Abstract
In this chapter, we investigate the rectangular summability of d-dimensional Fourier series. We consider two types of convergence, the so-called restricted and unrestricted convergence. In the first case, \(n \in {\mathbb N}^{d}\) is in a cone or a cone-like set and \(n\rightarrow \infty \) while in the second case, we have \(n \in {\mathbb N}^{d}\) and \(\min (n_1,\ldots ,n_d)\rightarrow \infty \), which is called Pringsheim’s convergence. Similarly, we consider two types of maximal operators, the restricted one defined on a cone or cone-like set and the unrestricted one defined on \({\mathbb N}^{d}\). We prove similar results as for the \(\ell _q\)-summability. In the restricted case, we use the Hardy space \(H_p^\Box ({\mathbb T}^d)\) and in the unrestricted case a new Hardy space \(H_p({\mathbb T}^d)\).
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In this chapter, we investigate the rectangular summability of d-dimensional Fourier series. We consider two types of convergence, the so-called restricted and unrestricted convergence. In the first case, \(n \in {\mathbb N}^{d}\) is in a cone or a cone-like set and \(n\rightarrow \infty \) while in the second case, we have \(n \in {\mathbb N}^{d}\) and \(\min (n_1,\ldots ,n_d)\rightarrow \infty \), which is called Pringsheim’s convergence. Similarly, we consider two types of maximal operators, the restricted one defined on a cone or cone-like set and the unrestricted one defined on \({\mathbb N}^{d}\). We prove similar results as for the \(\ell _q\)-summability. In the restricted case, we use the Hardy space \(H_p^\Box ({\mathbb T}^d)\) and in the unrestricted case a new Hardy space \(H_p({\mathbb T}^d)\).
In the first section, we present the basic definitions for the rectangular summability and verify some estimations for the kernel functions. In the next section, we can find the \(L_p({\mathbb T}^{d})\) convergence of the rectangular Cesàro and Riesz means. In Sect. 3.3, we investigate the restricted maximal operators of the rectangular Cesàro and Riesz means by taking the supremum over a cone. We show that these operators are bounded from the Hardy space \(H_p^\Box ({\mathbb T}^d)\) to \(L_p({\mathbb T}^d)\) for any \(p>p_0\), where \(p_0<1\) is depending again on the summation and on the dimension. As a consequence, we obtain the restricted almost everywhere convergence of the summability means. Similar results are also shown for cone-like sets.
We introduce the product Hardy spaces \(H_p({\mathbb T}^d)\) and present the atomic decomposition and a boundedness result for these spaces. Moreover, we show that the unrestricted maximal operator of the rectangular Cesàro and Riesz means is bounded from \(H_p({\mathbb T}^d)\) to \(L_p({\mathbb T}^d)\) for any \(p>p_0\). This implies the almost everywhere convergence of the summability means in Pringsheim’s sense. In the last section, we consider the rectangular \(\theta \)-summability and prove similar results as mentioned above. We give a sufficient and necessary condition for the uniform and \(L_1({\mathbb T}^{d})\) convergence of the rectangular \(\theta \)-means.
3.1 Summability Kernels
Definition 3.1.1
For \(f\in L_1({\mathbb T}^{d})\) and \(n \in {\mathbb N}^{d}\), the nth rectangular Fejér means \(\sigma _nf\) of the Fourier series of f and the nth rectangular Fejér kernel \(K_n\) are introduced by
and
respectively.
Again, we generalize this definition as follows.
Definition 3.1.2
Let \(f\in L_1({\mathbb T}^{d})\), \(n \in {\mathbb N}^{d}\) and \(\alpha \ge 0\). The nth rectangular Cesàro means \(\sigma _n^{\alpha }f\) of the Fourier series of f and the nth rectangular Cesàro kernel \(K_n^{\alpha }\) are introduced by
and
respectively.
The Cesàro means are also called rectangular \((C,\alpha )\)-means . If \(\alpha =1\), then these are the rectangular Fejér means and if \(\alpha =0\), then the rectangular partial sums (see Fig. 3.1).
The rectangular Fejér kernel \(K_n\) with \(d=2\), \(n_1=3\), \(n_2=5\)
Definition 3.1.3
For \(f\in L_1({\mathbb T}^{d})\), \(n \in {\mathbb N}^{d}\) and \(0<\alpha ,\gamma <\infty \), the nth rectangular Riesz means \(\sigma _n^{\alpha ,\gamma }f\) of the Fourier series of f and the nth rectangular Riesz kernel \(K_n^{\alpha ,\gamma }\) are given by
and
respectively.
For \(\alpha =\gamma =1\), we get back the rectangular Fejér means. The next results follow from
and
where \(K_{n_j}^{\alpha }\) and \(K_{n_j}^{\alpha ,\gamma }\) are the corresponding one-dimensional kernels.
Lemma 3.1.4
If \(0 \le \alpha , \gamma <\infty \) and \(n \in {\mathbb N}^{d}\), then
and
Lemma 3.1.5
If \(0 \le \alpha , \gamma <\infty \) and \(n \in {\mathbb N}^{d}\), then
Lemma 3.1.6
For \(f\in L_1({\mathbb T}^{d})\), \(n \in {\mathbb N}^{d}\) and \(0<\alpha ,\gamma <\infty \),
and
The rectangular Cesàro means are the weighted arithmetic means of the rectangular partial sums.
Lemma 3.1.7
For \(f\in L_1({\mathbb T}^{d})\), \(\alpha >0\) and \(n \in {\mathbb N}^{d}\), we have
and
We will use the next estimation of the derivatives of the one-dimensional kernel functions.
Theorem 3.1.8
For \(0<\alpha \le r+1\), \(n \in {\mathbb P}\) and \(t \in {\mathbb T}\), \(t\ne 0\),
Proof
Similar to Lemma 1.2.4 and Theorem 1.4.16, we have
which implies the first inequality.
We have seen in Theorem 1.4.16 and Lemma 1.4.14 that
In this proof, we use the notation
Abel rearrangement and Lemma 1.4.8 imply
where
Then
Iterating this result s-times \((s \in {\mathbb N})\),
Writing \(\beta =\alpha -1\) and using (1.4.11), we conclude
The equality
follows from (1.4.5). Suppose that \(|t| \ge 1/n\). The rth derivative of \(I_1\) can be estimated as
To estimate the second term, we choose \(s>\alpha +r\). Then the r times termwise differentiated series in \(I_2\) is absolutely convergent. Thus
Similarly,
because \(0<\alpha \le r+1\). Finally, if \(|t| < 1/n\), then the first inequality of our theorem implies the second one. \(\blacksquare \)
The next lemma can be proved as Lemma 1.4.13.
Lemma 3.1.9
For \(\alpha >-1\) and \(h>0\), we have
The same results hold if we choose different exponents \(\alpha _i\) and \(\gamma _i\) in the products.
3.2 Norm Convergence of Rectangular Summability Means
The next results follow from (3.1.1), (3.1.2), Theorem 2.3.3 and from the one-dimensional theorems.
Theorem 3.2.1
If \(0<\alpha \le 1\), then
If \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\), then
Theorem 3.2.2
If \(1 \le p<\infty \), \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\), then
and
Moreover, for all \(f\in L_p({\mathbb T}^{d})\),
and
Here, the convergence is understood in Pringsheim’s sense as in Theorem 2.1.8.
3.3 Almost Everywhere Restricted Summability over a Cone
In this section, we investigate the convergence of the rectangular Cesàro and Riesz summability means taken in a cone. For a given \(\tau \ge 1\), we define a cone by
The choice \(\tau =1\) yields the diagonal. The definition of the Cesàro and Riesz means can be extended to distributions as follows.
Definition 3.3.1
Let \(f\in D({\mathbb T}^{d})\), \(n \in {\mathbb N}^{d}\) and \(0 \le \alpha ,\gamma <\infty \). The nth rectangular Cesàro means \(\sigma _n^{\alpha }f\) and rectangular Riesz means \(\sigma _n^{\alpha ,\gamma }f\) of the Fourier series of f are given by
and
respectively.
Definition 3.3.2
We define the restricted maximal Cesàro and restricted maximal Riesz operator by
and
respectively .
If \(\alpha = 1\), we obtain the restricted maximal Fejér operator \(\sigma _\Box f\). As we can see on Fig. 3.2, in the restricted maximal operator the supremum is taken on a cone only. Marcinkiewicz and Zygmund [234] were the first who considered the restricted convergence. We show that the restricted maximal operator is bounded from \(H_{p}^\Box ({\mathbb T}^d)\) to \(L_{p}({\mathbb T}^d)\).
The cone for \(d=2\)
The next result follows easily from Theorem 3.2.1.
