Applications of an intersection formula to dual cones

TL;DR

This paper presents a new proof of a duality theorem involving extremal quantities of trigonometric polynomials using an intersection formula on dual cones, with applications to integral estimates of positive definite functions.

Contribution

It introduces a succinct proof of Révész's duality theorem utilizing an intersection formula on dual cones, offering new insights and applications in harmonic analysis.

Findings

Provided a new proof of Révész's duality theorem

Applied the intersection formula to integral estimates of positive definite functions

Demonstrated the utility of dual cone intersection formulas in harmonic analysis

Abstract

We give a succinct proof of a duality theorem obtained by R\'ev\'esz in 1991 which concerns extremal quantities related to trigonomertic polynomials. The key tool of our new proof is an intersection formula on dual cones in real Banach spaces. We show another application of this intersection formula which is related to the integral estimates of non-negative positive definite functions.

Equations

$$D^{+}=\left\{\phi \in X^{\prime }|\phi (x)\ge 0\forall x\in D\right\},$$

$${\left(\mathcal{C}\cap \mathcal{P}\right)}^{+}={\mathcal{C}}^{+}+{\mathcal{P}}^{+}.$$

$${\mu }_{(\cdot )}:L^1(G)\to M(G),f\mapsto {\mu }_f;{\mu }_f(E)={\int }_{E}f\mathrm{d}\lambda \text{ for any Borel set }E\subseteq G.$$

$$\hat{\mu }\in {\mathbb{C}}^{\hat{G}};\hat{\mu }(\gamma )={\int }_G\overline{\gamma }\mathrm{d}\mu \left(\gamma \in \hat{G}\right)$$

$$\hat{f}\in {\mathbb{C}}^{\hat{G}};\hat{f}(\gamma )={\int }_{G}f\overline{\gamma }\mathrm{d}\lambda \left(\gamma \in \hat{G}\right).$$

$${\int }_G\overline{\varphi }\mathrm{d}\mu ={\int }_{\hat{G}}\hat{\mu }\mathrm{d}\overline{\nu }.$$

$${\mathbb{Z}}_{+}^d=\left\{\mathbf{n}=(n_1,\dots ,n_d)\in {\mathbb{Z}}^d|\exists j\in \{1,2,\dots ,d\}\text{ such that }{n}_k=0\text{ for any }k<j\text{ and }{n}_j>0\right\}.$$

$$\mathcal{C}:=\{f\in L_{\mathbb{R},\mathrm{s}}^1\left({\mathbb{Z}}^d\right)|f\text{ has finite support, }$$

$$\mathrm{supp}\left(f_{+}\right)\subseteq \{0\}\cup M\cup -M\text{ and }\mathrm{supp}\left(f_{-}\right)\subseteq \{0\}\cup L\cup -L\}.$$

$${\mathcal{C}}^{+}=\left\{t\in L_{\mathbb{R},\mathrm{s}}^{\infty }\left({\mathbb{Z}}^d\right)|\mathrm{supp}\left(t_{+}\right)\subseteq L^c\cup -L^c\text{ and }\mathrm{supp}\left(t_{-}\right)\subseteq M^c\cup -M^c\right\}.$$

$${\mathcal{C}}^{-}=\left\{t\in L_{\mathbb{R},\mathrm{s}}^{\infty }\left({\mathbb{Z}}^d\right)|\mathrm{supp}\left(t_{+}\right)\subseteq M^c\cup -M^c\text{ and }\mathrm{supp}\left(t_{-}\right)\subseteq L^c\cup -L^c\right\}.$$

$$\mathcal{P}:=\left\{f\in L_{\mathbb{R},\mathrm{s}}^1\left({\mathbb{Z}}^d\right)|\hat{f}(\mathbf{x})=\sum _{\mathbf{n}\in {\mathbb{Z}}^d}f(\mathbf{n})e^{-i\mathbf{n}\cdot \mathbf{x}}=f(0)+2\sum _{\mathbf{n}\in {\mathbb{Z}}_{+}^d}f(\mathbf{n})\cos \left(\mathbf{n}\cdot \mathbf{x}\right)\ge 0\forall \mathbf{x}\in {\mathbb{T}}^d\right\}.$$

