackrel{1}{\lambda }}_{2p}=2p$ at $p\ne r$, then in ${\stackrel{1}{\theta }}_{r\mathrm{,}s}$, a cancellation occurs, and instead of m, we have in fact $m-1$.

In Section 1.5 we noted that knowledge of a finite number of coefficients of the exponential Fourier series of a function f from the infinite-dimensional space of quasi-polynomials leads to an exact reconstruction of f.

Here, we have four similar spaces. Let’s start with the cosine series (see Section 2.1).

Definition 3. Let $m\ge 1$ be a fixed integer. Consider a system of functions ${\stackrel{1}{U}}_m=\left\{\cos \left(\pi {\lambda }_k\left(x-\frac{1}{2}\right)\right)\right\}$, ${\lambda }_k\in ℂ$, $x\in \left[\mathrm{0,1}\right]$, $k=\mathrm{1,2,}\cdots \mathrm{,}m$, where $\left\{{\lambda }_k\right\}$ are arbitrary parameters. Consider the linear span ${\stackrel{1}{Q}}_m=span\left\{{\stackrel{1}{U}}_m\right\}$. We call a function $q\in {\stackrel{1}{Q}}_m$ as a first quasi-cosine of degree at most m.

Similar to Definition 2, it is clear that $q\in {\stackrel{1}{Q}}_m$ if and only if either $q\left(x\right)\equiv 0$ or $q\left(x\right)={\displaystyle {\sum }_k}{P}_{{\beta }_k}\left(x\right)\cos /\sin \left(\pi {\lambda }_{k}x\right)$, where the polynomials $P_{{\beta }_k}\left(x\right)\cancel{\equiv }0$ are of degree exactly ${\beta }_k$, and ${\displaystyle {\sum }_k}\left(1+{\beta }_k\right)\le m$ (sin/cos here means: “either sine or cosine”).

Likewise

Definition 4. Let $m\ge 1$ be a fixed integer. Consider a system of functions ${\stackrel{2}{U}}_m=\left\{\sin \left(\pi {\lambda }_k\left(x-\frac{1}{2}\right)\right)\right\}$, ${\lambda }_k\in ℂ$, $x\in \left[\mathrm{0,1}\right]$, $k=\mathrm{1,2,}\cdots \mathrm{,}m$, where $\left\{{\lambda }_k\right\}$ are arbitrary parameters. Consider the linear span ${\stackrel{2}{U}}_m=span\left\{{\stackrel{2}{V}}_m\right\}$. We call a function $q\in {\stackrel{2}{Q}}_m$ as a second quasi-cosine of degree at most m.

Similarly, we define the spaces of the first and second-kind quasi-sines. Here the sines and cosines are simply swapped in the same two spaces.

An essential detail of our approach is that the functions ${\stackrel{j}{\Theta }}_r\left(x\right)$ used above have an explicit, elementary form. The evidence for this is similar to the case of the Fourier exponential series (see [25] - [30] ).

Let’s focus on the case of space ${\stackrel{1}{Q}}_m$ (see previous section). In Formula (2.12), the right-hand side depends only on s². Let us denote $z=4s^2$ and ${\stackrel{1}{\theta }}_{r\mathrm{,}s}={\stackrel{1}{t}}_{r\mathrm{,}z}$. The functions ${\stackrel{1}{t}}_{r\mathrm{,}z}$, considered for $z\in ℂ$, are rational with ordinary poles at $z={\left({\stackrel{1}{\lambda }}_{2p}\right)}^2$. Let $U\subset ℂ$ be a bounded simply connected open subset containing all points $\left\{{\stackrel{1}{\lambda }}_{2p}\right\}$, with the positively oriented simple boundary curve $\gamma =\partial U$.

