Determination of cusp forms by central values of Rankin–Selberg <Emphasis Type="BoldItalic">L</Emphasis>-functions<Superscript>*</Superscript>

Let f be a fixed holomorphic Hecke eigen cusp form of weight k for \( SL\left( {2,{\mathbb Z}} \right) \), and let \( {\mathcal U} = \left\{ {{u_j}:j \geqslant 1} \right\} \) be an orthonormal basis of Hecke–Maass cusp forms for \( SL\left( {2,{\mathbb Z}} \right) \). We prove an asymptotic formula for the twisted first moment of the Rankin–Selberg L-functions \( L\left( {s,f \otimes {u_j}} \right) \) at \( s = \frac{1}{2} \) as u _j runs over \( {\mathcal U} \). It follows that f is uniquely determined by the central values of the family of Rankin–Selberg L-functions \( \left\{ {L\left( {s,f \otimes {u_j}} \right):{u_j} \in {\mathcal U}} \right\} \).     

On the application of linear positive operators for approximation of functions

For the linear positive Korovkin operator \(f\left( x \right) \to {t_n}\left( {f;x} \right) = \frac{1}{\pi }\int_{ - \pi }^\pi {f\left( {x + t} \right)E\left( t \right)dt} \), where E(x) is the Egervary–Szász polynomial and the corresponding interpolation mean \({t_{n,N}}\left( {f;x} \right) = \frac{1}{N}\sum\limits_{k = - N}^{N - 1} {{E_n}\left( {x - \frac{{\pi k}}{N}} \right)f\left( {\frac{{\pi k}}{N}} \right)} \), the Jackson-type inequalities \(\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant \left( {1 + \pi } \right){\omega _f}\left( {\frac{1}{n}} \right),\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant 2{\omega _f}\left( {\frac{\pi }{{n + 1}}} \right)\), where ωf (x) denotes the modulus of continuity, are proved for N > n/2. For ωf (x) ≤ Mx, the inequality \(\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant \frac{{\pi M}}{{n + 1}}\). is established. As a consequence, an elementary derivation of an asymptotically sharp estimate of the Kolmogorov width of a compact set of functions satisfying the Lipschitz condition is obtained.     

Markoff–Lagrange spectrum and extremal numbers

Let \( \gamma = \frac{1}{2}\left( {1 + \sqrt {5} } \right) \) denote the golden ratio. H. Davenport and W. M. Schmidt showed in 1969 that, for each non-quadratic irrational real number ξ, there exists a constant c > 0 with the property that, for arbitrarily large values of X, the inequalities\( \left| {{x_0}} \right| \leqslant X,\,\,\,\left| {{x_0}\xi - {x_1}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}}\,\,\,{\text{and}}\,\,\,\left| {{x_0}{\xi^2} - {x_2}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}} \)admit no non-zero solution \( \left( {{x_0},{x_1},{x_2}} \right) \in {\mathbb{Z}^3} \). Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers ξ such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each X ≥ 1. Such extremal numbers are transcendental and their set is stable under the action of \( {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) \) on \( \mathbb{R}\backslash \mathbb{Q} \) by linear fractional transformations. In this paper, it is shown that there exist extremal numbers ξ for which the Lagrange constant ν(ξ) = liminf _q→∞ q||qξ|| is \( \frac{1}{3} \), the largest possible value for a non-quadratic number, and that there is a natural bijection between the \( {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) \)-equivalence classes of such numbers and the non-trivial solutions of Markoff’s equation.     

The Frobenius problem for repunit numerical semigroups

A repunit is a number consisting of copies of the single digit 1. The set of repunits in base b is \(\big \{\frac{b^n-1}{b-1} ~|~ n\in {\mathbb N}\backslash \{0\}\big \}\). A numerical semigroup S is a repunit numerical semigroup if there exist integers \(b\in {\mathbb N}\backslash \left\{ 0,1\right\} \) and \(n\in {\mathbb N}\backslash \left\{ 0\right\} \) such that \(S=\big \langle \big \{\frac{b^{n+i}-1}{b-1} ~|~ i\in {\mathbb N}\big \}\big \rangle \). In this work, we give formulas for the embedding dimension, the Frobenius number, the type and the genus for a repunit numerical semigroup.     

