1. Introduction

We say that a series $\sum _{n\in Z^{N} }c_{n} \varphi _{n}$ converges on rectangulars if there exists the limit of the partial sums $S_{m}=\sum_{|n_{1}|\leq m_{1}}\sum_{|n_{2}|\leq m_{2}}...\sum_{|n_{N}|\leq m_{N}}c_{n}\varphi _{n}$ as $\ \underset{_{1\leq j\leq N}}{\min }m_{j}\rightarrow \infty$.

Let us remind that a system of elements $\varphi =\left\{ \varphi _{n}\right\} _{n\in Z^{N}}$ is called a basis of the Banach space $E$ at summation on rectangulars if any vector $x\in E$ decomposes uniquely in the series

$x=\sum\limits\limits_{n\in Z^{N}}c_{n}\varphi _{n}$

(1.1)

which is convergent with respect to the norm of the space $E$ at summation by rectangulars. Hence we exclude from consideration Banach spaces which do not possess the property of approximation (see ^[1] and ^[2]).

Factors $c_{n}$ in (1.1) are linear functionals:

$c_{n}=f_{n}(x)\, \, , \, \, \, \, \, \, n\in Z^{N}$

and, according to the well known Banach theorem (see, for example, ^[3], ^[4]), there is a constant $C_{\varphi }$ such that

$\left\Vert \, \varphi _{n}^{{}}\right\Vert ^{-1}\leq \left\Vert f_{n}\right\Vert \leq C_{\varphi }\left\Vert \_\varphi _{n}\right\Vert ^{-1}.$

A system of elements $\psi =\left\{ \psi _{n}\right\} _{n\in Z^{N}}$ from the Banach space is said to be $\omega$-linear independent at summation by rectangulars if the equality $\sum_{n\in Z^{N}}c_{n}\psi _{n}=0$ at summation on rectangulars is impossible at

$\sum\limits\limits_{n=1}^{\infty }\left\vert c_{n}\right\vert ^{2}\cdot \left\Vert \, \psi _{n}\, \right\Vert ^{2}>0.$

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2. Main Results

Theorem 2.1. Let $\left\{ \varphi _{n}\right\} _{n\in Z^{N}}$ be a normed basis in the Banach space E at summation by rectangulars. Further, let the system $\left\{ \psi _{n}\right\} _{n\in Z^{N}}$ be $\omega$-linear independent at summation by rectangulars and $\sum_{n\in Z^{N}}\left\Vert \varphi _{n}-\psi _{n}\right\Vert < \infty$. Then $\left\{ \psi _{n}\right\} _{n\in Z^{N}}$ is also a basis in E at summation by rectangulars.

Proof. Fix an $N$-dimensional vector $\beta =(\beta _{1}, \, \beta _{2}, \, ...\, , \, \beta _{N})$ with nonnegative integer components $\beta _{1}, \, \beta _{2}, \, ...\, , \, \beta _{N}$ and define as,

$%\begin{array}{c} \tilde{\psi }_{n} =\left\{\begin{array}{l} {\varphi _{n} \, \, \, \, as\, \, \, \, \, \, \, \, \, \, \left|n_{1} \right|\le \beta _{1}, \, \, \, \, \left|n_{2} \right|\le \beta _{2}, \, ..., \left|n_{N} \right|\le \beta _{N}, \, } \\ {\psi _{n} \, \, \, \, as\, \, \, \, \, \, \, \, \, or\, \, \left|n_{1} \right|>\beta _{1}, \, \, or\, \, \, \, \left|n_{2} \right|>\beta _{2}, \, \, \, ..., \, or\, \, \left|n_{N} \right|>\beta _{N} \, , \, \, \, \, \, here\, \, n\in Z^{N} .\, } \end{array}\right.$

Let us introduce the operator $S:E\rightarrow E$ which compares to each element

$x=\sum\limits_{n\in Z^{N}}f_{n}(x)\, \varphi _{n}=\lim\limits_{\underset{_{1\leq j\leq N}}{\min }m_{j}\rightarrow \infty }\, \, \, \, \sum\limits_{|n_{1}|\leq m_{1}}\sum\limits_{|n_{2}|\leq m_{2}}...\sum\limits_{|n_{N}|\leq m_{N}}f_{n}(x)\, \varphi _{n}$

to the element

$S\, x=\sum\limits_{n\in Z^{N}}f_{n}(x)\, \left( \varphi _{n}-\widetilde{\psi }% _{n}\right) . \label{GrindEQ___A205_}$

Obviously, for sufficiently large $\mu =\underset{1\leq j\leq N}{\min }\beta$, we have

$\left\| Sx\right\| \le C_{\varphi } \left\| x\right\| \sum\limits _{or\, \, \left|n_{1} \right|>\beta _{1} } \sum\limits _{\, or\, \, \left|n_{2} \right|>\beta _{2} } ...\sum\limits _{\, or\, \, \left|n_{N} \right|>\beta _{N} } \left\| \varphi _{n} -\psi _{n} \right\| <\varepsilon \left\| x\right\| .$