We have ${\stackrel{1}{t}}_{r\mathrm{,}z}=O\left(1/z\right)$, $z\to \infty$, therefore, according to Cauchy’s residue theorem

$0=\frac{1}{2\pi i}{\displaystyle {\int }_{\gamma }}\frac{{\stackrel{1}{t}}_{r\mathrm{,}t}}{t-s^2}\text{d}t={\stackrel{1}{t}}_{r\mathrm{,}{s}^2}+{\displaystyle \underset{\forall p}{\sum }}{\text{Res}}_{z={\left({\stackrel{1}{\lambda }}_{2p}\right)}^2}\left(\frac{{\stackrel{1}{t}}_{r\mathrm{,}z}}{z-s^2}\right)\mathrm{,}s\in U\left\{{\stackrel{1}{\lambda }}_{2p}\right\}\mathrm{.}$

So

${\stackrel{1}{t}}_{r\mathrm{,}s}=-{\displaystyle \underset{\forall p}{\sum }}{\text{Res}}_{z={\left({\stackrel{1}{\lambda }}_{2p}\right)}^2}\left(\frac{{\stackrel{1}{t}}_{r\mathrm{,}z}}{z-s^2}\right)\mathrm{,}s\in U\left\{{\stackrel{1}{\lambda }}_{2p}\right\}\mathrm{.}$(3.8)

These residues can be explicitly calculated. To avoid clattering by detailed formulas in the presentation, we present corresponding results only for cosine series in ${\stackrel{1}{L}}_2$, when all parameters $\left\{{\stackrel{1}{\lambda }}_{2p}\right\}$ are different.

Theorem 2. When for $p\ne q$, ${\stackrel{1}{\lambda }}_{2p}\ne {\stackrel{1}{\lambda }}_{2q}$, the system $\left\{{\stackrel{1}{\Theta }}_r\right\}$ is expressed directly in the form

${\stackrel{1}{\Theta }}_r\left(x\right)={\displaystyle \underset{k=n_1-m+1}{\overset{n_1}{\sum }}}{\stackrel{1}{c}}_{r\mathrm{,}k}\cos \left(\pi {\stackrel{1}{\lambda }}_{2k}\left(x-1/2\right)\right)\mathrm{,}x\in \left[\mathrm{0,1}\right]\mathrm{,}$(3.9)

where $r=n_1-m+\mathrm{1,}\cdots \mathrm{,}{n}_1$ and³

$\begin{array}{l}{\stackrel{1}{c}}_{r,k}=\frac{\pi \left(r+{\stackrel{1}{\lambda }}_{2k}/2\right)}{4{\stackrel{1}{\lambda }}_{2k}\text{sinc}\left(\pi \left(r-{\stackrel{1}{\lambda }}_{2k}/2\right)\right)}{\displaystyle \underset{\begin{array}{c}p=n_1-m+1\\ p\ne k\end{array}}{\overset{n_1}{\prod }}}\frac{4r^2-{\left({\stackrel{1}{\lambda }}_{2k}\right)}^2}{{\left({\stackrel{1}{\lambda }}_{2k}\right)}^2-{\left({\stackrel{1}{\lambda }}_{2p}\right)}^2}\\ {\displaystyle \underset{\begin{array}{c}q=n_1-m+1\\ q\ne r\end{array}}{\overset{n_1}{\prod }}}\frac{{\left({\stackrel{1}{\lambda }}_{2k}\right)}^2-4q^2}{4r^2-4q^2},r,k=n_1-m+1,\cdots ,n_1.\end{array}$

The proof is similar to the case of the exponential Fourier series (see [25] [27] ). Similar formulas are valid in the other three cases (for ${\stackrel{2}{\Theta }}_r\left(x\right)$ in the cosine series and ${\stackrel{j}{\Theta }}_r\left(x\right)\mathrm{,}\left(j=\mathrm{1,2}\right)$ in the case of the sine series.

Remark 4. In the above three cases, all singularities are eliminated using the sinc-function. However, Formula (3.9) shows that at ${\stackrel{1}{\lambda }}_{2k}=0$ in the cosine series, the singularity is preserved. For a solution to this problem numerically, please look at Section 4.1.3 below.