$${\mathcal{P}}^{+}=\left\{h\in L_{\mathbb{R},\mathrm{s}}^{\infty }\left({\mathbb{Z}}^d\right)|h\gg 0,\text{ that is, }h\text{ is positive definite}\right\}.$$

$$h(\mathbf{n})={\int }_{{\mathbb{T}}^d}{e}^{i\mathbf{n}\cdot \mathbf{x}}\mathrm{d}\nu (\mathbf{x})\left(\mathbf{n}\in {\mathbb{Z}}^d\right).$$

$$\sum _{\mathbf{n}\in {\mathbb{Z}}^d}f(\mathbf{n})h(\mathbf{n})={\int }_{{\mathbb{T}}^d}\hat{f}(\mathbf{x})\mathrm{d}\nu (\mathbf{x})\ge 0$$

$$\sum _{i,j=1}^m{c}_i{c}_{j}g\left({\mathbf{n}}_{\mathbf{i}}-{\mathbf{n}}_{\mathbf{j}}\right)<0$$

$$\sum _{\mathbf{n}\in {\mathbb{Z}}^d}\left(x\ast \tilde{x}\right)(\mathbf{n})g(\mathbf{n})<0,$$

$$\mathcal{H}:=\left\{f\in L_{\mathbb{R},\mathrm{s}}^1\left({\mathbb{Z}}^d\right)|f(0)=1\right\}.$$

$$\alpha :=\inf \left\{\sum _{\mathbf{n}\in {\mathbb{Z}}^d}f(\mathbf{n})r(\mathbf{n})|f\in \mathcal{C}\cap \mathcal{P}\cap \mathcal{H}\right\}$$

$$\omega :=\sup \left\{\delta \in \mathbb{R}|\exists t\in {\mathcal{C}}^{-}\text{ such that }r+t-\delta {\chi }_{\{0\}}\in {\mathcal{P}}^{+}\right\}$$

$$=\sup \left\{\delta \in \mathbb{R}|\exists t\in {\mathcal{C}}^{-}\text{ such that }\delta {\chi }_{\{0\}}-r-t\in {\mathcal{P}}^{-}\right\}$$

$$=\sup \left\{\delta \in \mathbb{R}|\delta {\chi }_{\{0\}}-r\in {\mathcal{C}}^{-}+{\mathcal{P}}^{-}\right\}$$

$${\left(\mathcal{C}\cap \mathcal{P}\right)}^{+}={\mathcal{C}}^{+}+{\mathcal{P}}^{+}$$

$$\omega =\sup \left\{\delta \in \mathbb{R}|\delta {\chi }_{\{0\}}-r\in {\left(\mathcal{C}\cap \mathcal{P}\right)}^{-}\right\}.$$

$$\alpha =\omega .$$

$$\sum _{\mathbf{n}\in {\mathbb{Z}}^d}f(\mathbf{n})r(\mathbf{n})\ge \delta \text{ for any }f\in \mathcal{C}\cap \mathcal{P}\cap \mathcal{H},$$

$$0\ge \sum _{\mathbf{n}\in {\mathbb{Z}}^d}f(\mathbf{n})\left(\delta {\chi }_{\{0\}}(\mathbf{n})-r(\mathbf{n})\right)=\delta f(0)-\sum _{\mathbf{n}\in {\mathbb{Z}}^d}f(\mathbf{n})r(\mathbf{n})=\delta -\sum _{\mathbf{n}\in {\mathbb{Z}}^d}f(\mathbf{n})r(\mathbf{n}).$$

$$\sum _{\mathbf{n}\in {\mathbb{Z}}^d}f(\mathbf{n})\left(\beta {\chi }_{\{0\}}(\mathbf{n})-r(\mathbf{n})\right)>0.$$

$$\beta >\sum _{\mathbf{n}\in {\mathbb{Z}}^d}f(\mathbf{n})r(\mathbf{n})$$

Full text

The author was supported by the “Lendület” Program (LP2012-46/2012) of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office – NKFIH, Grant No. K104206.