Hence, for the operator U defined by equality

$U\, x=x-S\, x=\sum\limits_{n\in Z^{N}}f_{n}(x)\, \widetilde{\psi }_{n},$

there is an inverse linear operator $U^{-1}$. Acting on both parts of the equality

$U^{-1}x=\sum\limits_{n\in Z^{N}}f_{n}\left( U^{-1}x\right) \varphi _{n}$

with the operator U, we obtain

$x=\sum\limits_{n\in Z^{N}}f_{n}\left( U^{-1}x\right) \widetilde{\psi }_{n}, \label{GrindEQ__6_}$

which implies that the system $\left\{ \widetilde{\psi }_{n}\right\} _{n\in Z^{N}}$ forms a basis in $E$ at summation on rectangulars, i.e. each vector $x\in E$ is decomposed uniquely in the series

$x=\sum\limits_{n\in Z^{N}}f_{n}\left( U^{-1}x\right) \widetilde{\psi }% _{n}=\lim\limits_{\underset{_{1\leq j\leq N}}{\min }m_{j}\rightarrow \infty }\, \, \, \, \sum\limits_{|n_{1}|\leq m_{1}}\sum\limits_{|n_{2}|\leq m_{2}}...\sum\limits_{|n_{N}|\leq m_{N}}f_{n}\left( U^{-1}x\right) \widetilde{\psi }% _{n}$

which is convergent with respect to the norm of the space $E$ at summation on rectangulars.

Since the system $\left\{ \widetilde{\psi }_{n}\right\} _{n\in Z^{N}}$forms a basis in E at summation on rectangulars, then

$\ {\psi _{k} =\sum\limits _{\left|n_{1} \right|\le \beta _{1} } \sum\limits _{\, \, \, \left|n_{2} \right|\le \beta _{2} } ...\sum\limits _{\, \, \, \left|n_{N} \right|\le \beta _{N} } f_{n} \left(U^{-1} \psi _{k} \right)\varphi _{n} } \\ {+\sum\limits _{\, or\, \, \left|n_{1} \right|>\beta _{1} } \sum\limits _{\, \, or\, \, \, \left|n_{2} \right|>\beta _{2} } ...\sum\limits _{\, \, \, \, or\, \, \left|n_{N} \right|>\beta _{N} } f_{n} \left(U^{-1} \psi _{k} \right)\psi _{n} } \\ =x_{k}^{1}+x_{k}^{2},$

here $k=(k_{1}, \, k_{2}, \, ...\, , \, k_{N})$ is a multi-index with components $% \, |k_{1}|\leq \beta _{1}$, $|k_{2}|\leq \beta _{2}\,$, $...\, , \, \, \, |k_{N}|% \leq \beta _{N}$, and

$x_{k}^{1} =\sum\limits_{|n_{1}|\leq \beta _{1}}\sum\limits_{|n_{2}|\leq \beta _{2}}...\sum\limits_{|n_{N}|\leq \beta _{N}}f_{n}\left( U^{-1}\psi _{k}\right) \varphi _{n}, \\x_{k}^{2} =\sum\limits _{\, or\, \, \left|n_{1} \right|>\beta _{1} } \sum\limits _{\, or\, \, \, \left|n_{2} \right|>\beta _{2} } ...\sum\limits _{\, \, or\, \, \left|n_{N} \right|>\beta _{N} } f_{n} \left(U^{-1} \psi _{k} \right)\psi _{n}$

$\omega$-linear independence of $\left\{\psi _{n} \right\}_{n\in Z^{N} }$ at summation on rectangulars implies linear independence of $% \left\{x_{k}^{1} \right\}$. As concepts of linear independence and baseness are equivalent in finite dimensional space,

$\varphi _{n} =\sum\limits _{|k_{1} |\le \beta _{1} }\sum\limits _{|k_{2} |\le \beta _{2} }... \sum\limits _{|k_{N} |\le \beta _{N} }\alpha _{nk} x_{k}^{1}$

is a multi-index with components $\, |n_{1} |\le \beta _{1}, \, \, \, \, |n_{2} |\le \beta _{2} \, , \, \, ...\, , \, \, \, |n_{N} |\le \beta _{N}$ for $n=(n_{1}, \, n_{2}, \, ...\, , \, n_{N})\in Z^{N}$.

