We will write the Fourier series of a function \(f\in L_1(\mathbb T^d)\), where \(\mathbb T^d=[-\pi,\pi)^d\) is the torus, in the form (\(x=(x_1,\dots,x_d)\), \((x,y)=\sum_{i=1}^dx_jy_j,\,|x|=\sqrt{(x,x)}\) )

$$f\sim\sum_{k\in\mathbb Z^d}\widehat f_ke_k,\qquad \widehat f_k=\frac{1}{(2\pi)^d} \int_{\mathbb T^d}f(x)e^{-i(k,x)}\,dx,\qquad e_k=e^{i(k,x)}.$$

If \(\mu\) is a finite Borel (complex-valued) measure on \(\mathbb T^d\), then we write its Fourier series in the form (see [1, Chap. 3])

$$\sum_{k\in\mathbb Z^d}\widehat f_ke_k,\qquad \widehat f_k=\int_{\mathbb T^d}e^{-i(k,x)}\,d\mu(x).$$

If \(\mu\) is a finite Borel measure on \(\mathbb R^d\) and \(|\mu|\) is its variation (see, for example, [2, Chap. XI]), then Wiener Banach algebras are defined as follows:

$$\begin{aligned} \, W &=W(\mathbb R^d) =\biggl\{f:f(x)=\int_{\mathbb R^d}e^{-i(x,y)}\,d\mu(y),\,\|f\|_W=|\mu|(\mathbb R^d)\biggr\}, \\ W_0 &=W_0(\mathbb R^d) =\biggl\{f:f(x)=\int_{\mathbb R^d}g(y)e^{-i(x,y)}\,dy,\,\|f\|_{W_0}=\|g\|_{L_1(\mathbb R^d)}\biggr\}; \end{aligned}$$

see [3, Chap. 6] and, most importantly, the survey [4], in which the list of references contains 175 titles.

The set of continuous positive definite functions on \(\mathbb R^d\) will be denoted by \(W^+(\mathbb R^d)\). These are functions from \(W(\mathbb R^d)\) defined by the condition \(\|f\|_W=f(0)\).

Let \(d=1\). Denote by \(l_f\) a piecewise linear continuous function defined by the conditions \(l_f(k)=f(k)\), \(k\in\mathbb Z\) (a polygonal line). Further, it was noted in the book [5, Chaps. XIX, 16] that, together with \(f\), also \(l_f\) belongs to \(W^+(\mathbb R^1)\). Therefore,

$$\|l_f\|_{W^+}=l_f(0)=f(0)=\|f\|_{W^+}. $$
(1)

It follows that always

$$\|l_f\|_W\le 6\|f\|_W. $$
(2)

Indeed, if the measure \(\mu\) in the representation of \(f\) is real, then

$$f(x)=\int_{\mathbb R}e^{-ixy}\,d\mu(y) =\int_{\mathbb R}e^{-ixy}\,d|\mu| -\int_{\mathbb R}e^{-ixy}\,d(|\mu|-\mu)=f_1(x)-f_2(x)$$

(\(f_1,f_2\in W^+(\mathbb R)\)). Obviously, \(l_{f_1}-l_{f_2}=l_f\) and \(\|l_f\|_W\le 3\|f\|_W\). In the general case, \(\|l_f\|_W\le 6\|f\|_W\).

In the recent paper [6], this inequality with coefficient 1 (instead of 6) was proved and different applications were given (from Wiener algebras to Fourier series and from Fourier series to algebras). These applications are completely new and cannot be obtained without theorems of this kind. Incidently, similar arguments were given in [7], but without applications, because an important theorem was lacking (see Theorem 1 below).

The purpose of this paper is to prove the following two theorems and their application to Fourier series in \(d\) variables.

FormalPara Theorem 1.