Let’s denote (see above Section)

${\stackrel{1}{a}}_k={\displaystyle \underset{r=-m+n_1+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{c}}_{r,k},{\stackrel{2}{a}}_k={\displaystyle \underset{r=-m+n_2+1}{\overset{n_2}{\sum }}}{f}_{2r+1}{\stackrel{2}{c}}_{r,k}.$(3.10)

Consider functions (see Section 2.1 and Theorem 2)

$\begin{array}{l}{\stackrel{1}{O}}_{n,m}\left(x\right)={\displaystyle \underset{k=-m+n_1+1}{\overset{n_1}{\sum }}}{\stackrel{1}{a}}_k\cos \left(\pi {\stackrel{1}{\lambda }}_{2k}\left(x-\frac{1}{2}\right)\right),x\in \left[0,1\right]\\ {\stackrel{2}{O}}_{n,m}\left(x\right)={\displaystyle \underset{k=-m+n_1+1}{\overset{n_1}{\sum }}}{\stackrel{2}{a}}_k\cos \left(\pi {\stackrel{2}{\lambda }}_{2k+1}\left(x-\frac{1}{2}\right)\right),x\in \left[0,1\right].\end{array}$(3.11)

Let’s denote now

$\begin{array}{l}{\stackrel{1}{A}}_{n\mathrm{,}m}\left(x\right)={\displaystyle \underset{s=0}{\overset{n_1-m}{\sum }}}\left(f_{2s}-{\displaystyle \underset{r=-m+n_1+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\theta }}_{r\mathrm{,}s}\right)\cos \left(2\pi sx\right)+{\stackrel{1}{O}}_{n\mathrm{,}m}\left(x\right)\mathrm{,}x\in \left[\mathrm{0,1}\right]\\ {\stackrel{2}{A}}_{n\mathrm{,}m}\left(x\right)={\displaystyle \underset{s=0}{\overset{n_2-m}{\sum }}}\left(f_{2s+1}-{\displaystyle \underset{r=-m+n_2+1}{\overset{n_2}{\sum }}}{f}_{2r+1}{\stackrel{2}{\theta }}_{r\mathrm{,}s}\right)\cos \left(\left(2s+1\right)\pi x\right)+{\stackrel{2}{O}}_{n\mathrm{,}m}\left(x\right)\mathrm{,}x\in \left[\mathrm{0,1}\right]\mathrm{.}\end{array}$(3.12)

The function $f\left(x\right)\in L_2\left[\mathrm{0,1}\right]$ is approximated by the formula

$f\left(x\right)\simeq A_{n\mathrm{,}m}\left(x\right)={\stackrel{1}{A}}_{n\mathrm{,}m}\left(x\right)+{\stackrel{2}{A}}_{n\mathrm{,}m}\left(x\right)\mathrm{,}x\in \left[\mathrm{0,1}\right]$(3.13)

Similar formulas were applied to the case of the sine series. This approximations was, in fact, constructed according to Krylov’s scheme (see Section 1.2). We call the algorithms corresponding to the above summation formulas adaptive since the terms ${\stackrel{j}{O}}_{n\mathrm{,}m}\left(x\right)\mathrm{,}j=\mathrm{1,2}$ appearing as the function g from (1.3) depends on the function $f\left(x\right)$ (“adapt” to it). Unlike the KE-method, which can only use a few g-functions (Section 1.3), here a broadest class of trigonometric functions that depend on parameters from $ℂ$ are used (see Section 3.3).

Remark 5. When solving linear equations in Section 3.1, sometimes the determinant of the matrix turns out to be zero. (see Remark 2 and beginning of Section 3.2). Formulas (3.12) take into account the resulting discrepancies. Above in (3.12), this is the next part of the sum

$d_1\left(x\right)={\displaystyle \underset{s\in {\Sigma }_m}{\sum }}\left(f_{2s}-{\displaystyle \underset{r=-m+n_1+1}{\overset{n_1}{\sum }}}{f}_{2r}{\stackrel{1}{\theta }}_{r,s}\right)\cos \left(2\pi sx\right),x\in \left[0,1\right]$

which is equal to zero in $\left[\mathrm{0,1}\right]$ if the determinant differs from zero.