Corona problem with data in ideal spaces of sequences

Let E be a Banach lattice on \({\mathbb {Z}}\) with order continuous norm. We show that for any function \(f = \{f_j\}_{j \in {\mathbb {Z}}}\) from the Hardy space \(\mathrm H_{\infty }\left( E \right) \) such that \(\delta \leqslant \Vert f (z)\Vert _E \leqslant 1\) for all z from the unit disk \({\mathbb {D}}\) there exists some solution \(g = \{g_j\}_{j \in {\mathbb {Z}}} \in \mathrm H_{\infty }\left( E' \right) \), \(\Vert g\Vert _{\mathrm H_{\infty }\left( E' \right) } \leqslant C_\delta \) of the Bézout equation \(\sum _j f_j g_j = 1\), also known as the vector-valued corona problem with data in \(\mathrm H_{\infty }\left( E \right) \).     

Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the Hormander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k≥ 3 using the multiparameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k≥ 3:■where x =(x_1,x_2,x_3)∈R~(n_1)×R~(n_2)×R~(n_3) and ξ =(ξ_1,ξ_2,ξ_3)∈R~(n_1)×R~(n_2)×R~(n_3). One of our main results is the following:Assume that m(ξ) is a function on R~(n_1+n_2+n_3) satisfying ■ with s_i n_i(1/p-1/2) for 1≤i≤3. Then T_m is bounded from H~p(R~(n_1)×R~(n_2)×R~(n_3) to H~p(R~(n_1)×R~(n_2)×R~(n_3)for all 0 p≤1 and ■ Moreover, the smoothness assumption on s_i for 1≤i≤3 is optimal. Here we have used the notations m_(j,k,l)(ξ)=m(2~jξ_1,2~kξ_2,2~lξ_3)Ψ(ξ_1)Ψ(ξ_2)Ψ(ξ_3) and Ψ(ξ_i) is a suitable cut-off function on R~(n_i) for1≤i≤3, and W~(s_1,s_2,s_3) is a three-parameter Sobolev space on R~(n_1)×R~(n_2)× R~(n_3).Because the Fefferman criterion breaks down in three parameters or more, we consider the L~p boundedness of the Littlewood-Paley square function of T_mf to establish its boundedness on the multi-parameter Hardy spaces.     

On one complement to the Hölder inequality: I

Let m ≥ 2, the numbers p ₁,…, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ...\frac{1}{{{p_m}}} < 1\), and γ₁ ∈ L ^p1(?¹), …, γ _m ∈ \({L^{{p_m}}}\)(?¹). We prove that, if the set of “resonance” points of each of these functions is nonempty and the “nonresonance” condition holds (both concepts have been introduced by the author for functions of spaces L ^p (?¹), p ∈ (1, +∞]), we have the inequality \(\mathop {\sup }\limits_{a,b \in {R^1}} \left| {\int\limits_a^b {\prod\limits_{k = 1}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]} d\tau } } \right| \leqslant C{\prod\limits_{k = 1}^m {\left\| {{\gamma _k} + \Delta {\gamma _k}} \right\|} _{L_{{a_k}}^{{p_k}}}}\left( {{\mathbb{R}^1}} \right)\), where the constant C > 0 is independent of functions \(\Delta {\gamma _k} \in L_{{a_k}}^{{p_k}}\left( {{\mathbb{R}^1}} \right)\) and \(L_{{a_k}}^{{p_k}}\left( {{\mathbb{R}^1}} \right) \subset {L^{{p_k}}}\left( {{\mathbb{R}^1}} \right)\), 1 ≤ k ≤ m are some specially constructed normed spaces. In addition, we give a boundedness condition for the integral of the product of functions over a subset of ?¹.     

