Fejér–heat generators and Lipschitz control for the Weil quadratic functional

Fej´er–heat generators and Lipschitz control for the Weil quadratic functional Eugen Malamutmann∗ December 8, 2025 Abstract We record two basic analytic properties of the Weil quadratic functional Q associated with the Riemann zeta function in the Guinand–Weil normalization. On the frequency side, Q acts on even, compactly supported test functions Φ with supp Φ ⊂ [ −K, K ] and decomposes as an Archimedean term minus an explicit prime contribution. For each K > 0 we introduce a Fej´er–heat cone GK of frequency windows obtained by convolving classical Fej´er kernels with Gaussian heat kernels and taking finite nonnegative linear combinations of even translates. We prove that GK is dense in the cone C+ even ([ −K, K ]) of continuous, even, nonnegative functions with respect to the supremum norm. Furthermore, we show that on each window [ −K, K ] the functional Q is Lipschitz continuous with respect to the supremum norm, with an explicit Lipschitz constant of the form LQ(K) = Larch(K) + Lprime(K), where Larch ( K ) arises from the Archimedean contribution and Lprime ( K ) from a truncated sum over primes and prime powers. These results yield a compact–by–compact approximation and continuity framework for the localized Weil functional in Guinand–Weil normalization. They are proved independently of any global positivity assumptions and should be viewed as a self-contained technical note providing tools that may be useful in operator–theoretic approaches to the Weil positivity criterion. 2020 Mathematics Subject Classification: 11M26, 11M06, 47B35, 46E22. Keywords: Riemann zeta function, Weil criterion, explicit formula, Toeplitz operators, reproducing kernel Hilbert spaces, Fej´er kernel, heat kernel. 1 Introduction The Weil positivity criterion is one of the classical reformulations of the Riemann Hypothesis. In its standard form it associates to the Riemann zeta function a quadratic functional Q on a suitable cone of even test functions Φ and asserts that Riemann Hypothesis ⇐⇒ Q(Φ) ≥0 for all admissible Φ. In this way the location of the nontrivial zeros—whether they all lie on the critical line or not—is encoded in the positivity of a single quadratic form. This point of view goes back to Weil’s seminal work on explicit formulas for global fields [ 11 , 12 ] and has since appeared in various guises in analytic number theory and in operator–theoretic approaches to zeta and L –functions. In this note we restrict attention to unconditional local analytic properties of the associated Weil functional. ∗University of Duisburg–Essen. ORCID: 0000-0003-4624-5890 1 We work throughout in the Guinand–Weil normalization of the explicit formula and adopt a localized, compact–by–compact perspective on the Weil functional Q . On the frequency side we consider even, compactly supported test functions Φ with supp Φ ⊂ [ −K, K ], and we view Q as a functional on the corresponding cone of test functions. In this setting Q decomposes naturally as Q(Φ) = Qarch(Φ) −Qprime(Φ), where Qarch is an Archimedean contribution involving the gamma factor and Qprime is an explicit prime contribution built from the von Mangoldt function. The precise definitions are recalled in section 3. The purpose of this note is not to address the global positivity problem for Q , but to isolate two basic analytic tools on each compact window [ −K, K ]: a flexible dense cone of frequency generators and a quantitative supremum–norm Lipschitz bound for Q . The note should be viewed as a self-contained technical contribution; the results are formulated so as to be directly usable in operator–theoretic treatments of the Weil criterion developed elsewhere. Fej´er–heat generators on compact windows For