For the trigonometric series \(\sum_{k\in\mathbb Z^d}c_ke_k\) to be the Fourier series of a Borel measure \(\mu\) on \(\mathbb T^d\) (of a function \(f\in L_1(\mathbb T^d)\) ), it is necessary and sufficient that there exist a function \(\varphi\in W(\mathbb R^d)\) ( \(\varphi\in W_0(\mathbb R^d)\) ) with the condition \(\varphi(k)=c_k\) , \(k\in\mathbb Z^d\) . In addition,

$$|\mu|(\mathbb T^d)=\min_\varphi\|\varphi\|_W$$

(the minimum over such functions) and this minimum is attained at

$$\varphi_0(x)=\int_{\mathbb T^d}e^{-i(x,y)}\,d\mu(y).$$

In the class of entire functions of exponential type at most \(\pi\) in each variable, there is only one such function in \(W_0\) . Further, the measure \(\mu\ge 0\) if and only if such a function \(\varphi\in W^+(\mathbb R^d)\) exists.

FormalPara Theorem 2.

1) Consider the cube

$$\Pi_k=\{x\in\mathbb R^d:k_j\le x_j\le k_j+1,\,1\le j\le d\}.$$

Any function \(\mathbb R^d\to\mathbb C\) which is linear in each variable \(x_1,\dots,x_d\) on each such cube is completely determined by the values at the vertices of such cubes \((k\in\mathbb Z^d)\) and is continuous on \(\mathbb R^d\) .

2) If \(f\in W\) , and \(l_f\) is a function from 1) defined by the condition \(l_f(k)=f(k)\) , \(k\in\mathbb Z^d\) , then

$$\|l_f\|_W\le\|f\|_W,\qquad\|l_f\|_{W_0}\le\|f\|_{W_0},\qquad\|l_f\|_{W^+}=\|f\|_{W^+}.$$
FormalPara Proof of Theorem 1.

In the case of the Fourier series of a measure \(\mu\), we have

$$\min_\varphi\|\varphi\|_W\le\|\varphi_0\|_W=|\mu|(\mathbb T^d). $$
(3)

On the other hand, if

$$\varphi(x)=\int_{\mathbb R^d}e^{-i(x,y)}\,d\mu(y),\qquad \|\varphi\|_W=|\mu|(\mathbb R^d),$$

then, for \(k\in\mathbb Z^d\), by virtue of the periodicity and Fubini’s theorem,

$$\begin{aligned} \, c_k &=\varphi(k)=\int_{\mathbb R^d}e^{-i(k,y)}\,d\mu(y) =\sum_{m\in\mathbb Z^d} \int_{\mathbb T^d+2\pi m}e^{-i(k,y)}\,d\mu(y) \\& =\sum_{m\in\mathbb Z^d}\int_{\mathbb T^d}e^{-i(k,y)}\,d\mu(y+2\pi m) =\int_{\mathbb T^d}e^{-i(k,y)}\sum_{m\in\mathbb Z^d}\,d\mu(y+2\pi m), \end{aligned}$$

i.e.,

$$c_k=\int_{\mathbb T^d}e^{-i(k,y)}\,d\mu_1(y),\qquad |\mu_1|(\mathbb T^d)\le|\mu|(\mathbb R^d),$$

and this series is the Fourier series of the measure \(\mu_1\).

It remains to take into account that, for any extension of \(\varphi\) from \(\mathbb Z^d\) to \(\mathbb R^d\) (see also (3)),

$$\|\varphi_0\|_W=|\mu_1|(\mathbb T^d)\le|\mu|(\mathbb R^d)=\|\varphi\|_W.$$

But if \(\varphi\) is an entire function of type at most \(\pi\) in \(z_1,\dots,z_d\), then the uniqueness of \(\varphi_0\) follows from the fact that any such function vanishing for \(z=k\), \(k\in\mathbb Z^d\), after division by \(\prod_{j=1}^d\sin\pi z_j\), is entire and bounded on \(\mathbb C^d\) (see, for example, [3, 3.4.4]) and, therefore, it is a constant. But then, for some \(\lambda\in\mathbb C\),

$$\varphi(x)=\varphi_0(x)+\lambda\prod_{j=1}^d\sin\pi x_j,$$

and if the limits of \(\varphi\) and \(\varphi_0\) exist as \(|x|\to\infty\), then we have \(\lambda=0\).