Theorem 3.3.3
If \(0<\alpha \le 1\), then
If \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\), then
Theorem 3.3.4
If \(0<\alpha \le 1\) and
then
Proof
We have seen in Theorem 3.1.8 that
and
Let a be an arbitrary \(H_p^\Box \)-atom with support \(I=I_1\times I_2\) and
We can suppose again that the center of I is zero. In this case,
Choose \(s\in {\mathbb N}\) such that \(2^{s-1}< \tau \le 2^s\). It is easy to see that if \(n_1 \ge k\) or \(n_2 \ge k\), then we have \(n_1,n_2 \ge k2^{-s}\). Indeed, since \((n_1,n_2)\) is in a cone, \(n_1 \ge k\) implies \(n_2 \ge \tau ^{-1} n_1 \ge k 2^{-s}\). By Theorem 2.4.19, it is enough to prove that
First we integrate over \(({\mathbb T}\setminus 4I_1) \times 4I_2\). Obviously,
We can suppose that \(i>0\). Using that
(see Corollary 1.5.3), (3.3.2) and the definition of the atom, we conclude
For \(x_1\in [{\pi i 2^{-K}},{\pi (i+1) 2^{-K}})\) \((i \ge 1)\) and \(t_1\in I_1\), we have
From this, it follows that
Since \(n_1 \ge 2^K 2^{-s}\), we obtain
which is a convergent series if \(p>1/(\alpha +1)\).
To consider (B), let \(I_1=I_2= (-\mu ,\mu )\) and
Then
Integrating by parts, we get that
Recall that the one-dimensional kernel \(K_{n_2}^\alpha \) satisfies
For \(x_1\in [{\pi i 2^{-K}},{\pi (i+1) 2^{-K}})\), the inequalities (3.3.2), (3.3.5) and (3.3.7) imply
Moreover, by (3.3.3), (3.3.5) and (3.3.7),
Consequently,
because \(p>1/(\alpha +1)\). Hence, we have proved that in this case
Next, we integrate over \(({\mathbb T}\setminus 4I_1) \times ({\mathbb T}\setminus 4I_2)\):
We may suppose again that \(i,j>0\). For \(x_1\in [{\pi i 2^{-K}},{\pi (i+1) 2^{-K}})\) and \(x_2\in [{\pi j 2^{-K}},{\pi (j+1) 2^{-K}})\), we have by (3.3.2) and (3.3.5) that
This implies that
Using (3.3.8) and integrating by parts in both variables, we get that
Note that \(A(\mu ,-\mu )=A(\mu ,\mu )=0\). Since \(|K_{n_1}^\alpha |\le Cn_1\) and (3.3.2) holds as well, we obtain
for all \(0\le \eta \le 1\). Moreover, the inequality
and (3.3.3) imply
for all \(0\le \zeta \le 1\). We use inequalities (3.3.5) and (3.3.7) to obtain
whenever \(x_1\in [{\pi i 2^{-K}},{\pi (i+1) 2^{-K}})\), \(x_2\in [{\pi j 2^{-K}}, {\pi (j+1) 2^{-K}})\) and \(0\le \eta ,\zeta \le 1\). If
then
because \((n_1,n_2)\) is in a cone. Choosing
we can see that
which is a convergent series. The analogous estimate for \(\left| D_{n_1,n_2}^2(x_1,x_2) \right| \) can be similarly proved.
For \(x_1\in [{\pi i 2^{-K}},{\pi (i+1) 2^{-K}})\) and \(x_2\in [{\pi j 2^{-K}},{\pi (j+1) 2^{-K}})\), we conclude that
So
by the hypothesis. The integration over \(4I_1 \times ({\mathbb T}\setminus 4I_2)\) can be done as above. This finishes the proof of (3.3.4) as well as the theorem. \(\blacksquare \)
Remark 3.3.5
In the d-dimensional case, the constant \(d/(d+1)\) appears if we investigate the corresponding term to \(D_{n}^1\). More exactly, if we integrate the term
over \(({\mathbb T}\setminus 4I_1) \times \cdots \times ({\mathbb T}\setminus 4I_d)\) similar to (3.3.11), then we get that \(p>d/(d+1)\).
Corollary 3.3.6
If \(0<\alpha \le 1\) and \(1<p<\infty \), then
Let us turn to the Riesz means.
Theorem 3.3.7
If \(0<\alpha <\infty \), \(\gamma \in {\mathbb P}\) and
then
Proof
Let
By the one-dimensional version of Corollary 2.2.28,
Taking into account (2.2.34), we conclude that
and
For \(0< \alpha \le 1\), the inequality can be proved as in Theorem 3.3.4. Now let \(\alpha >1\). Since
trivially and since \(|t|^{-\alpha -1}\le |t|^{-2}\) if \(|t|\ge 1\), we conclude that
Hence
and the theorem can be proved as above. \(\blacksquare \)
Corollary 3.3.8
Suppose that \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\). If \(1<p<\infty \), then
As we have seen in Theorems 2.5.6 and 2.5.12, in the one-dimensional case, the operators \(\sigma _\Box ^{\alpha }\) and \(\sigma _\Box ^{\alpha ,\gamma }\) are not bounded from \(H_p^\Box ({\mathbb T})\) to \(L_p({\mathbb T})\) if \(0<p\le 1/(\alpha +1)\) and \(\alpha =1\). Using interpolation, we obtain the weak type (1, 1) inequality.
Corollary 3.3.9
If \(0<\alpha \le 1\), then
If \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\), then
The density argument of Marcinkiewicz and Zygmund (Theorem 1.3.6) implies
Corollary 3.3.10
Suppose that \(f \in L_1({\mathbb T}^d)\). If \(0<\alpha \le 1\), then
If \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\), then
This result was proved by Marcinkiewicz and Zygmund [234] for the two-dimensional Fejér means. The general version of Corollary 3.3.10 is due to the author [328, 329].
3.4 Almost Everywhere Restricted Summability over a Cone-Like Set
Now we generalize the results of Sect. 3.3 to so-called cone-like sets (see Fig. 3.3). Suppose that for all \(j=2,\ldots ,d\), \(\kappa _j:{\mathbb R}_+\rightarrow {\mathbb R}_+\) are strictly increasing and continuous functions such that
Moreover, suppose that there exist \(c_{j,1},c_{j,2},\xi >1\) such that
Note that this is satisfied if \(\kappa _j\) is a power function. Let us define the numbers \(\omega _{j,1}\) and \(\omega _{j,2}\) via the formula
For convenience, we extend the notations for \(j=1\) by \(\kappa _1:=\mathcal {I}\), \(c_{1,1}=c_{1,2}=\xi \). Here \(\mathcal {I}\) denotes the identity function \(\mathcal {I}(x)=x\). Let \(\kappa =(\kappa _1,\ldots ,\kappa _d)\) and \(\tau =(\tau _1,\ldots ,\tau _d)\) with \(\tau _1=1\) and fixed \(\tau _j\ge 1\) \((j=2,\ldots ,d)\). We define the cone-like set (with respect to the first dimension) by
Figure 3.3 shows a cone-like set for \(d=2\).
Cone-like set for \(d=2\)
If \(\kappa _j=\mathcal {I}\) for all \(j=2,\ldots ,d\), then we get a cone investigated above. The condition on \(\kappa _j\) seems to be natural, because Gát [119] proved in the two-dimensional case that to each cone-like set with respect to the first dimension there exists a larger cone-like set with respect to the second dimension and conversely, if and only if (3.4.1) holds.
Here we have to consider a new Hardy space. We modify slightly the definition of \(H_p^\Box ({\mathbb T}^d)\). Fix \(\psi \in S({\mathbb R})\) such that \(\int _{{\mathbb R}}\psi (x)dx\ne 0\). For \(f\in D({\mathbb T}^d)\), let
Definition 3.4.1
For \(0<p<\infty \) the Hardy spaces \(H_{p}^\kappa ({\mathbb T}^{d})\) and weak Hardy spaces \(H_{p,\infty }^\kappa ({\mathbb T}^{d})\) consist of all distributions \(f\in D({\mathbb T}^d)\) for which
We can prove all the theorems of Sect. 2.4 for \(H_p^\kappa ({\mathbb T}^d)\). Among others,
where \(P_t\) is the one-dimensional Poisson kernel and
If each \(\kappa _j=\mathcal {I}\), we get back the Hardy spaces \(H_p^\Box ({\mathbb T}^d)\). We have to modify slightly the definition of atoms, too.
Definition 3.4.2
A bounded function a is an \(H_p^\kappa \)-atom if there exists a rectangle \(I:=I_1\times \cdots \times I_d\subset {\mathbb T}^d\) with \(|I_j|=\kappa _j(|I_1|^{-1})^{-1}\) such that
-
(i)
\(\mathrm{supp} \ a \subset I\),
-
(ii)
\(\Vert a\Vert _\infty \le |I|^{-1/p}\),
-
(iii)
\(\int _{I} a(x) x^{k}\, dx = 0\) for all multi-indices k with \(|k|\le \lfloor d(1/p-1) \rfloor \).