a fixed K > 0, let C+ even ([ −K, K ]) denote the cone of continuous, even, nonnegative functions on the interval [ −K, K ]. Motivated by applications to Toeplitz operators and reproducing kernel Hilbert spaces (RKHS), we introduce a cone of “elementary windows” obtained by convolving classical Fej´er kernels with Gaussian heat kernels. More precisely, we consider functions of the form ωB,t =FB∗ρt, where FB is a Fej´er kernel of order B and ρt is a one–dimensional heat kernel at time t > 0. The Fej´er–heat windows ωB,t are smooth positive functions on all of R ; we restrict them to [ −K, K ] when forming the generator cone GK . The cone GK consists of finite nonnegative linear combinations of such restricted windows and their even symmetrizations. Our first basic result shows that these Fej´er–heat windows generate a dense cone on each compact interval [−K, K]. Theorem 1.1 (Fej´er–heat density on [ −K, K ]).Let K > 0. Let GK be the cone generated by finite nonnegative linear combinations of even Fej´er–heat windows restricted to [ −K, K ]. Then GK is dense in C+ even ([ −K, K ]) with respect to the supremum norm. In other words, for every ψ∈C+ even([−K, K]) and every ε>0there exists g∈ GKsuch that ∥ψ−g∥∞< ε. This result provides a flexible family of generators for the frequency cones that is particularly suited to the operator–theoretic constructions developed elsewhere. The proof is based on classical approximation properties of Fej´er kernels and heat kernels, combined with a cone–preserving symmetrization argument. The full details are given in section 4. Lipschitz control of the Weil functional The second ingredient we develop is a quantitative continuity estimate for the Weil functional Q on each compact window. On the frequency side it is natural to equip our spaces of test functions with the supremum norm. We show that, after fixing K > 0, the functional Q is Lipschitz continuous on the corresponding frequency cone, with an explicit Lipschitz constant depending on K. Theorem 1.2 (Lipschitz continuity of Q on [ −K, K ]).Let K > 0be fixed. Then there exists a finite constant LQ ( K ) > 0such that for any two admissible test functions Φ 1, Φ 2∈C∞ c,evenK (see section 2) we have |Q(Φ1)−Q(Φ2)| ≤ LQ(K)∥Φ1−Φ2∥∞. 2 Moreover, the constant LQ(K)admits an explicit decomposition LQ(K) = Larch(K)+Lprime(K) = ZK −K|a∗(ξ)|dξ + 2 X 2≤n≤e2πK Λ(n) √n, where Larch ( K )arises from the Archimedean density a∗ and Lprime ( K )is a finite sum over primes and prime powers up to e2πK weighted by the von Mangoldt function. The proof of theorem 1.2 is elementary: since test functions Φ ∈C∞ c,evenK are compactly supported in [ −K, K ], the Archimedean integral reduces to a finite integral and the prime sum becomes a finite sum. The resulting bounds are completely explicit. The details are presented in section 5. Context and motivation Theorems 1.1 and 1.2 may serve as building blocks for more global operator–theoretic treatments of the Weil functional. On each compact window [ −K, K ], the density of Fej´er–heat generators and the Lipschitz control of Q provide an approximation and stability framework well adapted to Toeplitz operators and reproducing kernel Hilbert spaces. These tools can be combined with Toeplitz–symbol lower bounds and reproducing kernel estimates for the prime contribution to formulate operator–theoretic frameworks for studying the Weil positivity criterion on the full Weil cone. The present article is logically independent of any global positivity claims: all results here are unconditional analytic statements about the localized Weil functional in the Guinand–Weil normalization, and nothing in the proofs uses conjectural information about zeros. In Weil’s adelic reformulation of the explicit formulas [ 12 ], the positivity