1. Introduction

Let $X$ be a real Banach space and let $X^{\prime}$ denote its topological dual space endowed with the weak-∗ topology. For any set $D\subseteq X,$ the dual cone of $D$ is denoted by $D^{+}$ and is defined as

[TABLE]

see, e.g., [2, Section 2]. The polar cone (denoted by $D^{-}$) is defined as $D^{-}:=-D^{+}.$ Note that both $D^{+}$ and $D^{-}$ are weak-∗ closed convex cones in $X^{\prime},$ no matter what the set $D$ is. Moreover, by [2, Lemma 2.1.], if $\mathcal{C}$ and $\mathcal{P}$ are convex sets in $X$ such that $0\in\mathcal{C}\cap\mathcal{P}$ and $\mathcal{C}\cap\mathrm{int}\,\mathcal{P}\neq\emptyset$, then

[TABLE]

Consequently, in this case we have $\left(\mathcal{C}\cap\mathcal{P}\right)^{-}=\mathcal{C}^{-}+\mathcal{P}^{-}.$

In this short note we show two applications of the formula (1) that describes the structure of the dual cone of the intersection of cones. Both applications are of a Fourier-analytic nature, hence we collect some basic facts and notation of this topic below.

1. (i)

For a locally compact abelian group $G,$ the symbol $M(G)$ denotes the set of all complex-valued regular Borel measures on $G$ with finite total variation. $M(G)$ is a commutative, unital Banach algebra, where the norm is defined as $\left|\left|\mu\right|\right|=\left|\mu\right|(G),$ and the multiplication is defined by the convolution [7, 1.3.2. Corollary]. 2. (ii)

The symbol $L^{1}(G)$ stands for the set of all integrable functions on $G$ (with respect to the Haar measure, which is denoted by $\lambda.$) We may consider $L^{1}(G)$ as a subset of $M(G)$ by the embedding

[TABLE]

In fact $L^{1}(G)$ is a Banach subalgebra of $M(G)$ [7, 1.3.5. Theorem]. Moreover, $L^{1}(G)$ is unital if and only if $L^{1}(G)=M(G)$ if and only if $G$ is discrete [7, 1.7.3. Theorem]. 3. (iii)

$L^{\infty}(G)$ stands for the set of all essentially bounded measurable functions on $G.$ 4. (iv)

A continuous group homomorphism from the locally compact abelian group $G$ into the multiplicative group $\mathbb{T}=\{z\in C|\left|z\right|=1\}$ is called a character of $G.$ The set of all characters of $G$ forms a group (with pointwise multiplication) which is called the dual group of $G,$ and it is denoted by $\hat{G}.$ 5. (v)

For any $\mu\in M(G)$ (or $f\in L^{1}(G)$), the symbol $\hat{\mu}$ (or $\hat{f}$) denotes the Fourier transform of $\mu$ (or $f$), that is,

[TABLE]

and

[TABLE]

The Fourier transform is a continuous linear transformation from $L^{1}(G)$ into $C_{0}\left(\hat{G}\right),$ where $C_{0}\left(\hat{G}\right)$ denotes the set of all functions on $\hat{G}$ vanishing at infinity (the topology on $\hat{G}$ is the weak topology induced by the set of all functions $\hat{f}$ obtained as Fourier-transforms of $L^{1}$ functions on $G$). Moreover, it is a contraction as $\left|\left|\hat{f}\right|\right|_{\infty}\leq\left|\left|f\right|\right|_{1}.$ (For details, see [7, 1.2.4. Theorem].)

The following useful formula is an easy consequence of Fubini’s theorem. If $\mu\in M(G),\nu\in M\left(\hat{G}\right)$ and $\phi(x)=\int_{\hat{G}}\gamma(x)\mathrm{d}\nu(\gamma)\,(x\in G),$ then

[TABLE]

Now we turn to the detailed descriptions of the applications of the intersection formula (1). Section 2 is devoted to describe the first one, and Section 3 contains the second one.

2. A new proof of a duality theorem

In 1991, Révész proved a duality theorem on certain extremal quantities related to multivariable trigonometric polynomials [6]. That theorem is general enough to cover the duality statements appearing in [4], [5] and [8]. The setting of the theorem is as follows.