The following two results can be proved as Theorems 2.4.18 and 2.4.19.
Theorem 3.4.3
A distribution \(f\in D({\mathbb T}^d)\) is in \(H_p^\kappa ({\mathbb T}^d)\) \((0<p\le 1)\) if and only if there exist a sequence \((a_k,k \in {{\mathbb N}})\) of \(H_p^\kappa \)-atoms and a sequence \((\mu _k,k \in {{\mathbb N}})\) of real numbers such that
Moreover,
where the infimum is taken over all decompositions of f of the form (3.4.3).
Theorem 3.4.4
For each \(n\in {\mathbb N}^d\), let \(K_n \in L_1({\mathbb T}^d)\) and \(V_nf:= f*K_n\). Suppose that
for all \(H_{p_0}^\kappa \)-atoms a and for some fixed \(r\in {\mathbb N}\) and \(0<p_0\le 1\), where the rectangle I is the support of the atom. If \(V_*\) is bounded from \(L_{p_1}({\mathbb T}^d)\) to \(L_{p_1}({\mathbb T}^d)\) for some \(1<p_1 \le \infty \), then
for all \(p_0\le p\le p_1\).
Definition 3.4.5
For given \(\kappa ,\tau \) satisfying the above conditions, we define the restricted maximal Cesàro and restricted maximal Riesz operator by
and
respectively.
The next theorem holds obviously.
Theorem 3.4.6
If \(0<\alpha \le 1\), then
If \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\), then
Let H be an arbitrary subset of \(\{1,\ldots ,d\}\), \(H\ne \emptyset \), \(H\ne \{1,\ldots ,d\}\) and \(H^c:=\{1,\ldots ,d\}\setminus H\). Define
where the numbers \(\omega _{j,1}\) and \(\omega _{j,2}\) are defined in (3.4.2).
Theorem 3.4.7
If \(0<\alpha \le 1\) and
then
Proof
Since we will prove the result for \(d=2\), we simplify the notation. Instead of \(c_{2,1},c_{2,2}\) and \(\omega _{2,1},\omega _{2,2}\), we will write \(c_{1},c_{2}\) and \(\omega _{1},\omega _{2}\), respectively. Let a be an arbitrary \(H_p^\kappa \)-atom with support \(I=I_1\times I_2\), \(|I_2|^{-1}=\kappa (|I_1|^{-1})\) and
for some \(K\in {\mathbb N}\). We can suppose that the center of I is zero. In this case
and
To prove
first we integrate over \(({\mathbb T}\setminus 4I_1)\times 4I_2\):
If \(n_1\ge 2^K\) and \(x\in [{\pi i 2^{-K}},{\pi (i+1) 2^{-K}})\) \((i \ge 1)\), then by (3.3.5),
From this, it follows that
which is a convergent series if \(p>1/(1+\alpha )\).
We estimate (B) by
If \((n_1,n_2)\in {\mathbb R}_{\kappa ,\tau }^d\) and \(\frac{2^K}{\xi ^{k+1}}\le n<\frac{2^K}{\xi ^k}\), then \(n_2<\tau \kappa (\frac{2^K}{\xi ^k})\). The inequality \(|K_{n_2}^{\alpha }|\le Cn_2\) and (3.3.2) imply
Hence
Since \(\kappa (x) \le c_1^{-1}\kappa (\xi x)\) by (3.4.1), we conclude
which is convergent if \(p>1/(1+\omega _1)\). Note that
For \((B_2)\), we obtain similarly that
and, moreover,
which was just considered. Hence, we have proved that
The integral over \(4I_1 \times ({\mathbb T}\setminus 4I_2)\) can be handled with a similar idea. Indeed, let us denote the terms corresponding to \((A), (B), (B_1), (B_2)\) by \((A'), (B'), (B_1'), (B_2')\). If we take the integrals in \((A')\) over
then we get in the same way that \((A')\) is bounded if \(p>1/(1+\alpha )\). For \((B_1')\), we can see that
Thus
and this converges if \(p>\omega _2/(1+\omega _2)\), which is less than
Using (3.4.5), we establish that
Hence
Integrating over \(({\mathbb T}\setminus 4I_1) \times ({\mathbb T}\setminus 4I_2)\), we decompose the integral as
Notice that
For \(x_1\in [{\pi i 2^{-K}},{\pi (i+1) 2^{-K}})\) and \(x_2\in [\pi j \kappa (2^{K})^{-1},\pi (j+1) \kappa (2^{K})^{-1})\), we have by (3.3.2) and (3.3.5) that
Then
if \(p>1/(1+\alpha )\).
To consider (D) let us define \(A_1(x_1,x_2)\), \(A_2(x_1,x_2)\), \(D_{n_1,n_2}^1(x_1,x_2)\), \(D_{n_1,n_2}^2(x_1,x_2)\) and \(D_{n_1,n_2}^3(x_1,x_2)\) as in (3.3.6) and (3.3.9), respectively, and let \(I_1=[-\mu ,\mu ]\), \(I_2=[-\nu ,\nu ]\). Then
Obviously,
It follows from (3.3.5), (3.3.10) and (3.4.7) that
where \(0\le \zeta \le 1\). This leads to
which is convergent if
After some computation, we can see that the optimal bound is reached if
which means that
Considering \((D_2)\), we estimate as follows:
and
as above.
The term \(D_{n_1,n_2}^2\) can be handled similarly. We obtain
if
Using (3.3.3), we estimate \(D_{n_1,n_2}^3\) in the same way as (C) in (3.4.6). Now the exponents of \(n_1\) and \(n_2\) are non-negative and so they can be estimated by \(2^K\) and \(\kappa (2^K)\) as in (3.4.6). This proves that
which completes the proof. \(\blacksquare \)
Remark 3.4.8
In the d-dimensional case, the constant \(p_1\) appears if we investigate the terms corresponding to \(D_{n_1,n_2}^1\) and \(D_{n_1,n_2}^2\). Indeed, let \(\prod _{j=1}^{d}I_j\) be centered at 0 and the support of the atom a, A be the integral of a, \(I_j=:[-\mu _j,\mu _j]\) and
\(H\subset \{1,\ldots ,d\}\), \(H\ne \emptyset \), \(H\ne \{1,\ldots ,d\}\). If we integrate the term
over \(\prod _{j=1}^{d}({\mathbb T}\setminus 4I_j)\), then we get that
Moreover, considering the integral
we obtain
However, this bound is less than \(p_1\).
Remark 3.4.9
If \(\omega _{j,1}=\omega _{j,2}=1\) for all \(j=1,\ldots ,d\), then we obtain in Theorem 3.4.7 the bound
In particular, this holds if \(\kappa _j=\mathcal {I}\) for all \(j=1,\ldots ,d\), i.e., if we consider a cone. This bound was obtained for cones in Theorem 3.3.4.
Corollary 3.4.10
If \(0<\alpha \le 1\) and \(1<p<\infty \), then
We obtain similar results for the Riesz means (cf. Theorem 3.3.7). The details are left to the reader.
Theorem 3.4.11
If \(0<\alpha <\infty \), \(\gamma \in {\mathbb P}\) and
then
Corollary 3.4.12
Suppose that \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\). If \(1<p<\infty \), then
Corollary 3.4.13
If \(0<\alpha \le 1\), then
If \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\), then
Corollary 3.4.14
Suppose that \(f \in L_1({\mathbb T}^d)\). If \(0<\alpha \le 1\), then
If \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\), then
In the two-dimensional case, Corollaries 3.4.13 and 3.4.14 were proved by Gát [119] for Fejér summability. In this case, he verified also that if the cone-like set \({\mathbb R}_{\kappa ,\tau }^d\) is defined by \(\tau _j(n_1)\) instead of \(\tau _j\) and if \(\tau _j(n_1)\) is not bounded, then Corollary 3.4.14 does not hold and the largest space for the elements of which we have almost everywhere convergence is \(L\log L\). This means that under these conditions Theorem 3.4.7 cannot be true for any \(p<1\).
3.5 \(H_p({\mathbb T}^d)\) Hardy spaces
For the investigation of the unrestricted almost everywhere convergence of the rectangular summability means, we need a new type of Hardy spaces, the so-called product Hardy spaces.
Fix \(\psi \in S({\mathbb R})\) such that \(\int _{{\mathbb R}}\psi (x)dx\ne 0\). We define the product radial maximal function, the product non-tangential maximal function and the hybrid maximal function of \(f\in D({\mathbb T}^d)\) by
and
respectively, \((i=1,\ldots ,d)\).