problem is encoded in a distribution ∆ acting on test functions on the id`ele class group, and the positivity of ∆ is equivalent to the Riemann Hypothesis together with Artin’s conjecture for the corresponding L -functions. Our quadratic functional Q is the one–dimensional, frequency-side avatar of this distribution, specialized to the Riemann zeta function in the Guinand–Weil normalization. Related work. The use of explicit formulas of Guinand–Weil type to encode the zero distribution of ζ ( s ) and more general L -functions goes back to the classical works of Guinand and Weil [ 7 , 11 , 12 ], and is systematically developed in standard references such as Edwards and Iwaniec–Kowalski [ 4 , 8 ]. In that framework the positivity problem for suitable quadratic forms attached to ζ is usually treated either globally, at the level of adelic distributions, or via test functions adapted to particular analytic inequalities. The present note takes a more localized, operator–oriented point of view: we restrict to compact frequency windows, introduce a Fej´er–heat cone of generators tailored to Toeplitz and reproducing kernel Hilbert space methods, and establish quantitative Lipschitz control of the Weil functional on each window. As far as we are aware, this specific combination of Fej´er–heat generators and Lipschitz bounds for the Guinand–Weil functional has not been formulated explicitly in this localized setting. 2 Preliminaries and notation In this section we fix the basic function spaces, norms and transform conventions used throughout the paper. We work on the real line R on the frequency side, and on compact windows [ −K, K ] with K > 0 fixed. 3 2.1 Function spaces and norms For K > 0 we write C ([ −K, K ]) for the space of continuous complex–valued functions on the compact interval [−K, K], equipped with the supremum norm ∥ψ∥∞:= sup x∈[−K,K]|ψ(x)|. We denote by Ceven([−K, K]) the subspace of even functions, Ceven([−K, K]) := {ψ∈C([−K, K]) : ψ(−x) = ψ(x) for all x∈[−K, K]}, and by C+ even([−K, K]) the associated cone of nonnegative even functions, C+ even([−K, K]) := {ψ∈Ceven([−K, K]):ψ(x)≥0 for all x∈[−K, K]}. Unless explicitly stated otherwise, all function spaces on [ −K, K ] will be equipped with the supremum norm ∥·∥∞. On the frequency side we work with admissible test functions on a compact window. For K > 0 we define the space of even compactly supported test functions C∞ c,evenK:= Φ∈C∞(R) : supp Φ ⊂[−K, K],Φ(−ξ) = Φ(ξ) for all ξ,(1) i.e., the space of even, smooth functions on R whose support is contained in the compact window [ −K, K ]. Every Φ ∈C∞ c,evenK is continuous, even, and vanishes identically outside [ −K, K ]. When we say “let Φ be an admissible test function on [−K, K]” we mean Φ ∈C∞ c,evenK. Remark 2.1 (Choice of test class).Working with compactly supported test functions is a deliberate simplification that makes all convergence issues trivial: for Φ ∈C∞ c,evenK the Archimedean integral reduces to an integral over [ −K, K ], and the prime sum becomes a finite sum over those n with ξn = ( log n ) / (2 π ) ∈ [ −K, K ]. This is the natural setting for the “compact–by–compact” approach to the Weil criterion adopted here. For more general test classes (e.g., Paley–Wiener functions arising as Fourier transforms of compactly supported distributions), see [7,12]. 2.2 Fourier transform conventions We use the following Fourier transform convention on R: b ϕ(ξ) = ZR ϕ(t)e−2πitξ dt, ϕ(t) = ZRb ϕ(ξ)e2πitξ dξ, (2) whenever the integrals make sense. In the rest of the paper we work primarily on the frequency side, and we will often refer to Φ( ξ )asafrequency window. When needed, we identify Φ with a time–side test function g via (2): Φ(ξ) = bg(ξ), g(t) = ZR Φ(ξ)e2πitξ dξ. We will always take Φ to be even, so that the corresponding gis real–valued. 