Hence we have

$\begin{split} \ {x=\sum\limits _{\left|n_{1} \right|\le \beta _{1} } \sum\limits _{\, \, \, \left|n_{2} \right|\le \beta _{2} } ...\sum\limits _{\, \, \, \left|n_{N} \right|\le \beta _{N} } f_{n} \left(U^{-1} x\right)\varphi _{n} +\sum\limits _{\, or\, \, \left|n_{1} \right|>\beta _{1} } \sum\limits _{\, or\, \, \left|n_{2} \right|>\beta _{2} } ...\sum\limits _{\, or\, \, \left|n_{N} \right|>\beta _{N} } f_{n} \left(U^{-1} x\right)\psi _{n} =} \\ {=\sum\limits _{\left|k_{1} \right|\le \beta _{1} } \sum\limits _{\, \, \, \left|k_{2} \right|\le \beta _{2} } ...\sum\limits _{\, \, \, \left|k_{N} \right|\le \beta _{N} } \left(T\right)\psi _{k} +\sum\limits _{\, or\, \, \left|n_{1} \right|>\beta _{1} } \sum\limits _{\, \, or\, \, \left|n_{2} \right|>\beta _{2} } ...\sum\limits _{\, \, \, or\, \, \, \left|n_{N} \right|>\beta _{N} } f_{n} \left(U^{-1} x\right)\psi _{n} =} \\ {=\sum\limits _{\left|k_{1} \right|\le \beta _{1} } \sum\limits _{\, \, \, \left|k_{2} \right|\le \beta _{2} } ...\sum\limits _{\, \, \, \left|k_{N} \right|\le \beta _{N} } \left(T\right)\psi _{k} +\sum\limits _{\, or\, \, \left|s_{1} \right|>\beta _{1} } \sum\limits _{\, \, or\, \, \, \left|s_{2} \right|>\beta _{2} } ...\sum\limits _{\, \, or\, \, \left|s_{N} \right|>\beta _{N} } } \\ {\left(\sum\limits _{\left|k_{1} \right|\le \beta _{1} } \sum\limits _{\, \, \, \left|k_{2} \right|\le \beta _{2} } ...\sum\limits _{\, \, \, \left|k_{N} \right|\le \beta _{N} } \left(T\right)f_{s} \left(U^{-1} \psi _{k} \right)\right)\psi _{s} +\sum\limits _{or\, \, \, \left|n_{1} \right|>\beta _{1} } \sum\limits _{\, \, or\, \, \, \left|n_{2} \right|>\beta _{2} } ...\sum\limits _{\, \, or\, \, \, \left|n_{N} \right|>\beta _{N} } f_{n} \left(U^{-1} x\right)\psi _{n} } \end{split}$

Here,

$T=\sum\limits_{|n_{1}|\leq \beta _{1}}\sum\limits_{|n_{2}|\leq \beta _{2}}...\sum\limits_{|n_{N}|\leq \beta _{N}}f_{n}\left( U^{-1}x\right) a_{nk}$

It means that the system $\left\{ \psi _{n}\right\} _{n\in Z^{N}}$ is a basis in the Banach space $E$ at summation on rectangulars. Hence, Theorem (2.1) is proved.

When $N=1$ Theorem (2.1) was proved in ^[6]

Remark 1. At absence of $\omega$-linear independence of the system $% \left\{\psi _{n} \right\}_{n\in Z^{N} }$ at summation on rectangulars, one states that the system $\left\{\psi _{n} \right\}_{n\in Z^{N} }$ is a basis (probably, overfilling) with the finite defect in the Banch space $E$.

A function $f (x)\in L_{p}\left (T^{N}\right)$ belongs to the space $% W_{p}^{s}\left (T^{N}\right)$, if all its partial derivatives $D^{\alpha }f$ (in the sense of the theory of distributions) of the order $|\alpha |=s$ belong to $L_{p}\left (T^{N}\right)$, i.e. the norm

$\left\Vert f\right\Vert _{W_{p}^{s}\left( T^{N}\right) }=\left\Vert f\right\Vert _{L_{p}\left( T^{N}\right) }+\sum\limits_{|\alpha |=s}\left\Vert D^{\alpha }f\right\Vert _{L_{p}\left( T^{N}\right) },$

where $1\leq p < \infty$, $s=0, \, \, 1, \, \, 2, \, \, ...$, is finite.

In the case of $N=1$ belonging of a function $f (x)$ to the class $W_{p}^{s} \left (T^{N} \right)$ it means that $f (x)$ has $s-1$ continuous derivatives, $% f^{(s-1)} (x)$ is absolutely continuous, and $f^{(s)} (x)$ belongs to $L_{p} (T)$.

Corollary 1. Let $\psi _{n}(x)=(2\pi)^{-\frac{N}{p}}\cdot \left (1+\sum_{|\alpha |=s}\left\vert n^{\alpha }\right\vert \right) ^{-1}\cdot \exp \left (i\lambda _{n}x\right) +\alpha _{n}(x)$, where $\lambda _{n}\neq \lambda _{m}$ as $n\neq m$, be an $\omega$-linear independent system of functions satisfying the following conditions:

1. $\sum_{n\in Z^{N}}\left\vert \lambda _{n}-n\right\vert < \infty;$

2. $\sum_{n\in Z^{N}}\left\Vert \alpha _{n}(x)\right\Vert _{W_{p}^{s}\left (T^{N}\right) } < \infty.$

Then the system of functions $\left\{ \psi _{n}\right\} _{n\in Z^{N}}$ forms at summation on rectangulars a basis in $W_{p}^{s}\left (T^{N}\right)$, $% 1 < p < \infty$, $s=0, \, \, 1, \, \, 2, \, \, ...$.