Definition 3.5.1
For \(0<p<\infty \), the product Hardy spaces \(H_{p}({\mathbb T}^{d})\) , product weak Hardy spaces \(H_{p,\infty }({\mathbb T}^{d})\) and the hybrid Hardy spaces \(H_{p}^{i}({\mathbb T}^{d})\) \((i=1,\ldots ,d)\) consist of all distributions \(f\in D({\mathbb T}^d)\) for which
and
The Hardy spaces are independent of \(\psi _i\), more exactly, different functions \(\psi _i\) give the same space with equivalent norms. For \(f\in D({\mathbb T}^d)\), let
and
respectively \((i=1,\ldots ,d)\), where the Poisson kernel \(P_{t_i}\) was defined before Theorem 2.4.14. The next theorems were proved in Chang and Fefferman [54, 55], Gundy and Stein [155] or Weisz [346], so we omit the proofs.
Theorem 3.5.2
Let \(0<p<\infty \). Fix \(\psi \in S({\mathbb R})\) such that \(\int _{{\mathbb R}}\psi (x)dx\ne 0\). Then \(f\in H_{p}({\mathbb T}^d)\) if and only if \(\psi _{\triangledown }^*(f)\in L_{p}({\mathbb T}^d)\) or \(P_+^*(f)\in L_{p}({\mathbb T}^d)\) or \(P_{\triangledown }^*(f)\in L_{p}({\mathbb T}^d)\). We have the following equivalences of norms:
The same holds for the weak Hardy spaces:
and for the hybrid Hardy spaces:
As we can see from the next theorem, in the theory of product Hardy spaces, the hybrid Hardy spaces \(H_p^{i}({\mathbb T}^d)\) will play the role of the \(L_1({\mathbb T}^{d})\) spaces in some sense.
Theorem 3.5.3
If \(1<p<\infty \) and \(i=1,\ldots ,d\), then \(H_p({\mathbb T}^d)\sim H_p^{i}({\mathbb T}^d)\sim L_p({\mathbb T}^d)\) and
For \(p=1\), \(H_1({\mathbb T}^d)\subset H_1^{i}({\mathbb T}^d) \subset H_{1,\infty }^\Box ({\mathbb T}^d)\cap L_1({\mathbb T}^d)\) and
Definition 3.5.4
The set \(L(\log L)^{d-1}({\mathbb T}^d)\) contains all measurable functions for which
Theorem 3.5.5
\(H_1^i({\mathbb T}^d) \supset L(\log L)^{d-1}({\mathbb T}^d)\) for all \(i=1,\ldots ,d\) and
A straightforward generalization of the atoms would be the following:
-
(i)
\(\mathrm{supp} \ a \subset I\), \(I \subset {\mathbb T}^d\) is a rectangle,
-
(ii)
\({\Vert a \Vert }_\infty \le |I|^{-1/p}\),
-
(iii)
\(\int _{{\mathbb T}} a(x) x_i^k\, dx_i = 0\), for all \(i=1,\ldots ,d\).
However, the space \(H_p({\mathbb T}^d)\) do not have atomic decomposition with respect to these atoms (see Weisz [327]). The atomic decomposition for \(H_p({\mathbb T}^d)\) is much more complicated. One reason of this is that the support of an atom is not a rectangle but an open set. Moreover, here we have to choose the atoms from \(L_2({\mathbb T}^d)\) instead of \(L_ \infty ({\mathbb T}^d)\).
First of all, we introduce some notations. By a dyadic interval we mean one of the form \([k2^{-n},(k+1)2^{-n})\) for some \(k,n \in {\mathbb Z}\). A dyadic rectangle is the Cartesian product of d dyadic intervals. Suppose that \(F \subset {\mathbb T}^d\) is an open set. Let \(\mathcal {M}_1(F)\) denote those dyadic rectangles \(R=I\times S \subset F\), \(I\subset {\mathbb T}\) is a dyadic interval, \(S \subset {\mathbb T}^{d-1}\) is a dyadic rectangle that are maximal in the first direction. In other words, if \(I'\times S \supset R\) is a dyadic subrectangle of F (where \(I'\subset {\mathbb T}\) is a dyadic interval) then \(I=I'\). Define \(\mathcal {M}_i(F)\) similarly. Denote by \(\mathcal {M}(F)\) the maximal dyadic subrectangles of F in the above sense.
Recall that if \(I\subset {\mathbb T}\) is an interval, then rI is the interval with the same center as I and with length r|I| \((r\in {\mathbb N})\). For a rectangle \(R=I_1\times \ldots \times I_{d} \subset {\mathbb T}^d\) let \(rR:=rI_1\times \ldots \times rI_{d}\). Instead of \(2^r R\) we write \(R^r\) \((r\in {\mathbb N})\).
Definition 3.5.6
A function \(a\in L_2({\mathbb R}^d)\) is an \(H_p\)-atom \((0<p\le 1)\) if
-
(i)
\(\mathrm{supp} \ a \subset F\) for some open set \(F\subset {\mathbb T}^d\) with finite measure,
-
(ii)
\({\Vert a \Vert }_2 \le |F|^{1/2-1/p}\),
-
(iii)
a can be decomposed further into the sum of “elementary particles” \(a_R\in L_2({\mathbb R}^d)\),
$$ a=\sum _{R\in \mathcal {M}(F)} a_R, $$satisfying
-
(a)
\(\mathrm{supp} \ a_R \subset 5R\),
-
(b)
for all \(R\in \mathcal {M}(F)\), \(i=1,\ldots ,d\) and almost every fixed \(x_1,\ldots ,x_{i-1},x_{i+1},\ldots ,x_d\),
$$ \int _{{\mathbb T}} a_R(x) x_i^k \, dx_i = 0 \qquad (k=0,\ldots ,M(p) \ge \lfloor 2/p-3/2 \rfloor ), $$ -
(c)
for every disjoint partition \(\mathcal {P}_l\) \((l\in {\mathbb P})\) of \(\mathcal {M}(F)\),
$$ \left( \sum _{l\in {\mathbb P}} \left\| \sum _{R\in \mathcal {P}_l} a_R \right\| _2^2 \right) ^{1/2} \le |F|^{1/2-1/p}. $$
-
(a)
Theorem 3.5.7
A distribution \(f\in D({\mathbb T}^d)\) is in \(H_p({\mathbb T}^d)\) \((0<p \le 1)\) if and only if there exist a sequence \((a^k,k \in {{\mathbb N}})\) of \(H_p\)-atoms and a sequence \((\mu _k,k \in {{\mathbb N}})\) of real numbers such that
Moreover,
where the infimum is taken over all decompositions of f.
The result corresponding to Theorem 2.4.19 for the \(H_p({\mathbb T}^d)\) space is much more complicated. Since the definition of the \(H_p\)-atom is very complex, to obtain a usable condition about the boundedness of the operators, we have to introduce simpler atoms (see also the definition right after Theorem 3.5.5).
Definition 3.5.8
A function \(a\in L_2({\mathbb T}^d)\) is a simple \(H_p\)-atom or a rectangle \(H_p\)-atom if
-
(i)
\(\mathrm{supp} \ a \subset R\) for a rectangle \(R\subset {\mathbb T}^d\),
-
(ii)
\(\Vert a\Vert _2 \le |R|^{1/2-1/p}\),
-
(iii)
\(\int _{{\mathbb T}} a(x)x_i^k \, dx_i = 0\) for \(i=1,\ldots ,d\), \(k=0,\ldots ,M(p) \ge \lfloor 2/p-3/2 \rfloor \) and for almost every fixed \(x_j\), \(j=1,\ldots ,d\), \(j\ne i\).
Note that \(H_p({\mathbb T}^d)\) cannot be decomposed into rectangle p-atoms, a counterexample can be found in Weisz [327]. However, the following result says that for an operator V to be bounded from \(H_p({\mathbb T}^2)\) to \(L_p({\mathbb T}^2)\) \((0<p\le 1)\), it is enough to check V on simple \(H_p\)-atoms and the boundedness of V on \(L_2({\mathbb T}^2)\). We omit the proof because it can be found for all dimensions in Weisz [332, 346] (see also Fefferman [98]).
Theorem 3.5.9
Let \(d=2\), \(0<p_0\le 1\), \(K_n \in L_1({\mathbb T}^2)\) and \(V_nf:= f*K_n\) \((n\in {\mathbb N}^2)\). Suppose that there exists \(\eta >0\) such that for every simple \(H_{p_0}\)-atom a and for every \(r \ge 1\)
where R is the support of a. If \(V_*\) is bounded from \(L_2({\mathbb T}^2)\) to \(L_2({\mathbb T}^2)\), then
for all \(p_0\le p\le 2\).
Note that Theorem 2.4.16 holds also for \(H_p({\mathbb T}^{d})\) spaces with a very similar proof.
Theorem 3.5.10
If \(K\in L_1({\mathbb T}^{d})\), \(0<p<\infty \) and
then
Corollary 3.5.11
If \(p_0<1\) in Theorem 3.5.9, then for all \(f \in H_1^i({\mathbb T}^2)\) and \(i=1,2\),
Proof
Using the preceding theorem and interpolation, we conclude that the operator
when \(p_0<p<2\). Thus, it holds also for \(p=1\). By Theorem 3.5.3,
for all \(f \in H_1^i({\mathbb T}^2)\), \(i=1,2\). \(\blacksquare \)
Note that for higher dimensions, we have to modify slightly Theorem 3.5.9, Corollary 3.5.11 as well as the definition of simple \(H_p\)-atoms (see Weisz [332, 346]).