The same argument applies to Fourier series of functions \(f\in L_1(\mathbb T^d)\).

FormalPara Proof of Theorem 2.

1) The boundary of \(\Pi_k\) consists of cubes of dimension from \(1\) to \(d-1\). On the edges of \(\Pi_k\) (all the coordinates, except one, of the points are fixed), such a linear function is uniquely defined by the values at the endpoints (these are the vertices of \(\Pi_k\)). In the case of squares (all the coordinates, except two, are fixed), we draw a segment parallel to the coordinate axis and again use linearity, etc.

The coefficients of such a polynomial of degree \(d\) are easy to find.

For example, for \(d=2\), we can express this polynomial as

$$\begin{aligned} \, &a_1(x_1-k_1-1)(x_2-k_2-1)+a_2(x_1-k_1)(x_2-k_2-1) \\&\qquad{}+a_3(x_1-k_1-1)(x_2-k_2)+a_4(x_1-k_1)(x_2-k_2) \end{aligned}$$

and, substituting the vertices \((k_1,k_2)\), \((k_1+1,k_2)\), \((k_1,k_2+1)\), and \((k_1+1,k_2+1)\), we successively obtain \(a_1\), \(a_2\), \(a_3\), and \(a_4\). Further, the constant mixed derivative of the polynomial \(a_1\) is equal to the mixed difference over \(2^d\) vertices of \(\Pi_k\).

2) For \(x\in\mathbb R^d\) (\(h\in\mathbb R\), \(h_+=\max\{h,0\}\)), we assume

$$l_f(x)=\sum_{k\in\mathbb Z^d}f(k)\prod_{j=1}^d(1-|x_j-k_j|)_+. $$
(4)

Let

$$\Pi_k\subset\widetilde\Pi_k:=\{x:|x_j-k_j|\le 1,\,1\le j\le d\}.$$

If \(m\in\mathbb Z^d\), and \(x\notin\widetilde\pi_m\), then there exists a \(j_0\) such that \(|x_{j_0}-k_{j_0}|>1\), and hence \(l_f(x)=0\). Therefore, for \(x\in\widetilde\Pi_m\),

$$l_f(x)=f(m)\prod_{j=1}^d(1-|x_j-m_j|),\qquad l_f(m)=f(m)$$

and, for \(x\in\Pi_m\),

$$l_f(x)=f(m)\prod_{j=1}^d(m_j+1-x_j).$$

It follows from the same linearity property that \(l_f\in C(\mathbb R^d)\).

Let us now prove that if \(f\in W(\mathbb R^d)\), then \(\|l_f\|_W\le\|f\|_W\). Let us first assume that the function \(f\) is compactly supported. Then the sum in the definition of \(l_f\) (see (4)) is finite.

We have

$$\begin{aligned} \, \int_{\mathbb R^d}l_f(x)e^{i(x,y)}\,dx &=\sum_kf(k)\int_{\mathbb R^d}(1-|x_j-k_j|)_+ e^{i(x,y)}\,dx \\& =\sum_kf(k)\prod_{j=1}^d\int_{\mathbb R^1}(1-|x_j-k_j|)_+e^{ix_jy_j}\,dx_j. \end{aligned}$$

Since, for \(k\) and \(y\in\mathbb R^1\),

$$\int_{\mathbb R^1}(1-|x-k|)_+e^{ixy}\,dx =e^{iky}\int_{\mathbb R^1}(1-|x|)_+e^{ixy}\,dx =e^{iky}\biggl(\frac{2\sin (y/2)}{y}\biggr)^2,$$

it follows that

$$\int_{\mathbb R^d}l_f(x)e^{i(x,y)}\,dx =\sum_{k\in\mathbb Z^d}f(k)e^{i(k,y)} \prod_{j=1}^d\biggl(\frac{2\sin(y_j/2)}{y_j}\biggr)^2$$

and, by the inverse formula for the Fourier transform,

$$\|l_f\|_W=\frac{1}{(2\pi)^d}\int_{\mathbb R^d} \biggl|\sum_kf(k)e^{i(k,y)}\prod_{j=1}^d \biggl(\frac{2\sin(y_j/2)}{y_j}\biggr)^2\biggr|\,dy.$$