Let $d$ be a positive integer. Let us use the notation $\mathbb{T}^{d}=\left(\mathbb{R}/2\pi\mathbb{Z}\right)^{d}$ and

[TABLE]

Let $M\subseteq\mathbb{Z}_{+}^{d},\,L\subseteq\mathbb{Z}_{+}^{d},$ and let $M^{c}:=\mathbb{Z}_{+}^{d}\setminus M$ and $L^{c}:=\mathbb{Z}_{+}^{d}\setminus L.$

Consider the real Banach space $L_{\mathbb{R},\mathrm{s}}^{1}\left(\mathbb{Z}^{d}\right)$ of all symmetric real-valued absolutely summable functions on $\mathbb{Z}^{d}$ with its topological dual space $L_{\mathbb{R},\mathrm{s}}^{\infty}\left(\mathbb{Z}^{d}\right).$

Set

[TABLE]

[TABLE]

The set $\mathcal{C}\subseteq L_{\mathbb{R},\mathrm{s}}^{1}\left(\mathbb{Z}^{d}\right)$ is a convex set. It is easy to see that the dual cone of $\mathcal{C}$ is

[TABLE]

Therefore, the polar cone of $\mathcal{C}$ is

[TABLE]

Set

[TABLE]

Clearly, $\mathcal{P}$ is a convex set. The following Lemma is devoted to describe its dual cone.

Lemma 1**.**

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Proof of Lemma 1.

Recall that a function $h\in L_{\mathbb{R},\mathrm{s}}^{\infty}\left(\mathbb{Z}^{d}\right)$ is said to be positive definite if $\sum_{i,j=1}^{m}z_{i}\overline{z_{j}}g\left(\mathbf{n_{i}}-\mathbf{n_{j}}\right)\geq 0$ holds for any $\mathbf{n_{1}},\dots,\mathbf{n_{m}}\in\mathbb{Z}^{d}$ and $z_{1},\dots,z_{m}\in\mathbb{C}.$ However, for symmetric real functions, positive definiteness is equivalent to the a priori weaker condition $\sum_{i,j=1}^{m}c_{i}c_{j}g\left(\mathbf{n_{i}}-\mathbf{n_{j}}\right)\geq 0\,\left(\mathbf{n_{1}},\dots,\mathbf{n_{m}}\in\mathbb{Z}^{d},\,c_{1},\dots,c_{m}\in\mathbb{R}\right).$

Let us recall Bochner’s theorem [7, 1.4.3. Theorem] which says that a function $h\in L_{\mathbb{R},\mathrm{s}}^{\infty}\left(\mathbb{Z}^{d}\right)$ is positive definite if and only if there is a non-negative symmetric measure $\nu\in M_{\mathbb{R},\mathrm{s}}\left(\mathbb{T}^{d}\right)$ such that

[TABLE]

Therefore, the positive definite functions are in $\mathcal{P}^{+}$ as any positive definite $h\in L_{\mathbb{R},\mathrm{s}}^{\infty}\left(\mathbb{Z}^{d}\right)$ can be written in the form $h(\mathbf{n})=\int_{\mathbb{T}^{d}}e^{i\mathbf{n}\cdot\mathbf{x}}\mathrm{d}\nu(\mathbf{x})\,\,\left(\mathbf{n}\in\mathbb{Z}^{d}\right)$ for some $0\leq\nu\in M_{\mathbb{R},\mathrm{s}}\left(\mathbb{T}^{d}\right),$ and hence, by equation (2), the inequality

[TABLE]

for any $f\in\mathcal{P}\subset L_{\mathbb{R},\mathrm{s}}^{1}\left(\mathbb{Z}^{d}\right).$

Conversely, if $g\in L_{\mathbb{R},\mathrm{s}}^{\infty}\left(\mathbb{Z}^{d}\right)$ and $g$ is not positive definite, that is,

[TABLE]

for some $\{\mathbf{n_{1}},\dots,\mathbf{n_{m}}\}\subset\mathbb{Z}^{d}$ and $\{c_{1},\dots,c_{m}\}\subset\mathbb{R},$ then

[TABLE]

where $x=\sum_{i=1}^{m}c_{i}\chi_{\{\mathbf{n_{i}}\}}$ and $\tilde{x}$ is defined by $\tilde{x}(\mathbf{n})=x(-\mathbf{n})\,\left(\mathbf{n}\in\mathbb{Z}^{d}\right).$ Clearly, $x*\tilde{x}\in\mathcal{P}\subset L_{\mathbb{R},\mathrm{s}}^{1}\left(\mathbb{Z}^{d}\right),$ hence this means that $g\notin\mathcal{P}^{+}.$ ∎