This main theoretical result of this work is valid for all four proposed algorithms. Let’s look at the case of cosine series with even coefficients (see ${\stackrel{1}{A}}_{n\mathrm{,}m}$ in (3.12)). Denote $D_m=\left\{2\left(n_1-m+1\right)\mathrm{,}\cdots \mathrm{,2}{n}_1\right\}$ and take ${\Sigma }_m$ from (3.5).

Theorem 3 (The phenomenon of the over-convergence). Let $f\in {\stackrel{1}{Q}}_m$ (see Definition 3), and the Fourier coefficients $\left\{f_{2s}\right\}$, $s\in D_m\cup {\Sigma }_m$, of the function f be given. Denote by Λ the set of all integer parameters in $\left\{{\stackrel{1}{\lambda }}_{2j}\right\}$ (see Section

3.2). In order for the ${\stackrel{1}{A}}_{n\mathrm{,}m}\left(x\right)$ to be exact (that is, $f\left(x\right)\equiv {\stackrel{1}{A}}_{n\mathrm{,}m}\left(x\right)$, $x\in \left[\mathrm{0,1}\right]$), it is necessary and sufficient that $\Lambda \subseteq D_m\cup {\Sigma }_m$.

If we keep Formula (3.8) in mind, then the proof is quite similar to that given in [25] .

Definition 5. The algorithms underlying the summation formulas ${\stackrel{1}{A}}_{n\mathrm{,}m}\left(x\right)$ and ${\stackrel{2}{A}}_{n\mathrm{,}m}\left(x\right)$ (see (3.12)) will be denoted by ${ℭ}_1$ and ${ℭ}_2$, respectively. We denote the corresponding two algorithms for Fourier sine series by ${\mathfrak{S}}_1$ and ${\mathfrak{S}}_2$.

Accordingly, we denote the parallel algorithm for the cosine Fourier series by $ℭ$ and the sine series by $\mathfrak{S}$.

First, we’d like to present briefly a scheme for accelerating the convergence of work [29] . Let $m\ge 1$ be an integer and $D_m$ be a set of m different integers and , $z\in ℂ$, be a polynomial

(3.14)

where $\left\{{\lambda }_k\right\}$, $k\in D_m$, be the set of its zeros. Consider the sequence

(3.15)

(see Section 2.3.2).

$T_r\left(x\right)=\exp \left(i\pi rx\right)+{\displaystyle \underset{s\notin D_m}{\sum }}{t}_{r\mathrm{,}s}\exp \left(i\pi sx\right)\mathrm{,}r\in D_m$(3.16)

The space of quasi-polynomials $Q_m$ is defined (see Definition 2 above), and an explicit form of the functions $T_r\in Q_m$ is given.

The basic system of linear equations regarding the definition of $\left\{a_k\right\}$ (see (3.14)) has the following form

(3.17)

where ${\tilde{D}}_m$ is a set of other m different integers ( ${\tilde{D}}_m\cap D_m=\varnothing$).

The next step is finding the zeros of polynomial . Finally, function $f\left(x\right)$ is approximated by the summation formula

$f\simeq F_{n\mathrm{,}m}\left(x\right)={\displaystyle \underset{r\in D_m}{\sum }}{f}_r{T}_r\left(x\right)+{\displaystyle \underset{s\in D_n\backslash D_m}{\sum }}\left(f_s-{\displaystyle \underset{r\in D_m}{\sum }}{f}_r{t}_{r\mathrm{,}s}\right)\exp \left(i\pi sx\right)\mathrm{.}$(3.18)

with the error

$R_{n\mathrm{,}m}\left(x\right)\stackrel{def}{=}f\left(x\right)-F_{n\mathrm{,}m}\left(x\right)\mathrm{.}$(3.19)

This algorithm is called adaptive since equation (3.17) “adapts” solutions $\left\{a_k\right\}$ to the behavior of coefficients $\left\{f_s\right\}$. It can be noted that this scheme differs from our previous cases only in notations.