In the general case, we apply this equality to \(f_n(x)=f(x)\prod_{j=1}^d(1-|x_j|/n)_+\), obtaining

$$(2\pi)^d\|l_{f_n}\|_W =\int_{\mathbb R^d}\biggl|\sum_kf(k)\prod_{j=1}^d \biggl(1-\frac{|k_j|}{n}\biggr)_+e^{i(k,y)} \prod_{j=1}^d\biggl(\frac{2\sin(y_j/2)}{y_j}\biggr)^2\biggr|\,dy.$$

As before, \(\mathbb R^d=\bigcup_m(\mathbb T^d+2\pi m)\) and, due to periodicity,

$$(2\pi)^d\|l_{f_n}\|_W =\int_{\mathbb T^d}\biggl|\sum_kf(k)\prod_{j=1}^d \biggl(1-\frac{|k_j|}{n}\biggr)_+e^{i(k,y)} \sum_m\prod_{j=1}^d \biggl(\frac{2\sin(y_j/2)}{y_j+2\pi m_j}\biggr)^2\biggr|\,dy.$$

But always

$$\sum_m\prod_j|a_{m_j}|\le\prod_j\sum_m|a_{m_j}|$$

and, for \(y\in\mathbb R\),

$$\sum_{m=-\infty}^\infty\biggl(\frac{2\sin (y/2)}{y+2\pi m}\biggr)^2\equiv 1$$

(this well-known equality can be obtained, for example, from the partial fraction expansion of the meromorphic function \(1/\sin^2(z/2)\)). Thus,

$$(2\pi)^d\|l_{f_n}\|_W \le\int_{\mathbb T^d}\biggl|\sum_kf(k)\prod_{j=1}^d \biggl(1-\frac{|k_j|}{n}\biggr)_+e^{i(k,y)}\biggr|\,dy.$$

By virtue of Theorem 1 (under the condition \(f\in W(\mathbb R^d)\)), the series \(\sum_kf(k)e^{i(k,y)}\) is the Fourier series of a measure \(\mu\) on \(\mathbb T^d\). But then, under the modulus sign, we have the \((C,1)\)-means of \(\sigma_n\) of this series and, therefore,

$$\frac{1}{(2\pi)^d}\int_{\mathbb T^d} |\sigma_n(y)|\,dy \le|\mu|(\mathbb T^d)\le\|f\|_W.$$

Let us pass to the limit as \(n\to\infty\), taking into account the fact that \(l_{f_n}\to l_f\) everywhere and that \(l_f\in C(\mathbb R^d)\) (see [8, Theorem 2]). Then we see that \(\|l_f\|_W\le\|f\|_W\), and also that \(f\) and \(l_f\) belong to \(W^+(\mathbb R^d)\) (see (1)), and now, by virtue of (2), also that \(f\) and \(l_f\) belong to \(W_0(\mathbb R^d)\).

Also note that the condition \(l_f\in W\) needs to be checked only near \(\infty\), because \(W\) is an algebra with a local property and, for \(|x|\le N\), the function \(l_f\) has a bounded mixed derivative if the function itself is bounded (see [4, 7.2, 7.3]).

Turning to the applications, we denote by \(l_c\) the function from Theorem 2 with the conditions

$$l_c(k)=c_k,\qquad k\in\mathbb Z^d.$$
FormalPara Proposition 1.

For the series \(\sum_{k\in\mathbb Z^d}c_ke_k\) to be the Fourier series of a function (measure), it is necessary and sufficient that \(l_c\in W_0(\mathbb R^d)\) \((l_c\in W(\mathbb R^d))\) .