Theorem 2.2. Let

$\psi _{n}(x)=(2\pi )^{-\, \frac{N}{2}}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-\frac{1}{2}}\cdot \exp \left( i\lambda _{n}x\right) +\alpha _{n}(x),$

where $\lambda _{n}\neq \lambda _{m}$ as $n\neq m$, be an $\omega$-linear independent system of functions satisfying the following conditions:

1. $k=\sqrt{\underset{n}{\sup }\frac{\theta ^{2}+\sum_{|\alpha |=s}\left (\theta ^{2}\left\vert \lambda _{n}^{\alpha }\right\vert ^{2}+\left\vert \lambda _{n}^{\alpha }-n^{\alpha }\right\vert ^{2}\right) }{1+\sum_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}}} < 1$, here $\theta =\exp (MN\pi)-1$,

2. $\sum_{n\in Z^{N}}\left\Vert \alpha _{n}(x)\right\Vert _{W_{2}^{s}\left (T^{N}\right) }^{2} < \infty.$

$M=\underset{j}{\sup }\underset{n_{j}}{\sup }\left\vert \lambda _{n_{j}}-n_{j}\right\vert ;$

Then the system of functions $\left\{ \psi _{n}(x)\right\} _{n\in Z^{N}}$ forms the Riesz basis in the space $W_{2}^{s}\left (T^{N}\right)$.

Proof. It is known that the system of functions $\varphi _{n} (x)=(2\pi)^{-\frac{N% }{2} } \cdot \left (1+\sum _{|\alpha |=s}\left|n^{\alpha } \right|^{2} \right)^{-\frac{1}{2} } \cdot \exp (inx)$ forms an orthonormal basis in the space $W_{2}^{s} \left (T^{N} \right)$. The norm in this space is introduced in such a way at following:

$\left\| f\right\| _{W_{2}^{s} \left(T^{N} \right)}^{2} =\left\| f\right\| _{L_{2} \left(T^{N} \right)}^{2} +\sum\limits _{|\alpha |=s}\left\| D^{\alpha } f\right\| _{L_{2} \left(T^{N} \right)}^{2} .$

Let

$\tilde{\psi}_{n}(x)=(2\pi )^{-\frac{N}{2}}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-\frac{1}{2}}\cdot \exp (i\lambda _{n}x),$

where $\lambda _{n}\neq \lambda _{m}$ as $n\neq m$, be an $\omega$-linear independent system of functions satisfying the condition:

$k=\sqrt{\underset{n}{\sup }\frac{\theta ^{2}+\sum\limits_{|\alpha |=s}\left( \theta ^{2}\left\vert \lambda _{n}^{\alpha }\right\vert ^{2}+\left\vert \lambda _{n}^{\alpha }-n^{\alpha }\right\vert ^{2}\right) }{1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}}}<1,$

here $\theta =\exp (MN\pi)-1$, $M=\underset{j}{\sup }\underset{n_{j}}{% \sup }\left\vert \lambda _{n_{j}}-n_{j}\right\vert$.

Further, let $\left\{a_{n} \right\}$ be a finite system of complex numbers. Then

$\left\| \sum\limits _{n}a_{n} \left(\widetilde{\psi }_{n} -\varphi _{n} \right) \right\| _{W_{2}^{s} \left(T^{N} \right)}^{2} =\left\| \sum\limits _{n}a_{n} \left(% \widetilde{\psi }_{n} -\varphi _{n} \right) \right\| _{L_{2} \left(T^{N} \right)}^{2} +\sum\limits _{|\alpha |=s}\left\| D^{\alpha } \left(\sum\limits _{n}a_{n} \left(\widetilde{\psi }_{n} -\varphi _{n} \right) \right)\right\| _{L_{2} \left(T^{N} \right)}^{2} = \\ =(2\pi )^{-N} \left[\left\| \sum\limits _{n}a_{n} \cdot \left(\left(1+\sum\limits _{|\alpha |=s}\left|n^{\alpha } \right|^{2} \right)^{-\frac{1}{2} } \left(\exp \left(i\lambda _{n} x\right)-\exp (inx)\right)\right)\right\| \right. _{L_{2} \left(T^{N} \right)}^{2} + \\ +\left. \sum\limits _{|\alpha |=s}\left\| D^{\alpha } \left(\sum\limits _{n}a_{n} \cdot \left(1+\sum\limits _{|\alpha |=s}\left|n^{\alpha } \right| ^{2} \right)^{-\frac{1}{% 2} } \cdot \left(\exp \left(i\lambda _{n} x\right)-\exp (inx)\right)\right)\right\| _{L_{2} \left(T^{N} \right)}^{2} \right]$