3.6 Almost Everywhere Unrestricted Summability
For the almost everywhere unrestricted summability, we introduce the next maximal operators.
Definition 3.6.1
We define the unrestricted maximal Cesàro and unrestricted maximal Riesz operator by
and
respectively.
For \(\alpha = \gamma =1\), the operator is called unrestricted maximal Fejér operator and denoted by \(\sigma _* f\).
We will first prove that the operator \(\sigma _*^\alpha \) is bounded from \(L_p({\mathbb T}^d)\) to \(L_p({\mathbb T}^d)\) \((1<p\le \infty )\) and then that it is bounded from \(H_p({\mathbb T}^d)\) to \(L_p({\mathbb T}^d)\) \((1/(\alpha +1)<p\le 1)\). To this end, we introduce the next one-dimensional operators.
Definition 3.6.2
Let
and
Obviously,
The same holds for the operators \(\sigma _{*}^{\alpha ,\gamma }\) and \(\tau _{*}^{\alpha ,\gamma }\). The next result can be proved similar to Theorem 3.3.4.
Theorem 3.6.3
If \(0<\alpha \le 1\) and \(1/(\alpha +1)< p\le \infty \), then
Proof
It is easy to see that
Let a be an arbitrary \(H_p\)-atom with support \(I\subset {\mathbb T}\) and
Then
Using (3.3.2) and (3.3.5), we can see that
and
as in Theorem 3.3.4.
To estimate (B), observe that by (iii) of the definition of the atom,
Thus,
Using Lagrange’s mean value theorem and (3.3.3), we conclude
where \(\xi \in I\) and \(x\in [{\pi i 2^{-K}},{\pi (i+1) 2^{-K}})\). Consequently,
and
which proves the theorem. \(\blacksquare \)
We can verify in the same way
Theorem 3.6.4
If \(0<\alpha <\infty \), \(\gamma \in {\mathbb P}\) and \(1/(\alpha \wedge 1+1)< p\le \infty \), then
The next result can be obtained by interpolation.
Corollary 3.6.5
Suppose that \(1< p\le \infty \). If \(0<\alpha \le 1\), then
and
If \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\), then
and
Now, we turn to the higher dimensional case and verify the \(L_p({\mathbb T}^d)\) boundedness of \(\sigma _*^{\alpha }\) and \(\sigma _*^{\alpha ,\gamma }\).
Theorem 3.6.6
Suppose that \(1< p\le \infty \). If \(0<\alpha <\infty \), then
If \(0<\alpha <\infty \), \(\gamma \in {\mathbb P}\) and \(1<p\le \infty \), then
Proof
For \(0<\alpha \le 1\), let us apply Corollary 3.6.5 to obtain
The inequality for \(1<\alpha <\infty \) follows from Lemma 3.1.9. The result for \(\sigma _*^{\alpha ,\gamma }\) can be proved in the same way. \(\blacksquare \)
The next result is due to the author [331, 332].
Theorem 3.6.7
If \(0<\alpha <\infty \) and \(1/(\alpha +1)< p\le \infty \), then
Proof
By Theorem 3.1.8,
for \(0<\alpha \le s+1\), \(n_j \in {\mathbb P}\) and \(t \in {\mathbb T}\), \(t\ne 0\). Choose a simple \(H_p\)-atom a with support \(R=I_1\times I_2\), where \(I_1\) and \(I_2\) are intervals with
and
We assume that \(r\ge 2\) is an arbitrary integer. Theorem 3.6.6 implies that the operator \(\sigma ^\alpha _*\) is bounded from \(L_2({\mathbb T}^d)\) to \(L_2({\mathbb T}^d)\). By Theorem 3.5.9, we have to integrate \(\left| \sigma _*^{\alpha } a \right| ^p\) over
First, we integrate over \(({\mathbb T}\setminus I_1^{r}) \times I_2\):
Here we may suppose that \(i_1>0\). For \(k,l \in {\mathbb N}\) let \(A_{0,0}(x):= a(x)\),
and
By (iii) of the definition of the simple \(H_p\)-atom, we can show that \(\mathrm{supp} \ A_{k,l} \subset R\) and \(A_{k,l}(x_1,x_2)\) is zero if \(x_1\) is at the boundary of \(I_1\) or \(x_2\) is at the boundary of \(I_2\) for \(k,l=0,\ldots ,M(p)+1\) \((i=1,2)\), where \(M(p) \ge \lfloor 2/p-3/2 \rfloor \). Moreover, using (ii), we can compute that
We may suppose that \(M(p)\ge \alpha +1\) and choose \(N\in {\mathbb N}\) such that \(N<\alpha \le N+1\). For \(x_1\in [{\pi i_1 2^{-K_1}},{\pi (i_1+1) 2^{-K_1}})\), \(t_1\in [{-\pi 2^{-K_1-1}},{\pi 2^{-K_1-1}})\), inequality (3.6.1) implies
and
Integrating by parts, we can see that
whenever \(x_1\in [{\pi i_1 2^{-K_1}},{\pi (i_1+1) 2^{-K_1}})\). Hence, by Hölder’s inequality and (3.6.3),
Using again Hölder’s inequality and the fact that \(\sigma _*^\alpha \) is bounded on \(L_2({\mathbb T})\), we conclude
Then (3.6.2) implies
To estimate (B), we use (3.6.4):
and
Next, we integrate over \(({\mathbb T}\setminus I_1^{r}) \times ({\mathbb T}\setminus I_2)\):
We will only consider the term (D):
where we may suppose again that \(i_1>0\) and \(i_2>0\). Integrating by parts,
Thus
All other integrals can be handled in the same way. Consequently,
which finishes the proof of the theorem. \(\blacksquare \)
Theorem 3.6.8
If \(0<\alpha <\infty \), \(\gamma \in {\mathbb P}\) and \(1/(\alpha +1)< p\le \infty \), then
Proof
Similar to (3.3.13), for \(s \in {\mathbb N}\), \(n_j \in {\mathbb P}\) and \(t \in {\mathbb T}\), \(t\ne 0\), we have
The theorem can be proved as Theorem 3.6.7. \(\blacksquare \)
Corollary 3.5.11 implies
Corollary 3.6.9
Let \(f \in H_1^i({\mathbb T}^d)\) for some \(i=1,\ldots ,d\). If \(0<\alpha <\infty \), then
If \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\), then
By the density argument, we get here almost everywhere convergence for functions from the spaces \(H_1^i({\mathbb T}^d)\) instead of \(L_1({\mathbb T}^d)\). In some sense, the Hardy space \(H_1^i({\mathbb T}^d)\) plays the role of \(L_1({\mathbb T}^d)\) in higher dimensions.
Corollary 3.6.10
Let \(f \in H_1^i({\mathbb T}^d)\) for some \(i=1,\ldots ,d\). If \(0<\alpha <\infty \), then
If \(0<\alpha <\infty \) and \(\gamma \in {\mathbb P}\), then
The almost everywhere convergence is not true for all \(f \in L_1({\mathbb T}^d)\).
A counterexample, which shows that the almost everywhere convergence is not true for all integrable functions, is due to Gát [119]. Recall that
3.7 Rectangular \(\theta \)-Summability
In this section, we introduce some new function spaces and then we generalize the rectangular Cesàro and Riesz means. As we will see in Definition 3.7.4, instead of condition (2.6.2), we have to suppose here that \(\theta : {\mathbb R}^{d}\rightarrow {\mathbb R}\) is a d-dimensional function and
for all \(n \in {\mathbb P}^{d}\). We will see that it is more convenient to suppose that \(\theta \) is in the Wiener algebra \(W(C,\ell _1)({\mathbb R}^{d})\). All summability methods considered in the literature satisfy the condition \(\theta \in W(C,\ell _1)({\mathbb R}^d)\).
Definition 3.7.1
A measurable function \(f: {\mathbb R}^{d}\rightarrow {\mathbb R}\) belongs to the Wiener amalgam space \(W(L_\infty ,\ell _1)({\mathbb R}^d)\) if
The smallest closed subspace of \(W(L_\infty ,\ell _1)({\mathbb R}^d)\) containing continuous functions is denoted by \(W(C,\ell _1)({\mathbb R}^d)\) and is called Wiener algebra.
Lemma 3.7.2
If \(1\le p\le \infty \), then
Moreover, \(W(L_\infty ,\ell _1)({\mathbb R}^d)\) is dense in \(L_p(\mathbb {R}^{d})\) for \(1\le p<\infty \).