FormalPara Proof.

It follows from Theorems 1 and 2.

FormalPara Proposition 2.

For the series \(\sum_{k\in\mathbb Z^d}c_ke_k\) to be the Fourier series of a function of Vitali bounded variation on \(\mathbb T^d\) , it is necessary and sufficient that \(l_c(x)\in W_0(\mathbb R^d)\) and \(l_c(x)\prod_{j=1}^dx_j\in W(\mathbb R^d)\) .

FormalPara Proof.

By definition (see, for example, [4, 4.2]), the Vitali variation is

$$V_{\textrm{vit}}(f)=\sup\sum|\Delta_hf(x)|,$$

where (\(e_j\) is the unit vector on the axis \(ox_j\), \(|h_j|>0\), and the supremum is taken over all admissible \(x\) and \(h\))

$$\Delta_hf(x)=\biggl(\,\prod_{j=1}^d\Delta_{h_j}\biggr)f(x),\qquad \Delta_{h_j}f(x)=f(x+h_je_j)-f(x).$$

For example, for smooth functions,

$$V_{\textrm{vit}}(f) =\int\biggl|\frac{\partial^df(x)}{\partial x_1\dotsb\partial x_d}\biggr|\,dx.$$

It is only necessary to take into account that, for periodic functions \(f\),

$$\begin{aligned} \, &f\in V_{\textrm{vit}}(\mathbb T^d) \\&\quad\longleftrightarrow\quad \sup_n\int_{\mathbb T^d} \biggl|\frac{\partial^d\sigma_n(f)}{\partial x_1\dotsb\partial x_d}\biggr|\,dx =\sup_n\int_{\mathbb T^d} \biggl|\sum_kf(k)\prod_{j=1}^dk_j \biggl(1-\frac{|k_j|}{n}\biggr)_+e^{i(k,x)}\biggr|\,dx<\infty. \end{aligned}$$

The need for a condition involving \(\sigma_n\) is obvious if we proceed from the definition of the variation \(V_{\textrm{vit}}\) (the case \(d=1\) was considered in [9, Chaps. 1, 60]). To prove sufficiency, we apply either the Banach–Alaoglu theorem or simply Banach’s theorem, because the space \(C(\mathbb T^d)\) is separable. The condition in question means that the norms of \(\sigma_n\) in the space conjugate to \(C(\mathbb T^d)\) are, for example, bounded by the number \(M\). But the ball in such a space is weakly compact, i.e., there is a subsequence \(\sigma_n\) weakly converging to a function from \(C(\mathbb T^d)\) (converging pointwise everywhere). But then, also for the limit function, we have \(V_{\textrm{vit}}\le M\).

FormalPara Proposition 3.

If \(\sum c_ke_k\) is the Fourier series of a measure, \(\lim_{|k|\to\infty}c_k=0\) (which is also necessary), and

$$\int_{\mathbb R^d}\biggl|\frac{\partial^dl_c(x)}{\partial x_1\dotsb\partial x_d}\biggr|\,dx <\infty$$

or, which is the same, \(l_c(x)\prod_{j=1}^dx_j\in W(\mathbb R^d)\) , then \(\sum c_ke_k\) is the Fourier series of a function from \(L_1(\mathbb T^d)\) .

FormalPara Proof.

Let us use Proposition 2, Theorem 2 from [8] (if \(f\in W(\mathbb R^d)\): \(f(\infty)=0\) and, outside of some cube, \(f\) is a function of Vitali bounded variation, then \(f\in W_0(\mathbb R^d)\)), and Theorem 1.

Just as in Propositions 1 and 2 (which are criteria), it is possible to formulate a boundedness criterion for Fourier partial sums in \(L_1(\mathbb T^d)\) and a convergence criterion for Fourier series in \(L_1(\mathbb T^d)\). In the first case, the sequence of norms \(W_0\) will be bounded, while, in the second case, the zero limit will appear.