2.3 The Fej´er kernel We use a continuous analogue of the classical Fej´er kernel, adapted to the real line rather than the circle. For an integer B≥1 we define the Fej´er kernel on Rby FB(x) = Bsin(πBx) πBx 2 , x ∈R, x = 0,(3) 4 with the continuous extension FB (0) = B at the origin. Equivalently, FB ( x ) = Bsinc2 ( Bx ) where sinc(u) := sin(πu)/(πu) is the normalized sinc function. The kernel FB is smooth, even, and nonnegative on R , with FB ( x ) > 0 for |x|< 1 /B . It belongs to L1(R) and has total mass ZR FB(x)dx = 1.(4) The Fourier transform of FBis the triangular window b FB(ξ) = (1−|ξ|/B, |ξ| ≤ B, 0,|ξ|> B, (5) which has compact support in [−B, B] and satisfies b FB(0) = 1. As B→ ∞ , the family ( FB ) B≥1 forms a classical positive approximate identity on R ; see, e.g., [ 5 , 9 , 13 ] for background on Fej´er kernels and harmonic analysis. For every bounded, uniformly continuous function f:R→Cone has ∥f∗FB−f∥∞−→ 0 (B→ ∞), where ∗denotes convolution on R. 2.4 The heat kernel We also use the one–dimensional heat kernel (Gaussian kernel) on R, defined for t > 0 by ρt(x) = 1 √4πt exp−x2 4t, x ∈R.(6) The kernel ρtis smooth, even, and strictly positive, with total mass ZR ρt(x)dx = 1. As t→ 0 + , the family ( ρt ) t>0 forms a standard approximate identity on R : for every continuous function f on R with at most polynomial growth at infinity, the convolutions f∗ρt converge locally uniformly to f . For background on heat kernels and their analytic properties, see [ 3 , 6 ]. 2.5 Lipschitz continuity We record the basic notion of Lipschitz functional that will be used for the Weil functional Q on compact frequency windows. Definition 2.2 (Lipschitz functional).Let ( X, ∥·∥ ) be a normed space and let F : X→C be a functional. We say that Fis Lipschitz continuous with Lipschitz constant L≥0 if |F(x)−F(y)| ≤ L∥x−y∥ for all x, y ∈X. In later sections we will apply this notion to the Weil functional Q restricted to admissible test functions Φ ∈C∞ c,evenK, equipped with the supremum norm ∥·∥∞. 3 The Weil quadratic functional in Guinand–Weil normalization In this section we recall the Guinand–Weil explicit formula for the Riemann zeta function and use it to define the associated Weil quadratic functional Q on even, compactly supported test functions. We then decompose Qinto its Archimedean and prime contributions. 5 3.1 The Guinand–Weil explicit formula We start from the Guinand–Weil explicit formula for the Riemann zeta function. Let g : R→C be an even Schwartz function such that its Fourier transform bg is compactly supported. Then the Guinand–Weil explicit formula (see [7,11,12]) can be written schematically as X ρbg ρ−1 2 i!= (Archimedean term) −2X n≥2 Λ(n) √ng(log n),(7) where the sum on the left runs over the nontrivial zeros ρ of ζ ( s ) and the Archimedean term involves the logarithmic derivative of the gamma factor. We do not need the detailed form of the Archimedean contribution here; what matters for us is that, after a change of variables to the frequency side, the right-hand side of (7) can be written as a linear functional of a frequency window Φ with an explicit Archimedean density and an explicit prime sum. 3.2 Definition of the Weil functional We work on the frequency axis with coordinate ξ = η/ (2 π ). Following the conventions of [7,11,12], we define the Archimedean density a(ξ) = log π−ℜψ1 4+iπξ, a∗(ξ)=2π a(ξ),(8) where ψ= Γ′/Γ is the digamma function. Prime nodes are placed at ξn=log n 2π, n ≥2,(9) and we use the one-sided Weil weights wQ(n) = 2Λ(n) √n,(10) where Λ is the von Mangoldt function. Definition 3.1 (Weil quadratic functional).Let Φ ∈C∞ c,evenK be an admissible test function (see section 2). The Weil quadratic functional in Guinand–Weil normalization is the linear