Now, let $r\in L_{\mathbb{R},\mathrm{s}}^{\infty}\left(\mathbb{Z}^{d}\right)$ with $r(\mathbf{0})=1$ be fixed and let us define the affine subspace

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According to [6, eq. (5) and eq. (12)], let us define the extremal quantities

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and

[TABLE]

[TABLE]

[TABLE]

(It is clear that the definition on $\alpha$ coincides with the definition given in [6, eq. (5)]. It is less obvious that the definition of $\omega$ is the same as the one given in [6, eq. (12)]. However, the fact that the nonnegative symmetric measures on $\mathbb{T}^{d}$ are in one-to-one correspondence with the real positive definite functions on $\mathbb{Z}^{d}$ by the Fourier transform may convince the reader that the definition of $\omega$ is also correct.)

We mentioned before that $\mathcal{C}$ and $\mathcal{P}$ are convex sets in the real Banach space $L_{\mathbb{R},\mathrm{s}}^{1}\left(\mathbb{Z}^{d}\right)$. It is clear that $0\in\mathcal{C}\cap\mathcal{P}$ and $\mathcal{C}\cap\mathrm{int}\,\mathcal{P}\neq\emptyset,$ as $\chi_{\{\mathbf{0}\}}\in\mathcal{C}\cap\mathrm{int}\,\mathcal{P}.$ (The fact that $\chi_{\{\mathbf{0}\}}\in\mathrm{int}\,\mathcal{P}$ can be easily seen by the following. The Fourier transform is a contraction from $L_{\mathbb{R},\mathrm{s}}^{1}\left(\mathbb{Z}^{d}\right)$ into $C_{\mathbb{R},\mathrm{s}}\left(\mathbb{T}^{d}\right),$ and $\widehat{\chi_{\{\mathbf{0}\}}}=1.$) Therefore, by [2, Lemma 2.1.], the intersection formula

[TABLE]

holds. Consequently, we have $\left(\mathcal{C}\cap\mathcal{P}\right)^{-}=\mathcal{C}^{-}+\mathcal{P}^{-}.$ So, by this intersection formula, $\omega$ can be rewritten as

[TABLE]

Theorem 2** (Révész, [6]).**

[TABLE]

A short proof.

If $\delta\chi_{\{\mathbf{0}\}}-r\in\left(\mathcal{P}\cap\mathcal{C}\right)^{-},$ then

[TABLE]

as in this case

[TABLE]

Therefore, $\omega\leq\alpha.$

On the contrary, if $\beta>\omega,$ then $\beta\chi_{\{\mathbf{0}\}}-r\notin\left(\mathcal{C}\cap\mathcal{P}\right)^{-},$ that is, there exists some $f\in\mathcal{C}\cap\mathcal{P}$ such that

[TABLE]

This $f$ is necessarily a non-zero element of $\mathcal{P},$ hence $f(\mathbf{0})>0$. Therefore, without loss of generality, we can assume that $f(\mathbf{0})=1,$ so there exists some $f\in\mathcal{C}\cap\mathcal{P}\cap\mathcal{H}$ such that

[TABLE]

That is,

[TABLE]

for some $f\in\mathcal{C}\cap\mathcal{P}\cap\mathcal{H},$ which means that $\beta>\alpha.$ So, we deduced that $\beta>\omega$ implies $\beta>\alpha,$ therefore, $\alpha\leq\omega.$ The proof is done. ∎

3. Another application of the intersection formula

The second application concerns integral estimates of non-negative positive definite functions. This problem is related to Wiener’s problem [9, 11] and to the recent works [1, 10]. The arguments in this section are partially parallel to the arguments presented in the previous section.

Let $L_{\mathbb{R},\mathrm{s}}^{1}(\mathbb{Z})$ denote the real Banach space of all real-valued, symmetric, summable functions on $\mathbb{Z}$ and let us consider its topological dual space $L_{\mathbb{R},\mathrm{s}}^{\infty}\left(\mathbb{Z}\right)$ endowed with the weak-∗ topology.