Proof
For \(p=\infty \), the statement is trivial. If \(1\le p<\infty \), then
Since \(W(L_\infty ,\ell _1)({\mathbb R}^{d})\) contains the space of continuous functions with compact support, \(W(L_\infty ,\ell _1)({\mathbb R}^{d})\) is dense in \(L_p(\mathbb {R}^{d})\) if \(1\le p<\infty \). \(\blacksquare \)
The Wiener amalgam spaces and Wiener algebra are used quite often in Gabor analysis, because they provide convenient and general classes of windows (see, e.g., Walnut [323] and Gröchenig [152]).
Theorem 3.7.3
-
(a)
If \(\theta \in W(C,\ell _1)({\mathbb R}^d)\) then (3.7.1) holds.
-
(b)
If the one-dimensional function \(\theta \) is continuous and \(|\theta |\) can be estimated by an integrable function \(\eta \) which is non-decreasing on \((-\infty ,c)\) and non-increasing on \((c,\infty )\) then \(\theta \in W(C,\ell _1)({\mathbb R})\).
-
(c)
There exists \(\theta \not \in W(C,\ell _1)({\mathbb R})\) such that (3.7.1) holds.
Proof
It is easy to see that
which shows (a). Under the conditions of (b), \(\Vert \theta \Vert _{W(C,\ell _1)}\le \Vert \eta \Vert _1\).
To see (c), let \(\theta \ge 0\) be continuous and even on \({\mathbb R}\), \(\theta (0):=0\),
and
Then \(\theta \in L_1({\mathbb R})\),
and
This finishes the proof of Theorem 3.7.3. \(\blacksquare \)
Definition 3.7.4
Suppose that \(\theta \in W(C,\ell _1)({\mathbb R}^d)\). For \(f\in L_1({\mathbb T}^{d})\) and \(n \in {\mathbb N}^{d}\), the nth rectangular \(\theta \)-means \(\sigma _n^{\theta }f\) of the Fourier series of f and the nth rectangular \(\theta \)-kernel \(K_n^{\theta }\) are introduced by
and
respectively.
By Theorem 3.7.3, the \(\theta \)-kernels \(K_{n}^{\theta }\) and the \(\theta \)-means \(\sigma _n^{\theta }f\) are well defined. We suppose often that
where \(\theta _i \in W(C,\ell _1)({\mathbb R})\) for all \(i=1,\ldots ,d\). Then \(\theta \in W(C,\ell _1)({\mathbb R}^d)\) and
Lemma 3.7.5
Suppose that \(\theta \in W(C,\ell _1)({\mathbb R}^d)\). For \(f\in L_1({\mathbb T}^{d})\) and \(n \in {\mathbb N}^{d}\), we have
The \(\theta \)-means can also be written as a convolution of f and the Fourier transform of \(\theta \) in the following way.
Theorem 3.7.6
If \(\theta \in W(C,\ell _1)({\mathbb R}^d)\) and \(\widehat{\theta }\in L_1({\mathbb R}^d)\), then
for almost every \(x\in {\mathbb T}^d\) and for all \(n\in {\mathbb N}^d\) and \(f\in L_1({\mathbb T}^d)\).
Proof
If \(f(t)=e^{\imath k \cdot t}\) \((k\in {\mathbb Z}^d, t\in {\mathbb T}^d)\), then
Thus, the theorem holds also for trigonometric polynomials. The proof can be finished as in Theorem 2.2.30. \(\blacksquare \)
We extend again the definition of the rectangular \(\theta \)-means to distributions.
Definition 3.7.7
Suppose that \(\theta \in W(C,\ell _1)({\mathbb R}^d)\). For \(f\in D({\mathbb T}^{d})\) and \(n \in {\mathbb N}^{d}\), the nth rectangular \(\theta \)-means \(\sigma _n^{\theta }f\) of the Fourier series of f are given by
3.7.1 Feichtinger’s Algebra \(S_0({\mathbb R}^d)\)
Theorem 3.7.6 is a fundamental result, so the condition \(\widehat{\theta }\in L_1({\mathbb R}^d)\) is of great importance. In this subsection, we give some sufficient conditions for a function \(\theta \) to satisfy \(\widehat{\theta }\in L_1({\mathbb R}^d)\). In contrary to the other sections, we do not prove all results here. Some of them are presented without proof. Several such conditions are already known. The next one can be found in Bachman, Narici and Beckenstein [15, p. 323].
Theorem 3.7.8
If \(\theta \in L_1({\mathbb R})\) is bounded on a neighborhood of 0 and \(\widehat{\theta }\ge 0\), then \(\widehat{\theta }\in L_1({\mathbb R})\).
Obviously, \(\theta \) is bounded on a neighborhood of 0 if \(\theta \in L_\infty ({\mathbb R})\) or \(\theta \) is continuous at 0. Moreover, if \(\theta \in L_1({\mathbb R})\) has compact support and \(\theta \in \text{ Lip }(\alpha )\) for some \(\alpha >1/2\), then \(\widehat{\theta }\in L_1({\mathbb R})\) (see Natanson and Zuk [244, p. 176]).
Now we introduce a Banach space, called Feichtinger’s algebra, the Fourier transforms of the elements of which are all integrable. This space was first considered in Feichtinger [100].
Definition 3.7.9
The short-time Fourier transform of \(f\in L_2({\mathbb R}^d)\) with respect to a window function \(g\in L_2({\mathbb R}^d)\) is defined by
Definition 3.7.10
Let \(g_0(x):= e^{-\pi \Vert x\Vert _2^2}\) be the Gauss function. We define the Feichtinger’s algebra \(S_0({\mathbb R}^d)\) by
Any other non-zero Schwartz function defines the same space and an equivalent norm. It is known that \(S_0({\mathbb R}^d)\) contains all Schwartz functions. Moreover, \(S_0({\mathbb R}^d)\) is isometrically invariant under translation, modulation and Fourier transform (see Feichtinger and Zimmermann [100, 106]). Actually, \(S_0({\mathbb R}^{d})\) is the minimal Banach space having this property (see Feichtinger [100]). Furthermore, Feichtinger’s algebra is a subspace of the Wiener algebra, the embedding \(S_0({\mathbb R}^d)\hookrightarrow W(C,\ell _1)({\mathbb R}^d)\) is dense and continuous and
where \(\mathcal {F}\) denotes the Fourier transform and \(\mathcal {F}(W(C,\ell _1)({\mathbb R}^d))\) the set of Fourier transforms of the functions from \(W(C,\ell _1)({\mathbb R}^d)\) (see Feichtinger and Zimmermann [106], Losert [223] and Gröchenig [152]). Let us define the weight function
Theorem 3.7.11
-
(a)
If \(\theta \in S_0({\mathbb R}^d)\), then \(\widehat{\theta }\in S_0({\mathbb R}^d) \subset L_1({\mathbb R}^d)\).
-
(b)
If \(\theta \in L_1({\mathbb R}^d)\) and \(\widehat{\theta }\) has compact support, then \(\theta \in S_0({\mathbb R}^d)\).
-
(c)
If \(\theta \in L_1({\mathbb R}^d)\) has compact support and \(\widehat{\theta }\in L_1({\mathbb R}^d)\), then \(\theta \in S_0({\mathbb R}^d)\).
-
(d)
If \(\theta v_s,\widehat{\theta }v_s \in L_2({\mathbb R}^d)\) for some \(s>d\), then \(\theta \in S_0({\mathbb R}^d)\).
-
(e)
If \(\theta v_s,\widehat{\theta }v_s \in L_\infty ({\mathbb R}^d)\) for some \(s>3d/2\), then \(\theta \in S_0({\mathbb R}^d)\).
For more about Feichtinger’s algebra see Feichtinger and Zimmermann [100, 106]).
Sufficient conditions can also be given with the help of Sobolev, fractional Sobolev and Besov spaces. We do not give a detailed description of these spaces. For the interested readers, we refer to Triebel [313], Runst and Sickel [267], Stein [289] and Grafakos [143]. The Sobolev space \(W_p^k({\mathbb R}^d)\) \((1\le p\le \infty ,k\in {\mathbb N})\) is defined by
and endowed with the norm
where D denotes the distributional derivative.
This definition can be extended to every real s in the following way. The fractional Sobolev space \(\mathcal {L}_p^s({\mathbb R}^d)\) \((1\le p\le \infty ,s\in {\mathbb R})\) consists of all tempered distributions \(\theta \) for which
where \(\mathcal {F}\) denotes the Fourier transform. It is known that
with equivalent norms.