Joseph Fourier used the trigonometric form of the Fourier series, although Euler’s formula for the exponential was known long before this. Solving a practical problem, instead of approximation (1.1), he studied the formula

$\begin{array}{l}f\left(x\right)\simeq {\displaystyle \underset{k=-n}{\overset{n}{\sum }}}\left(a_k\cos \pi kx+b_k\sin \pi kx\right)\mathrm{,}x\in \left[-\mathrm{1,1}\right]\mathrm{.}\\ a_0=\frac{1}{2}{\displaystyle {\int }_{-1}^1}f\left(t\right)\text{d}t,a_k={\displaystyle {\int }_{-1}^1}f\left(t\right)\cos \pi kt\text{d}t,b_k={\displaystyle {\int }_{-1}^1}f\left(t\right)\sin \pi kt\text{d}t,\end{array}$(3.20)

where f is a real-valued function.

Do the acceleration algorithms proposed above work now similarly to the algorithm for (1.1)? The exact answer is not so trivial. First, we show that the partial sum (1.1) is not generally equivalent to the partial sum of the form (3.20). For example, if in (1.1) we go to trigonometric functions, then for even n, the summation over sines is carried out in the range from 1 to $\left(n-1\right)$, while over cosines - from 0 to n. In addition, even coefficients for sines and odd - for cosines are equal to zero. Therefore, we can apply algorithm ${ℭ}_1$ to the cosine sum and to the sine sum - ${\mathfrak{S}}_1$ (see Definition 5). Two algorithms are used here in parallel (see Sections 3.5.1 and 3.5.2).

As for the sum (3.20), we can consider the sums of cosines and sines separately on $\left[\mathrm{0,1}\right]$ and continue each of them to $\left[-\mathrm{1,0}\right)$, even and odd. All four of our algorithms work here in parallel. Thus, the numerical complexity of this algorithm is twice more at the same n.

Recall that the adaptive algorithm for the Fourier exponential series was designated $\mathfrak{A}$ (see [25] and Section 1.5 above). We’d like to introduce notations for the algorithms mentioned above.

Definition 6. For the algorithm of article [25] , we will leave the designation $\mathfrak{A}$. We denote the parallel algorithm for the trigonometric form of the exponential Fourier series by $ℭ{\mathfrak{S}}_{12}$ and the parallel algorithm for (3.20) - by $ℭ\mathfrak{S}$.

Numerical experiments were carried out with the following functions.

$f_1\left(x\right)=(\begin{array}{ll}\mathrm{0,}\hfill & 0\le x\mathrm{<mo>\; <\; </mo>}1/3\mathrm{<mo>\; ;\; </mo>}\hfill \\ \frac{29}{20}{\sin }^5\left(3x-1\right)\mathrm{,}\hfill & 1/3\le x\le \mathrm{1\; <mo>\; ;\; </mo>}\hfill \end{array}$(4.1)

$f_2\left(x\right)=\frac{63}{89}{\text{e}}^{2i{\left(x+\frac{2}{3}\right)}^2}\mathrm{,}{f}_3\left(x\right)=\frac{\cos \left(x\right)}{\left(2/3+i\right)-x}\mathrm{,}x\in \left[\mathrm{0,1}\right]$(4.2)

Here $f_1\left(x\right)\in C^5$ and $f_1{\left(x\right)}^{\left(5\right)}$ is a piece-wise smooth function with two jumps: at $x=1/3$ and $x=1$ (see (1.1)). The other two functions are analytic in a neighborhood of segment [0, 1]. Function $f_2\left(x\right)$ is entire, but not exponential type. Function $f_3\left(z\right)$ is rational and has only one pole at $z=3/2$. Functions $f_1\left(x\right)$ and $f_2\left(x\right)$ were tested in [29] (on $x\in \left[-\mathrm{1,1}\right]$ with the linear change of variable).