As we have,

$\left\| \sum\limits _{n}a_{n} \cdot \left(1+\sum\limits _{|\alpha |=s}\left|n^{\alpha } \right|^{2} \right)^{-\frac{1}{2} } \cdot \left(\exp \left(i\lambda _{n} x\right)-\exp (inx)\right)\right\| _{L_{2} \left(T^{N} \right)} \le \\ \le \sum\limits _{k=1}^{\infty }\frac{1}{k!} \left\| \sum\limits _{n}a_{n} \cdot \left(1+\sum\limits _{|\alpha |=s}\left|n^{\alpha } \right|^{2} \right)^{-\frac{1}{2% } } \cdot \left[i\left(\lambda _{n}-n\right)x\right]^{k} \cdot \exp (inx)\right\| _{L_{2} \left(T^{N} \right)} .$

Further,

$\left\Vert \sum\limits_{n}a_{n}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-\frac{1}{2}}\cdot \left[i\left( \lambda _{n}-n\right) x\right] ^{k}\cdot \exp (inx)\right\Vert _{L_{2}\left( T^{N}\right) }= \\ \begin{array}{l} {=\left\Vert \, \, \sum\limits_{n}a_{n}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-\frac{1}{2}}\left( \sum\limits_{\beta _{1}+\beta _{2}+...+\beta _{N}=k}\frac{k!}{\beta _{1}!\beta _{2}!...\beta _{N}!}\cdot \right. \right. } \\ {\cdot \left. \left. \prod\limits_{j=1}^{N}\left( \lambda _{n_{j}}-n_{j}\right) ^{\beta _{j}}\cdot x_{j}^{\beta _{j}}\right) \cdot \exp (inx)\right\Vert _{L_{2}\left( T^{N}\right) }=}% \end{array}% \\ =\left\Vert \sum\limits_{\beta _{1}+\beta _{2}+...+\beta _{N}=k}\frac{k!}{\beta _{1}!\beta _{2}!...\beta _{N}!}\cdot \prod\limits_{j=1}^{N}x_{j}^{\beta _{j}}\cdot \left( \sum\limits_{n}a_{n}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-\frac{1}{2}}\cdot \right. \right.$

$\left. \left. \cdot \prod\limits_{j=1}^{N}\left( \lambda _{n_{j}}-n_{j}\right) ^{\beta _{j}}\cdot \exp (inx)\right) \right\Vert _{L_{2}\left( T^{N}\right) }\leq \sum\limits_{\beta _{1}+\beta _{2}+...+\beta _{N}=k}\frac{k!}{\beta _{1}!\beta _{2}!...\beta _{N}!}\cdot \pi ^{k}\cdot \\ \cdot \left\Vert \sum\limits_{n}a_{n}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-\frac{1}{2}}\cdot \prod\limits_{j=1}^{N}\left( \lambda _{n_{j}}-n_{j}\right) ^{\beta _{j}}\cdot \exp (inx)\right\Vert _{L_{2}(T^{N})}\leq \\ \sum\limits_{\beta _{1}+\beta _{2}+...+\beta _{N}=k}\frac{k!}{\beta _{1}!\beta _{2}!...\beta _{N}!}\cdot \pi ^{k}\cdot \left( \sum\limits_{n}\left\vert a_{n}\right\vert ^{2}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-1}\cdot \prod\limits_{j=1}^{N}\left\vert \lambda _{n_{j}}-n_{j}\right\vert ^{2\beta _{j}}\cdot (2\pi )^{N}\right) ^{\frac{1}{2% }}\leq$

$\leq (2\pi )^{\frac{N}{2}}\sum\limits_{\beta _{1}+\beta _{2}+...+\beta _{N}=k}\frac{% k!}{\beta _{1}!\beta _{2}!...\beta _{N}!}\cdot \pi ^{k}\cdot M^{k}\cdot \left( \sum\limits_{n}\left\vert a_{n}\right\vert ^{2}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-1}\right) ^{\frac{1}{2}% }= \\ =(2\pi )^{\frac{N}{2}}\cdot \pi ^{k}\cdot M^{k}\cdot \left( \sum\limits_{n}\left\vert a_{n}\right\vert ^{2}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-1}\right) ^{\frac{1}{2}% }\cdot \sum\limits_{\beta _{1}+\beta _{2}+...+\beta _{N}=k}\frac{k!}{\beta _{1}!\beta _{2}!...\beta _{N}!}= \\ =(2\pi )^{\frac{N}{2}}\cdot \pi ^{k}\cdot M^{k}\cdot N^{k}\cdot \left( \sum\limits_{n}\left\vert a_{n}\right\vert ^{2}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-1}\right) ^{\frac{1}{2}% },$

where summation is carried out on all integer nonnegative $\beta _{1}, \beta _{2}, ..., \beta _{N}$ such that $\beta _{1}+\beta _{2}+...+\beta _{N}=k$,

$\left\Vert \sum\limits_{n}a_{n}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-\frac{1}{2}}\cdot \left( \exp \left( i\lambda _{n}x\right) -\exp (inx)\right) \right\Vert _{L_{2}\left( T^{N}\right) }\leq \\ \leq (2\pi )^{\frac{N}{2}}\cdot \sum\limits_{k=1}^{\infty }\frac{1}{k!}\cdot \pi ^{k}\cdot M^{k}\cdot N^{k}\cdot \left( \sum\limits_{n}\left\vert a_{n}\right\vert ^{2}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-1}\right) ^{\frac{1}{2}}= \\ =(2\pi )^{\frac{N}{2}}\left( \exp (MN\pi )-1\right) \cdot \left( \sum\limits_{n}\left\vert a_{n}\right\vert ^{2}\cdot \left( 1+\sum\limits_{|\alpha |=s}\left\vert n^{\alpha }\right\vert ^{2}\right) ^{-1}\right) ^{\frac{1}{2}% }.$