functional Q(Φ) = ZK −K a∗(ξ) Φ(ξ)dξ −X ξn∈[−K,K] wQ(n) Φ(ξn),(11) with a∗,ξnand wQ(n) as in (8)–(10). Strictly speaking, Q is linear in Φ; we call it “quadratic” because in applications to the Weil criterion one typically takes Φ to be a self-convolution Φ = g∗˜g of a test function g , so that Q(Φ) becomes a quadratic form in g. Remark 3.2 (Well-posedness).For Φ ∈C∞ c,evenK the functional Q (Φ) is trivially well-defined: since supp Φ ⊂ [ −K, K ], the Archimedean integral reduces to a finite integral over [ −K, K ], and the prime sum is a finite sum over those n≥ 2 with ξn = ( log n ) / (2 π ) ∈ [ −K, K ], i.e., over 2≤n≤e2πK. 6 3.3 Archimedean and prime decomposition It is convenient to split Qinto its Archimedean and prime parts. Proposition 3.3 (Decomposition of Q ).For any admissible test function Φas in Definition 3.1, the Weil functional decomposes as Q(Φ) = Qarch(Φ) −Qprime(Φ), where Qarch(Φ) = ZR a∗(ξ) Φ(ξ)dξ, (12) Qprime(Φ) = X n≥2 wQ(n) Φ(ξn).(13) Proof. This is a restatement of Definition 3.1 with the Archimedean integral and prime sum written as separate functionals. Remark 3.4 (Normalization and link to Weil’s explicit formulas).The normalization adopted here is compatible with the Guinand–Weil form of the explicit formula for the Riemann zeta function [ 7 , 11 , 12 ]. In Weil’s adelic framework, the positivity problem is encoded in a distribution ∆ acting on test functions on the id`ele class group, and the positivity of ∆ is equivalent to the Riemann Hypothesis together with Artin’s conjecture for the corresponding L -functions. Our functional Q is the one-dimensional frequency-side avatar of this distribution, specialized to the Riemann zeta function in the Guinand–Weil normalization. We define Q using the standard Archimedean density and Weil weights from the explicit formula; on the Paley–Wiener test class this coincides with the classical Weil functional, and (11) should be viewed as its localized extension to compactly supported frequency windows. This remark is included for context only; the results of the present note do not rely on the positivity of ∆ or on any unproved conjectures. Remark 3.5 (Relation to Paley–Wiener spaces).In operator–theoretic treatments of the Weil criterion one often works with Paley–Wiener test functions, i.e., functions g∈C∞ c ([ −T, T ]) on the time side and their Fourier transforms Φ = bg . By the Paley–Wiener theorem, such Φ are entire functions and do not have compact support on the frequency side; thus they do not belong to C∞ c,evenK as defined in (1) . Nonetheless, the density and Lipschitz results of this paper remain useful for the operator–theoretic setting: the space C∞ c,evenK is dense in natural topologies on Paley–Wiener functions restricted to [ −K, K ], so the quantitative estimates established here extend by continuity to broader test function classes. Remark 3.6 (Notation for weights).We use wQ ( n ) for the Weil weights appearing in the definition of Q . In later sections we will also encounter other weight systems (for example, in the Lipschitz estimates); these will be clearly distinguished by subscripts. 4 Fej´er–heat generators and density on [−K, K] In this section we introduce a cone of Fej´er–heat frequency windows and prove that it is dense in the cone of continuous, even, nonnegative functions on each compact interval [ −K, K ]. Throughout we fix K > 0. 