Supplement to Hölder’s inequality. II

Suppose that m ≥ 2, numbers p ₁, …, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + \cdots + \frac{1}{{{p_m}}} < 1\), and functions \({\gamma _1} \in {L^{{p_1}}}\left( {{?^1}} \right), \cdots ,{\gamma _m} \in {L^{{p_m}}}\left( {{?^1}} \right)\) are given. It is proved that if the set of “resonance” points of each of these functions is nonempty and the “nonresonance” condition holds (both notions were defined by the author for functions in L ^p (?¹), p ∈ (1, +∞]), then \(\mathop {\sup }\limits_{a,b \in {R^1}} \left| {\mathop \smallint \limits_a^b \prod\limits_{k = 1}^m {[{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)]} d\tau } \right| \leqslant C\prod\limits_{k = 1}^m {{{\left\| {{\gamma _k} + \Delta {\gamma _k}} \right\|}_{L_{ak}^{pk}\left( {{R^1}} \right)}}} \) where the constant C > 0 is independent of the functions \(\Delta {\gamma _k} \in L_{ak}^{pk}\left( {{?^1}} \right)\) and \(L_{ak}^{pk}\left( {{?^1}} \right) \subset {L^{pk}}\left( {{?^1}} \right)\), 1 ≤ k ≤ m, are special normed spaces. A condition for the integral over ?¹ of a product of functions to be bounded is also given.     

Generalized Kloosterman sum with primes

The work is devoted to generalized Kloosterman sums modulo a prime, i.e., trigonometric sums of the form \(\sum\nolimits_{p \leqslant x} {\exp \left\{ {2\pi i\left( {a\bar p + {F_k}\left( p \right)} \right)/q} \right\}} \) and \(\sum\nolimits_{n \leqslant x} {\mu \left( n \right)\exp \left\{ {2\pi i\left( {a\bar n + {F_k}\left( n \right)} \right)/q} \right\}} \), where q is a prime number, \(\left( {a,q} \right) = 1,m\bar m \equiv 1\left( {\bmod {\kern 1pt} q} \right)\), F _k (u) is a polynomial of degree k ≥ 2 with integer coefficients, and p runs over prime numbers. An upper estimate with a power saving is obtained for the absolute values of such sums for x ≥ q^1/2+ε.     

Some Berezin Number Inequalities for Operator Matrices

The Berezin symbol à of an operator A acting on the reproducing kernel Hilbert space H = H(Ω) over some (nonempty) set is defined by \(\tilde A(\lambda ) = \left\langle {A\hat k_\lambda ,\hat k_\lambda } \right\rangle \), λ ∈ Ω, where \(\hat k_\lambda = k_\lambda /\left\| {k_\lambda } \right\|\) is the normalized reproducing kernel of H. The Berezin number of the operator A is defined by \(ber(A) = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\tilde A(\lambda )} \right| = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\left\langle {A\hat k_\lambda ,\hat k_\lambda } \right\rangle } \right|\). Moreover, ber(A) ? w(A) (numerical radius). We present some Berezin number inequalities. Among other inequalities, it is shown that if \(T = \left[ {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right] \in \mathbb{B}(\mathcal{H}(\Omega _1 ) \oplus \mathcal{H}(\Omega _2 ))\), then

$$ber(T) \leqslant \frac{1}{2}(ber(A) + ber(D)) + \frac{1}{2}\sqrt {(ber(A) - ber(D))^2 + \left( {\left\| B \right\| + \left\| C \right\|} \right)^2 } .$$

     

On the densities of rational multiples

For two subsets of natural numbers \( A,B \subset \mathbb{N} \), define the set of rational numbers \( \mathcal{M}\left( {A,B} \right) \) with the elements represented by m/n, where m and n are coprime, m is divisible by some a ∈ A, and n is divisible by some b ∈ B. Let I be some interval of positive real numbers and \( \mathcal{F}_x^I \) denote the set of rational numbers m/n ∈ I such that m and n are coprime and n ? x. The analogue to the Erdös–Davenport theorem about multiples is proved: under some constraints on I, the limits \( {{{\sum {\left\{ {\frac{1}{{mn}}:\frac{m}{n} \in \mathcal{F}_x^I \cap \mathcal{M}\left( {A,B} \right)} \right\}} }} \left/ {{\sum {\left\{ {\frac{1}{{mn}}:\frac{m}{n} \in \mathcal{F}_x^I} \right\}} }} \right.} \) exist for all subsets \( A,B \subset \mathbb{N} \) as x → ∞.     