Let us define $\mathcal{C}=\left\{f\in L_{\mathbb{R},\mathrm{s}}^{1}(\mathbb{Z})\middle|f\geq 0\right\}$ and $\mathcal{P}=\left\{f\in L_{\mathbb{R},\mathrm{s}}^{1}(\mathbb{Z})\middle|\hat{f}\geq 0\right\}.$ Clearly, $\mathcal{C}$ and $\mathcal{P}$ are convex cones in $L_{\mathbb{R},s}^{1}(\mathbb{Z}).$ The closedness of $\mathcal{C}$ is obvious, and $\mathcal{P}$ is also closed as the Fourier transform is a continuous (moreover, norm-non-increasing) linear transformation from $L_{\mathbb{R},\mathrm{s}}^{1}(\mathbb{Z})$ into $C_{\mathbb{R},\mathrm{s}}(\mathbb{T}),$ and the nonnegative functions form a closed set of $C_{\mathbb{R},\mathrm{s}}(\mathbb{T})$ with respect to the maximum norm topology. (The symbol $\mathbb{T}$ denotes the additive group of real numbers modulo $2\pi$ and $C_{\mathbb{R},\mathrm{s}}(\mathbb{T})$ stands for the Banach space of all continuous, symmetric real functions on $\mathbb{T}.$)

Lemma 3**.**

[TABLE]

and

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Proof of Lemma 3.

The first statement of Lemma 3 is obvious. The proof of the second statement is very similar to the proof of Lemma 1. ∎

Let $L$ and $N$ be positive integers. Let us define the extremal quantities

[TABLE]

and

[TABLE]

Let us introduce

[TABLE]

and observe that $\mathcal{S}_{L,N}$ is closed in the weak-∗ topology as it is the intersection of weak-∗ closed sets.

Note that

[TABLE]

and by the result of Lemma 3,

[TABLE]

Remark 4*.*

Let us note that $K(L,N)$ is finite as the set $\mathcal{P}^{+}\cap\mathcal{S}_{L,N}$ is not empty. Indeed, one can easily check that the function

[TABLE]

is positive definite, and therefore, it is an element of $\mathcal{P}^{+}\cap\mathcal{S}_{L,N}.$

Theorem 5**.**

[TABLE]

Proof.

The key idea is the observation that $\mathcal{C}$ and $\mathcal{P}$ are convex sets in $L_{\mathbb{R},\mathrm{s}}^{1}(\mathbb{Z})$ such that $0\in\mathcal{C}\cap\mathcal{P}$ and $\mathcal{C}\cap\mathrm{int}\,\mathcal{P}\neq\emptyset$ as $\chi_{\{0\}}\in\mathcal{C}\cap\mathrm{int}\,\mathcal{P}.$ Therefore, the intersection formula

[TABLE]

holds.

On the one hand, if $h\in\mathcal{P}^{+}\cap\mathcal{S}_{L,N}$ then

[TABLE]

as $h(0)\geq h(n)\,\left(n\in\mathbb{Z}\right)$ holds for any positive definite function $h\in L_{\mathbb{R},\mathrm{s}}^{\infty}(\mathbb{Z}).$ Therefore, in this case

[TABLE]

hence $C(L,N)\leq K(L,N).$

On the other hand, by the intersection formula,

[TABLE]

holds, hence the following argument shows the opposite inequality. If

[TABLE]

then $(C+1)\chi_{\{-N,\dots,N\}}-\chi_{\{-LN,\dots,LN\}}=g+h$ for some $g\in\mathcal{C}^{+}$ and $h\in\mathcal{P}^{+}.$ Clearly, this $h$ is an element of $\mathcal{P}^{+}\cap\mathcal{S}_{L,N},$ and $h(0)\leq C,$ hence $K(L,N)\leq C(L,N).$ ∎

Remark 6*.*

We have noted (see Remark 4) that $K(L,N)$ is finite. Therefore, the result of Theorem 5 directly implies the finiteness of $C(L,N).$

Acknowledgement

The author is grateful to Szilárd Révész for proposing the problems discussed in this paper and for useful communication. The author is also grateful to the anonymous referee for his/her comments which helped to improve the presentation of the paper.