Further,

$\sum\limits _{|\alpha |=s}\left\| D^{\alpha } \left(\sum\limits _{n}a_{n} \cdot \left(1+\sum\limits _{|\alpha |=s}\left|n^{\alpha } \right| ^{2} \right)^{-\frac{1}{% 2} } \cdot \left(\exp \left(i\lambda _{n} x\right)-\exp (inx)\right)\right)\right\| _{L_{2} \left(T^{N} \right)} = \\ =\sum\limits _{|\alpha |=s}\left\| \sum\limits _{n}a_{n} \cdot \left(1+\sum\limits _{|\alpha |=s}\left|n^{\alpha } \right| ^{2} \right)^{-\frac{1}{2} } \cdot D^{\alpha } \left(\exp \left(i\lambda _{n} x\right)-\exp (inx)\right)\right\| _{L_{2} \left(T^{N} \right)} .$

Therefore,

$\left\| \sum\limits _{n}a_{n} \left(\widetilde{\psi }_{n} -\varphi _{n} \right) \right\| _{W_{2}^{s} \left(T^{N} \right)} \le \left(\exp \left(MN\pi \right)-1\right)^{2} \cdot \left(\sum\limits _{n}\left|a_{n} \right|^{2} \cdot \left(1+\sum\limits _{|\alpha |=s}\left|n^{\alpha } \right| ^{2} \right)^{-1} \right)+ \\ +\sum\limits _{|\alpha |=s}\left(\left(\exp \left(MN\pi \right)-1\right)^{2} \cdot \left(\sum\limits _{n}\left|a_{n} \right|^{2} \cdot \left|\lambda _{n}^{\alpha } \right|^{2} \cdot \left(1+\sum\limits _{|\alpha |=s}\left|n^{\alpha } \right| ^{2} \right)^{-1} \right)\right. + \\ \left. +\sum\limits _{n}\left|a_{n} \right|^{2} \cdot \left|\lambda _{n}^{\alpha } -n^{\alpha } \right|^{2} \cdot \left(1+\sum\limits _{|\alpha |=s}\left|n^{\alpha } \right| ^{2} \right)^{-1} \right) .$

Hence, we have

$\left\| \sum\limits _{n}a_{n} \left(\widetilde{\psi }_{n} -\varphi _{n} \right) \right\| _{W_{2}^{s} \left(T^{N} \right)} \le k\cdot \left(\sum\limits _{n}\left|a_{n} \right|^{2} \right)^{\frac{1}{2} } .$

Since $k < 1$, then by theorem by R. Paly and N. Winner (^[5], p.224) the system of functions $\left\{\widetilde{\psi }_{n} (x)\right\}_{n\in Z^{N} }$ forms a basis in the space $W_{2}^{s} \left (T^{N} \right)$. On the other hand, the theorem by N.K. Bary (see ^[3], p. 382) implies that the $\omega$-linear system of functions $\left\{\psi _{n} (x)\right\}_{n\in Z^{N} }$, quadratically close to the Riesz basis $\left\{\widetilde{\psi }_{n} (x)\right\}_{n\in Z^{N} }$ in $W_{2}^{s} \left (T^{N} \right)$, is a Riesz basis in $W_{2}^{s} \left (T^{N} \right)$. Hence, Theorem (2.2) is proved.

Theorem 2.3. Let $\psi _{n} (x)=(2\pi)^{-\frac{N}{p} } \cdot \left (1+|n|^{2} \right)^{-\frac{s}{2} } \cdot \exp \left (i\lambda _{n} x\right)+\alpha _{n} (x)$, $n\in Z^{N}$, where $\lambda _{n} \ne \lambda _{m}$, as $n\ne m$, $\omega$ be an linear independent system of functions at summation on rectangles that satisfies the following conditions:

1. $\sum _{n\in Z^{N} }\left\| \sum _{k\in Z^{N} }\left (\frac{1+|k|^{2} }{1+|n|^{2} } \right)^{\frac{s}{2} } \left (\prod _{j=1}^{N}\frac{\sin\left (\lambda _{n_{j} }-k_{j} \right)\pi }{\left (\lambda _{n_{j} }-k_{j} \right)\pi }-\delta _{nk}\right)\cdot \exp (ikx) \right\| _{L\, _{p} \, \left (T^{N} \right)} < \infty;$

2. $\sum _{n\in Z^{N} }\left\| \alpha _{n} (x)\right\| _{L\, _{p}^{s} \, \left (T^{N} \right)} < \infty.$

Then, the summation on rectangles system functions $\left\{\psi _{n} \right\}_{n\in Z^{N} }$ forms a basis $L\, _{p}^{s} \, \left (T^{N} \right)$, $1 < p < \infty$.