A Supplement to Hölder’s Inequality. The Resonance Case. I

Suppose that m ≥ 2, numbers p₁, …, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ... + \frac{1}{{{p_m}}} < 1\), and functions γ₁ ∈ \({L^{{p_1}}}\)(?¹), …, γ _m ∈ \({L^{{p_m}}}\)(?¹) are given. It is proved that if the set of “resonance points” of each of these functions is nonempty and the so-called “resonance condition” holds, then there are arbitrarily small (in norm) perturbations Δγ_k ∈ \({L^{{p_k}}}\)(?¹) under which the resonance set of each function γ_k + Δγ_k coincides with that of γ_k for 1 ≤ k ≤ m, but \({\left\| {\int\limits_0^t {\prod\limits_{k = 0}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]d\tau } } } \right\|_{{L^\infty }\left( {{\mathbb{R}^1}} \right)}} = \infty \). The notion of a resonance point and the resonance condition for functions in the spaces L ^p (?¹), p ∈ (1, +∞], were introduced by the author in his previous papers.     

Reconstruction of the potential of the Sturm–Liouville operator from a finite set of eigenvalues and normalizing constants

It is well known that the potential q of the Sturm–Liouville operator L_y = ?y? + q(x)y on the finite interval [0, π] can be uniquely reconstructed from the spectrum \(\left\{ {{\lambda _k}} \right\}_1^\infty \) and the normalizing numbers \(\left\{ {{\alpha _k}} \right\}_1^\infty \) of the operator L_D with the Dirichlet conditions. For an arbitrary real-valued potential q lying in the Sobolev space \(W_2^\theta \left[ {0,\pi } \right],\theta > - 1\), we construct a function q_N providing a 2N-approximation to the potential on the basis of the finite spectral data set \(\left\{ {{\lambda _k}} \right\}_1^N \cup \left\{ {{\alpha _k}} \right\}_1^N\). The main result is that, for arbitrary τ in the interval ?1 ≤ τ < θ, the estimate \({\left\| {q - \left. {{q_N}} \right\|} \right._\tau } \leqslant C{N^{\tau - \theta }}\) is true, where \({\left\| {\left. \cdot \right\|} \right._\tau }\) is the norm on the Sobolev space \(W_2^\tau \). The constant C depends solely on \({\left\| {\left. q \right\|} \right._\theta }\).     

Approximation of entire functions of exponential type by exponential sums<Superscript>*</Superscript>

Let K be a compact set in \( {{\mathbb R}^n} \). For \( 1 \leqslant p \leqslant \infty \), the Bernstein space \( B_K^p \) is the Banach space of all functions \( f \in {L^p}\left( {{{\mathbb R}^n}} \right) \)such that their Fourier transform in a distributional sense is supported on K. If \( f \in B_K^p \), then f is continuous on \( {{\mathbb R}^n} \) and has an extension onto the complex space \( {{\mathbb C}^n} \) to an entire function of exponential type K. We study the approximation of functions in \( B_K^p \) by finite τ -periodic exponential sums of the form

$ \sum\limits_m {{c_m}{e^{2\pi {\text{i}}\left( {x,m} \right)/\tau }}} $

in the \( {L^p}\left( {\tau {{\left[ { - 1/2,1/2} \right]}^n}} \right) \)-norm as τ → ∞ when K is a polytope in \( {{\mathbb R}^n} \).