4.1 The Fej´er–heat convolution Definition 4.1 (Fej´er–heat window).Let B≥ 1 be an integer and let t > 0. Let FB:R→R be the one–dimensional Fej´er kernel of order B (see section 2), and let ρt:R→R be the 7 one–dimensional heat kernel at time t (see section 2). We define the Fej´er–heat window of order (B, t) by ωB,t =FB∗ρt, where ∗denotes convolution on R. By construction, FB is smooth, nonnegative and even, while ρt is smooth, strictly positive and even; both belong to L1 ( R ) with total mass 1 (see section 2). As a convolution of a nonnegative function with a strictly positive one, ωB,t is itself strictly positive. Moreover, the families {FB}B≥1 and {ρt}t>0 form positive approximate identities on R : for every bounded, uniformly continuous function f:R→Cone has ∥f∗FB−f∥∞−→ 0 (B→ ∞),∥f∗ρt−f∥∞−→ 0 (t→0+). 4.2 Basic properties of Fej´er–heat windows Lemma 4.2 (Basic properties of Fej´er–heat windows).For every integer B≥ 1and t > 0the Fej´er–heat window ωB,t satisfies: (i) ωB,t(x)≥0for all x∈R(positivity); (ii) ωB,t(−x)=ωB,t(x)for all x∈R(evenness); (iii) ZR ωB,t(x)dx = 1 (normalization). Proof. By definition, ωB,t(x)=(FB∗ρt)(x) = ZR FB(y)ρt(x−y)dy. (i) Since FB ( y ) ≥ 0 and ρt ( z ) ≥ 0 for all y, z ∈R , the integrand is nonnegative, hence ωB,t(x)≥0 for all x. (ii) Using that both FBand ρtare even functions, we obtain ωB,t(−x) = ZR FB(y)ρt(−x−y)dy =ZR FB(−y)ρt(x−y)dy =ZR FB(y)ρt(x−y)dy =ωB,t(x), where we changed variables y7→ −yand used evenness. (iii) Using Fubini’s theorem and the normalization RRFB=RRρt= 1, we have ZR ωB,t(x)dx =ZRZR FB(y)ρt(x−y)dy dx =ZR FB(y)dyZR ρt(u)du= 1 ·1=1. This proves the three properties. 4.3 The Fej´er–heat cone In applications we want to work with even windows on the frequency axis. We obtain these by symmetrizing translates of Fej´er–heat windows. Definition 4.3 (Even Fej´er–heat windows).Let B≥ 1 be an integer, t > 0 and x0∈R . The associated even Fej´er–heat window centered at x0is the function ΩB,t,x0(x) = 1 2ωB,t(x−x0)+ωB,t(x+x0), x ∈R. By theorem 4.2 and the definition, each Ω B,t,x0 is continuous, nonnegative and even, with RRΩB,t,x0(x)dx = 1. 8 Definition 4.4 (Fej´er–heat cone).For K > 0 we define the Fej´er–heat cone GK to be the set of all functions on [−K, K] that can be written in the form g(x) = N X j=1 cjΩBj,tj,xj(x), x ∈[−K, K], for some integer N≥ 1, integers Bj≥ 1, parameters tj> 0, centers xj∈R , and coefficients cj≥ 0. Equivalently, GK is the cone generated by finite nonnegative linear combinations of even Fej´er–heat windows restricted to [−K, K]. By construction, every g∈ GKis continuous, even and nonnegative on [−K, K]. 4.4 Density theorem We now show that GK is dense in the cone of continuous, even, nonnegative functions on [ −K, K ] with respect to the supremum norm. This is the localized form of theorem 1.1 from the introduction. Theorem 4.5 (Fej´er–heat density).Let K > 0. Then the cone GK is dense in C+ even ([ −K, K ]) with respect to the supremum norm ∥·∥∞ . More precisely, for every ψ∈C+ even ([ −K, K ]) and every ε > 0there exists g∈ GKsuch that ∥ψ−g∥∞< ε. Proof. Fix K > 0, ψ∈C+ even ([ −K, K ]) and ε > 0. Throughout this proof, all supremum norms ∥·∥∞without explicit domain annotation are taken over [−K, K]. Step 1: even extension to the real line. Define an even, bounded, continuous extension e ψ:R→[0,∞) by e ψ(x) = (ψ(x),|x|≤K, ψ(K),|x|> K. Since ψ is continuous and even on [ −K, K ], the function e ψ is continuous, even and nonnegative on R. Let M:= ∥e ψ∥∞<∞. Step 2: approximation by Fej´er and heat kernels. By the approximate identity property of the heat kernels ρt, there exists t0>0 such that for all 0 <t≤t0we have ∥e ψ∗ρt−e ψ∥L∞([−K,K]) <ε 4.(14) Fix one such t∈(0, t0] and set f(x) := ( e ψ∗ρt)(x), x ∈R. Then fis continuous, bounded, even and nonnegative. Similarly, by the approximate identity property of the Fej´er kernels FB , there exists B0> 0 such that for all B≥B0we have ∥f∗FB−f∥∞<ε 4.(15) For such Bwe note that, by associativity and commutativity of convolution, (e ψ∗ωB,t)(x) = e ψ∗(FB∗ρt)(x) = (e ψ∗ρt)∗FB(x)=(f∗FB)(x). Combining (14) and (15), we obtain for all x∈[−K, K]: |e ψ∗ωB,t(x)−ψ(x)|≤|e ψ∗ωB,t(x)−f(x)|+|f(x)−e ψ(x)|<ε 4+ε 4=ε 2. 9