Proof. By Theorem Sokol-Sokolowski, the system functions $\varphi _{n} (x)=(2\pi)^{-\, \frac{N}{p} } \exp\left (inx\right)$ forms a normalized basis in $L_{p} \left (T^{N} \right)$ at summation on rectangles, i.e, for every $f\in L_{p} \left (T^{N} \right)$, there is a single row $\sum _{n\in Z^{N} }f_{n} \exp (inx)$ such that

$S_{m} (x)=\sum\limits _{|n_{1} |\le m_{1} }\sum\limits _{|n_{2} |\le m_{2} }... \sum\limits _{|n_{N} |\le m_{N} }f_{n} \exp (inx)$

which partial sums converges (on rectangles) to function $f (x)$ in $L_{p} \left (T^{N} \right)$ with respect to norm topology, while $\mathop{\min }\limits_{1\le j\le N} m_{j} \to \infty$.

Similarly, the system functions

$\varphi _{n} (x)=(2\pi )^{-\frac{N}{p} } \cdot \left(1+|n|^{2} \right)^{-\, \frac{s}{2} } \cdot \exp \left(inx\right)$

forms a normalized basis in $L\, _{p}^{s} \, \left (T^{N} \right)$ at summation on rectangles, ie, for every $f\in L\, _{p}^{s} \, \left (T^{N} \right)$, there is a single row

$\sum\limits _{n\in Z^{N} }\widetilde{f}_{n} \cdot \varphi _{n} (x)$

such that

$S_{m} (x)=\sum\limits _{|n_{1} |\le m_{1} }\sum\limits _{|n_{2} |\le m_{2} }... \sum\limits _{|n_{N} |\le m_{N} }\widetilde{f}_{n} \cdot \varphi _{n} (x)$

which partial sums converges (on rectangles) to function $f (x)$ in $L\, _{p}^{s} \, \left (T^{N} \right)$ with respect to norm topology, while $\mathop{\min }\limits_{1\le j\le N} m_{j} \to \infty$.

Consequently,

$\begin{array}{l} {\left\| {f(x) - {S_m}(x)} \right\|_{L{\kern 1pt} _p^s{\kern 1pt} \left( {{T^N}} \right)}} = \sum\limits\nolimits_{n \in {Z^N}} {{{(2\pi )}^{ - \frac{N}{p}}}} \cdot n{f_n} \cdot \exp (inx)\\ - {\left. {\sum\limits\nolimits_{|{n_1}| \le {m_1}} {\sum\limits\nolimits_{|{n_2}| \le {m_2}} . } ..\sum\limits\nolimits_{|{n_N}| \le {m_N}} {{{(2\pi )}^{ - \frac{N}{p}}}} \cdot {f_n} \cdot \exp (inx)} \right\|_{L{{\kern 1pt} _p}{\kern 1pt} \left( {{T^N}} \right)}} \to 0 \end{array}$

while $\mathop{\min }\limits_{1\le j\le N} m_{j} \to \infty$ where $p\ge 1, \, \, s\ge 0$, $\widetilde{f}_{n} =(2\pi)^{-\, \frac{N}{q} } \cdot \left (1+|n|^{2} \right)^{\frac{s}{2} } \cdot \int _{T^{N} }f (x)\cdot \exp (-inx) dx$, $\frac{1}{p} +\frac{1}{q} =1$. We have

$\widetilde{\psi }_{n} (x)=(2\pi )^{-\, \frac{N}{p} } \cdot \left(1+|n|^{2} \right)^{-\, \frac{s}{2} } \cdot \exp (i\lambda _{n} x)$

where $\lambda _{n} \ne \lambda _{m}$, while $n\ne m$, be an $\omega$-linear independent system of functions that satisfies the following conditions:

$\sum\limits _{n\in Z^{N} }\left\| \sum\limits _{k\in Z^{N} }\left(\frac{1+|k|^{2} }{1+|n|^{2} } \right)^{\frac{s}{2} } \left(\prod\limits _{j=1}^{N}\frac{\sin \left(\lambda _{n_{j} } -k_{j} \right)\pi }{\left(\lambda _{n_{j} } -k_{j} \right)\pi } -\delta _{nk} \right)\cdot \exp (ikx) \right\| _{L\, _{p} \, \left(T^{N} \right)} <\infty.$

as

$\left\| \varphi _{n} -\widetilde{\psi }_{n} \right\| _{L\, _{p}^{s} \, \left(T^{N} \right)} =\left\| \, \sum\limits _{k\in Z^{N} }\left(1+|k|^{2} \right)^{\frac{s}{2} } \left(\varphi _{n} -\widetilde{\psi }_{n} \right)_{k} \cdot \exp (ikx) \right\| _{L\, _{p} \, \left(T^{N} \right)}$