In order to define the Besov spaces, take a non-negative Schwartz function \(\psi \in \mathcal {S}({\mathbb R})\) with support [1/2, 2] that satisfies
For \(x\in {\mathbb R}^d\), let
The Besov space \(B_{p,r}^s({\mathbb R}^d)\) \((0<p,r\le \infty ,s\in {\mathbb R})\) is the space of all tempered distributions f for which
The Sobolev, fractional Sobolev and Besov spaces are all quasi-Banach spaces, and if \(1\le p,r\le \infty \), then they are Banach spaces. All these spaces contain the Schwartz functions. The following facts are known: in the case \(1\le p,r\le \infty \), one has
For two quasi-Banach spaces \({\mathbb X}\) and \({\mathbb Y}\), the embedding \({\mathbb X}\hookrightarrow {\mathbb Y}\) means that \({\mathbb X}\subset {\mathbb Y}\) and \(\Vert f\Vert _{{\mathbb Y}} \le C\Vert f\Vert _{{\mathbb X}}\).
The connection between Besov spaces and Feichtinger’s algebra is summarized in the next theorem.
Theorem 3.7.12
We have
-
(i)
If \(1\le p\le 2\) and \(\theta \in B_{p,1}^{d/p}({\mathbb R}^d)\), then \(\widehat{\theta }\in L_1({\mathbb R}^d)\) and
$$ \left\| \widehat{\theta }\right\| _1 \le C \left\| \theta \right\| _{B_{p,1}^{d/p}}. $$ -
(ii)
If \(s>d\), then \(\mathcal {L}_1^s({\mathbb R}^d)\hookrightarrow S_0({\mathbb R}^d)\).
-
(iii)
If \(d'\) denotes the smallest even integer which is larger than d and \(s>d'\), then
$$ B_{1,\infty }^s({\mathbb R}^d)\hookrightarrow W_1^{d'}({\mathbb R}^d)\hookrightarrow S_0({\mathbb R}^d). $$
Proof
(i) was proved in Girardi and Weis [130] and (ii) in Okoudjou [250]. The first embedding of (iii) follows from (3.7.3) and (3.7.4). If k is even, then \(W_1^{k}({\mathbb R}^d)\hookrightarrow \mathcal {L}_1^k({\mathbb R}^d)\) (see Stein [289, p. 160]). Then (ii) proves (iii). \(\blacksquare \)
It follows from (i) and (3.7.3) that \(\theta \in W_p^{j}({\mathbb R}^d)\) \((j>d/p, j\in {\mathbb N})\) implies \(\widehat{\theta }\in L_1({\mathbb R}^d)\). If \(j\ge d'\), then even \(W_1^{j}({\mathbb R}^d)\hookrightarrow S_0({\mathbb R}^d)\) (see (iii)). Moreover, if \(s>d'\) as in (iii), then
by (3.7.4) and (3.7.5). Theorem 3.7.12 says that \(B_{1,\infty }^s({\mathbb R}^d) \subset S_0({\mathbb R}^d)\) \((s>d')\) and if we choose \(\theta \) from the larger space \(B_{p,1}^{d/p}({\mathbb R}^d)\) \((1\le p\le 2)\), then \(\widehat{\theta }\) is still integrable.
The embedding \(W_1^{2}({\mathbb R})\hookrightarrow S_0({\mathbb R})\) follows from (iii). With the help of the usual derivative, we give another useful sufficient condition for a function to be in \(S_0({\mathbb R}^d)\). As usual, we denote by \(C^k({\mathbb R}^{d})\) the set of k times continuously differentiable functions.
Definition 3.7.13
A function \(\theta \) is in \(V_1^k({\mathbb R})\) if there are numbers \(-\infty =a_0<a_1<\cdots<a_n<a_{n+1}=\infty \), where \(n=n(\theta )\) depends on \(\theta \) and
for all \(i=0,\ldots ,n\) and \(j=0,\ldots ,k\). The norm of this space is defined by
where \(\theta ^{(k-1)}(a_i\pm 0)\) denotes the right and left limits of \(\theta ^{(k-1)}\).
These limits do exist and are finite because \(\theta ^{(k)}\in C(a_i,a_{i+1}) \cap L_1({\mathbb R})\) implies
for some \(a\in (a_i,a_{i+1})\). Since \(\theta ^{(k-1)}\in L_1({\mathbb R})\), we establish that
Similarly, \(\theta ^{(j)}\in C_0({\mathbb R})\) for \(j=0,\ldots ,k-2\).
Of course, \(W_1^2({\mathbb R})\) and \(V_1^2({\mathbb R})\) are not identical. For \(\theta \in V_1^2({\mathbb R})\), we have \(\theta '=D\theta \); however, \(\theta ''=D^2\theta \) only if \(\lim _{x\rightarrow a_i+0}\theta '(x)=\lim _{x\rightarrow a_i-0}\theta '(x)\) \((i=1,\ldots ,n)\).
Theorem 3.7.14
We have \(V_1^2({\mathbb R}) \hookrightarrow S_0({\mathbb R})\).
Proof
Integrating by parts, we have
Observe that the first sum is 0. In the second sum, we integrate by parts again to obtain
The first sum is equal to
Hence
On the other hand,
which finishes the proof of Theorem 3.7.14. \(\blacksquare \)
The next Corollary follows from the definition of \(S_0({\mathbb R}^d)\) and from Theorem 3.7.14.
Corollary 3.7.15
If each \(\theta _j \in V_1^2({\mathbb R})\) \((j=1,\ldots ,d)\), then
3.7.2 Norm Convergence of the Rectangular \(\theta \)-Means
First, we investigate the \(L_2({\mathbb T}^{d})\)-norm convergence of \(\sigma _n^\theta f\) as \(n\rightarrow \infty \) \((n \in {\mathbb N}^{d})\) in Pringsheim’s sense.
Theorem 3.7.16
If \(\theta \in W(C,\ell _1)({\mathbb R}^d)\) and \(\theta (0)=1\), then
Proof
It is easy to see that the norm of the operator
can be given by
Thus, the norms of \(\sigma _n^\theta \) \((n\in {\mathbb N}^d)\) are uniformly bounded. Since \(\theta \) is continuous, the convergence holds for all trigonometric polynomials. The set of the trigonometric polynomials are dense in \(L_2({\mathbb T}^d)\), so the usual density theorem proves Theorem 3.7.16. \(\blacksquare \)
Now, we give a sufficient and necessary condition for the uniform and \(L_1({\mathbb T}^{d})\) convergence \(\sigma _n^\theta f \rightarrow f\).
Theorem 3.7.17
If \(\theta \in W(C,\ell _1)({\mathbb R}^d)\) and \(\theta (0)=1\), then the following conditions are equivalent:
-
(i)
\(\widehat{\theta }\in L_1({\mathbb R}^d)\),
-
(ii)
\(\sigma _n^\theta f \rightarrow f\) uniformly for all \(f\in C({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb N}^{d}\),
-
(iii)
\(\sigma _n^\theta f(x) \rightarrow f(x)\) for all \(x\in {\mathbb T}^d\) and \(f\in C({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb N}^{d}\),
-
(iv)
\(\sigma _n^\theta f \rightarrow f\) in the \(L_1({\mathbb T}^d)\)-norm for all \(f\in L_1({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb N}^{d}\),
-
(v)
\(\sigma _n^\theta f \rightarrow f\) uniformly for all \(f\in C({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb R}_\tau ^d\),
-
(vi)
\(\sigma _n^\theta f(x) \rightarrow f(x)\) for all \(x\in {\mathbb T}^d\) and \(f\in C({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb R}_\tau ^d\),
-
(vii)
\(\sigma _n^\theta f \rightarrow f\) in the \(L_1({\mathbb T}^d)\)-norm for all \(f\in L_1({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb R}_\tau ^d\).
Recall the definition of \(R_\tau ^d\) from (3.3.1).
Proof
We may suppose that \(d=1\), since the multi-dimensional case is similar. First, we verify the equivalence between (i), (ii), (iii) and (iv). If (i) holds, then by Theorem 3.7.6,
and so the operators \(\sigma _n : C({\mathbb T})\rightarrow C({\mathbb T})\) are uniformly bounded. Since (ii) holds for all trigonometric polynomials and the set of the trigonometric polynomials are dense in \(C({\mathbb T})\), (ii) follows easily. (ii) implies (iii) trivially.
Suppose that (iii) is satisfied. We are going to prove (i). For a fixed \(x\in {\mathbb T}\), the operators
are uniformly bounded by the Banach-Steinhaus theorem. We get by Lemma 3.7.5 that
Hence
Since \(K_n^\theta \) is \(2\pi \)-periodic, we have for \(\alpha \le n/2\) that
For a fixed \(t\in {\mathbb R}\), let
and
It is easy to see that
Moreover,
and
Lebesgue’s dominated convergence theorem implies that
Obviously,
and so
Of course, this holds for all \(t\in {\mathbb R}\). We have by (3.7.2) that
Thus
Inequality (3.7.6) yields that
and so
which shows (i).