     

The prime-counting function and its analytic approximations

The paper describes a systematic computational study of the prime counting function π(x) and three of its analytic approximations: the logarithmic integral \({\text{li}}{\left( x \right)}: = {\int_0^x {\frac{{dt}}{{\log \,t}}} }\), \({\text{li}}{\left( x \right)} - \frac{1}{2}{\text{li}}{\left( {{\sqrt x }} \right)}\), and \(R{\left( x \right)}: = {\sum\nolimits_{k = 1}^\infty {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} \mathord{\left/ {\vphantom {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} k}} \right. \kern-\nulldelimiterspace} k} }\), where μ is the Möbius function. The results show that π(x)x) for 2≤x≤10¹⁴, and also seem to support several conjectures on the maximal and average errors of the three approximations, most importantly \({\left| {\pi {\left( x \right)} - {\text{li}}{\left( x \right)}} \right|} < x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}\) and \( - \frac{2}{5}x^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} < {\int_2^x {{\left( {\pi {\left( u \right)} - {\text{li}}{\left( u \right)}} \right)}du < 0} }\) for all x>2. The paper concludes with a short discussion of prospects for further computational progress.     

On the towers of torsion Bertrandias and Payan modules

For an odd prime p, let K/k be a Galois p-extension and S be a set of primes of k containing the primes lying over p. For the p ^r th roots \({\mu _{{p^r}}}\left( K \right)\) of unity in K, we describe the so-called Sha group Sha _S (G(K/k), \({\mu _{{p^r}}}\left( K \right)\)) in terms of the Galois groups of certain subfields of K corresponding to S. As an application, we investigate a tower of extension fields \({\left\{ {{k_{{T^i}}}} \right\}_i} \geqslant 0\) where \({k_{{T^{i + 1}}}}\) is defined as the fixed field of a free part of the Galois group of the Bertrandias and Payan extension of \({k_{{T^i}}}\) over \({k_{{T^i}}}\). This is called a tower of torsion parts of the Bertrandias and Payan extensions over k. We find a relation between the degrees \({\left\{ {\left[ {{k_{{T^{i + 1}}}}:{k_{{T^i}}}} \right]} \right\}_{i \geqslant 0}}\) over the towers. Using this formula we investigate whether the towers are stationary or not.     

Complementary inequalities to improved AM-GM inequality

Following an idea of Lin, we prove that if A and B are two positive operators such that 0 mI ≤ A ≤m'I≤ M'I ≤ B ≤ MI, then Φ~2(A+B/2)≤K~2(h)/(1+(logM'/m'/g))~2Φ~2(A≠B) and Φ~2(A+B/2)≤K~2(h)/(1+(logM'/m'/g))~2(Φ(A)≠Φ(B))~2 where K(h)=(h+1)~2/4 and h = M/m and Φ is a positive unital linear map.     

Invariant <Emphasis Type="Italic">K</Emphasis>-minimal Sets in the Discrete and Continuous Settings

A sufficient condition for a set \(\Omega \subset L^{1}\left( \left[ 0,1\right] ^{m}\right) \) to be invariant K-minimal with respect to the couple \(\left( L^{1}\left( \left[ 0,1\right] ^{m}\right) ,L^{\infty }\left( \left[ 0,1\right] ^{m}\right) \right) \) is established. Through this condition, different examples of invariant K-minimal sets are constructed. In particular, it is shown that the \(L^{1}\)-closure of the image of the \(L^{\infty }\)-ball of smooth vector fields with support in \(\left( 0,1\right) ^{m}\) under the divergence operator is an invariant K-minimal set. The constructed examples have finite-dimensional analogues in terms of invariant K-minimal sets with respect to the couple \(\left( \ell ^{1},\ell ^{\infty }\right) \) on \(\mathbb {R}^{n}\). These finite-dimensional analogues are interesting in themselves and connected to applications where the element with minimal K-functional is important. We provide a convergent algorithm for computing the element with minimal K-functional in these and other finite-dimensional invariant K-minimal sets.