where

$\left(\varphi _{n} -\widetilde{\psi }_{n} \right)_{k} =(2\pi )^{-N} \int _{T^{N} }\left[\varphi _{n} (x)-\widetilde{\psi }_{n} (x)\right]\cdot \exp (-ikx)dx$

are Fourier coefficients. Hence we get

$\left(\varphi _{n} -\widetilde{\psi }_{n} \right)_{k} =(2\pi)^{-N} \int _{T^{N} }\left[\varphi _{n} (x)-\widetilde{\psi }_{n} (x)\right]\cdot \exp (-ikx)dx \\ =(2\pi )^{-N-\, \frac{N}{p} } \left(1+|n|^{2} \right)^{-\, \frac{s}{2} } \int _{T^{N} }\left[\exp (inx)-\exp (i\lambda _{n} x)\right]\cdot \exp (-ikx)dx \\ \;\;\;\;\;\;\;\;=(2\pi )^{-N-\, \frac{N}{p} } \left(1+|n|^{2} \right)^{-\, \frac{s}{2} } \left[\int _{T^{N} }\exp \left(i\left(n-k\right)x\right)dx-\int _{T^{N} }\exp \left(i\left(\lambda _{n}-k\right)x\right)dx \right]=\\ \;\;\;\;\;\;\;\;=(2\pi )^{-\, \frac{N}{p} } \left(1+|n|^{2} \right)^{-\, \frac{s}{2} } \left[\delta _{nk}-(2\pi )^{-N} \int _{T^{N} }\exp \left(i\left(\lambda _{n}-k\right)x\right)dx \right]=\\ \;\;\;\;\;\;\;\;=(2\pi )^{-\, \frac{N}{p} } \left(1+|n|^{2} \right)^{-\, \frac{s}{2} } \left[\delta _{nk}-(2\pi )^{-N} \prod\limits _{j=1}^{N}\int _{-\pi }^{\pi }\exp \left(i\left(\lambda _{n_{j} } -k_{j} \right)x_{j} \right)dx_{j} \right]=\\ \;\;\;\;\;\;\;\;=(2\pi )^{-\, \frac{N}{p} } \left(1+|n|^{2} \right)^{-\, \frac{s}{2} } \left[\delta _{nk}-(2\pi )^{-N} \prod\limits _{j=1}^{N}\left. \frac{1}{i(\lambda _{n_{j} }-k_{j} )} \exp \left(i\left(\lambda _{n_{j} }-k_{j} \right)x_{j} \right)\right|_{-\pi }^{\pi }\right]=\\ \;\;\;\;\;\;\;\;=(2\pi )^{-\, \frac{N}{p} } \left(1+|n|^{2} \right)^{-\, \frac{s}{2} } \left[\delta _{nk}-\prod\limits _{j=1}^{N}\frac{\sin\left(\lambda _{n_{j} }-k_{j} \right)\pi }{\left(\lambda _{n_{j}}-k_{j} \right)\pi } \right]$

in this way,

$\left(\varphi _{n} -\widetilde{\psi }_{n} \right)_{k} =(2\pi )^{-\, \frac{N}{p} } \left(1+|n|^{2} \right)^{-\, \frac{s}{2} } \left[\delta _{nk}-\prod\limits _{j=1}^{N}\frac{\sin \left(\lambda _{n_{j} }-k_{j} \right)\pi }{\left(\lambda _{n_{j} }-k_{j} \right)\pi } \right]$

hence

$\sum\limits _{n\in Z^{N} }\left\| \varphi _{n} -\psi _{n} \right\| _{L\, _{p}^{s} \, \left(T^{N} \right)} =\sum\limits _{n\in Z^{N} }\left\| \varphi _{n} -\widetilde{\psi }_{n} -\alpha _{n} (x)\right\| _{L\, _{p}^{s} \, \left(T^{N} \right)} \le \sum\limits _{n\in Z^{N} }\left\| \varphi _{n} -\widetilde{\psi }_{n} \right\| _{L\, _{p}^{s} \, \left(T^{N} \right)}\\ \;\;\;\;\;\;\;+\sum\limits _{n\in Z^{N} }\left\| \alpha _{n} (x)\right\| _{L\, _{p}^{s} \, \left(T^{N} \right)} = \\ =\sum\limits _{n\in Z^{N} }\left\| \, \sum\limits _{k\in Z^{N} }\, \left(\frac{1+|k|^{2} }{1+|n|^{2} } \right)^{\, \frac{s}{2} } \cdot (2\pi )^{-\frac{N}{p} } \left[\delta _{nk}-\prod\limits _{j=1}^{N}\frac{\sin \left(\lambda _{n_{j} }-k_{j} \right)\pi }{\left(\lambda _{n_{j} }-k_{j} \right)\pi } \right]\cdot \exp (ikx) \right\| _{L\, _{p} \, \left(T^{N} \right)} \\ \;\;\;\;\;\;\;+\sum\limits _{n\in Z^{N} }\left\| \alpha _{n} (x)\right\| _{L\, _{p}^{s} \, \left(T^{N} \right)} <\infty .$

By Theorem (2.2) we have the proof of the Theorem (2.3).

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