If \(\widehat{\theta }\in L_1({\mathbb R})\), then Theorem 3.7.6 implies
Hence (iv) follows from (i) because the set of the trigonometric polynomials are dense in \(L_1({\mathbb T})\). The fact that (iv) implies (i) can be proved similarly as \((iii) \Rightarrow (i)\), since, by duality, the norm of the operator \(\sigma _n^\theta : L_1({\mathbb T})\rightarrow L_1({\mathbb T})\) is again
It is easy to see that the equivalence between (i), (v), (vi) and (vii) can be proved in the same way. \(\blacksquare \)
Note that the statement \((i) \Leftrightarrow (ii)\) was shown in the one-dimensional case by Natanson and Zuk [244] for \(\theta \) having compact support. The situation in our general case is much more complicated and can be found in Feichtinger and Weisz [103]. One part of the preceding result can be generalized for \(L_p({\mathbb T}^{d})\) spaces.
Theorem 3.7.18
Assume that \(\theta (0)=1\), \(\theta \in W(C,\ell _1)({\mathbb R}^d)\) and \(\widehat{\theta }\in L_1({\mathbb R}^d)\). If \(1 \le p<\infty \) and \(f\in L_p({\mathbb T}^{d})\), then
and
Proof
For simplicity, we show the theorem for \(d=1\). Using Theorem 3.7.6, we conclude
and
The theorem follows from the Lebesgue dominated convergence theorem. \(\blacksquare \)
Since \(\theta \in S_0({\mathbb R}^d)\) implies \(\theta \in W(C,\ell _1)({\mathbb R}^d)\) and \(\widehat{\theta }\in S_0({\mathbb R}^d) \subset L_1({\mathbb R}^d)\), the next corollary follows from Theorems 3.7.17 and 3.7.18.
Corollary 3.7.19
If \(\theta \in S_0({\mathbb R}^d)\) and \(\theta (0)=1\), then
-
(i)
\(\sigma _n^\theta f \rightarrow f\) uniformly for all \(f\in C({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb N}^{d}\),
-
(ii)
\(\sigma _n^\theta f \rightarrow f\) in the \(L_1({\mathbb T}^d)\)-norm for all \(f\in L_1({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb N}^{d}\),
-
(iii)
\(\sigma _n^\theta f \rightarrow f\) in the \(L_p({\mathbb T}^{d})\)-norm for all \(f\in L_p({\mathbb T}^{d})\) \((1<p<\infty )\) as \(n\rightarrow \infty \) and \(n\in {\mathbb N}^{d}\).
The next corollary follows from the fact that \(\theta \in S_0({\mathbb R}^d)\) is equivalent to \(\widehat{\theta }\in L_1({\mathbb R}^d)\), provided that \(\theta \) has compact support (see, e.g., Feichtinger and Zimmermann [106]).
Corollary 3.7.20
If \(\theta \in C({\mathbb R}^d)\) has compact support and \(\theta (0)=1\), then the following conditions are equivalent:
-
(i)
\(\theta \in S_0({\mathbb R}^d)\),
-
(ii)
\(\sigma _n^\theta f \rightarrow f\) uniformly for all \(f\in C({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb N}^{d}\),
-
(iii)
\(\sigma _n^\theta f(x) \rightarrow f(x)\) for all \(x\in {\mathbb T}^d\) and \(f\in C({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb N}^{d}\),
-
(iv)
\(\sigma _n^\theta f \rightarrow f\) in the \(L_1({\mathbb T}^d)\)-norm for all \(f\in L_1({\mathbb T}^d)\) as \(n\rightarrow \infty \),
-
(v)
\(\sigma _n^\theta f \rightarrow f\) uniformly for all \(f\in C({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb R}_\tau ^d\),
-
(vi)
\(\sigma _n^\theta f(x) \rightarrow f(x)\) for all \(x\in {\mathbb T}^d\) and \(f\in C({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb R}_\tau ^d\),
-
(vii)
\(\sigma _n^\theta f \rightarrow f\) in the \(L_1({\mathbb T}^d)\)-norm for all \(f\in L_1({\mathbb T}^d)\) as \(n\rightarrow \infty \) and \(n\in {\mathbb R}_\tau ^d\).
3.7.3 Almost Everywhere Convergence of the Rectangular \(\theta \)-Means
Definition 3.7.21
For given \(\kappa ,\tau \) satisfying the conditions given in Sect. 3.4, we define the restricted maximal \(\theta \)-operators by
The unrestricted maximal \(\theta \)-operator is defined by
In this subsection, we suppose that
For the restricted convergence, we suppose in addition that
Here \(\mathcal {I}\) denotes the identity function, so
Similar to (2.6.6), assume that \(\widehat{\theta }_j\) is \((N+1)\)-times differentiable \((N\ge 0)\) and there exists
such that
for \(i=N,N+1\) and all \(j=1,\ldots ,d\).
Theorem 3.7.22
Assume that (3.7.7), (3.7.8) and (3.7.9) are satisfied with \(N=0\). If
then
Moreover,
Proof
Inequality (3.7.2) implies that
Similarly,
from which we get immediately that
By Theorem 3.7.6,
as in (2.2.34). From this, it follows that
and
The proof can be finished as in Theorem 3.3.4. \(\blacksquare \)
Corollary 3.7.23
Assume that (3.7.7), (3.7.8) and (3.7.9) are satisfied with \(N=0\). If \(f \in L_1({\mathbb T}^d)\), then
Combining the proofs of Theorems 3.7.22 and 3.4.7, we obtain
Theorem 3.7.24
Assume that (3.7.7), (3.7.8) and (3.7.9) are satisfied with \(N=0\). If
then
Moreover,
We recall that \(p_1\) was defined in (3.4.4).
Corollary 3.7.25
Assume that (3.7.7), (3.7.8) and (3.7.9) are satisfied with \(N=0\). If \(f \in L_1({\mathbb T}^d)\), then
For the unrestricted convergence, we can allow more general conditions for \(\theta \). The next theorem can be shown as Theorems 2.6.7 and 3.6.7.
Theorem 3.7.26
If each \(\theta _j\) satisfies (2.6.2) and (2.6.3), then
for \(1/2<p \le \infty \). If (3.7.7), (3.7.8) and (3.7.9) are satisfied, then the preceding inequality holds for
In both cases
for all \(i=1,\ldots ,d\).
Corollary 3.7.27
Under the conditions of Theorem 3.7.26,
for all \(f \in H_1^i({\mathbb T}^d)\) and \(i=1,\ldots ,d\).
Note that these results are proved in Weisz [332, 333, 335].
3.7.4 Some Summability Methods
It is easy to see that \(\theta \in V_1^2({\mathbb R})\subset S_0({\mathbb R})\) for all examples 2.6.13–2.6.20 of Sect. 2.6.3 and Example 2.6.21 (the Riesz summation) with \(1 \le \alpha <\infty \). Moreover, in Example 2.6.21, \(\theta \in S_0({\mathbb R})\) for all \(0<\alpha <\infty \). In the next examples, \(\theta \) has d variables and \(\theta \in S_0({\mathbb R}^d)\).
Example 3.7.28
(Riesz summation] ). Let
for some \((d-1)/2<\alpha <\infty ,\gamma \in {\mathbb P}\) (see Fig. 3.4).
Riesz summability function with \(d=2\), \(\alpha =1\), \(\gamma =2\)
Example 3.7.29
(Weierstrass summation). Let
(see Fig. 3.5). In the first case \(\widehat{\theta }(x)=e^{-\Vert x\Vert _{2}^{2}/2}\) and in the second one, \(\widehat{\theta }(x) = c_d/(1+\Vert x\Vert _2^2)^{(d+1)/2}\) for some \(c_d\in {\mathbb R}\) (see Stein and Weiss [293, p. 6.]).
Weierstrass summability function \(\theta (t)=e^{-\Vert t\Vert _{2}^{2}/2}\)
Example 3.7.30
(Picard and Bessel summations ). Let
(see Fig. 3.6). Here \(\widehat{\theta _0}(x) = c_d e^{-\Vert x\Vert _{2}}\) for some \(c_d\in {\mathbb R}^d\).
Picard-Bessel summability function with \(d=2\)
Lemma 3.7.31
Let \(\theta \in W(C,\ell _1)({\mathbb R})\), \(\mathcal {I}\,\theta \in W(C,\ell _1)({\mathbb R})\) and \(\theta \) be even and twice differentiable on the interval (0, c), where \([-c,c]\) is the support of \(\theta \) \((0<c\le \infty )\). Suppose that
If \(\theta '\) and \(\max (\mathcal {I},1)\theta ''\) are integrable, then
i.e., (3.7.9) hold with \(N=0\) and \(\beta _j=0\).
Proof
By integrating by parts, we have
Similarly,
which proves the lemma. \(\blacksquare \)
Note that all examples 2.6.13–2.6.21 satisfy Lemma 3.7.31, (3.7.7), (3.7.8) and (3.7.9). Thus, all results of Sects. 3.7.2 and 3.7.3 hold.
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Weisz, F. (2021). Rectangular Summability of Higher Dimensional Fourier Series. In: Lebesgue Points and Summability of Higher Dimensional Fourier Series. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-74636-0_3
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