Abstract: In this paper, we investigate the uniqueness of difference operators concerning an entire function by using the method of complex difference equations. The results include the difference analogues of the Bruck conjecture. We also present some results on difference operators concerning an entire function with positive deficiency, which generalize the results of Heittokangas, Chen, Yi, et al.

Abstract: We prove that there is a sequence of trigonometric polynomials with positive integer coefficients, which converges to zero almost everywhere. We also prove that there is a function f: ℝ → ℝ such that the sums of its shifts are dense in all real spaces L p (ℝ) for 2 ≤ p < ∞ and also in the real space C0(R).

Abstract: We prove that a good average order on the Goldbach generating function implies that the real parts of the non-trivial zeros of the Riemann zeta function are strictly less than 1. This together with existing results establishes an equivalence between such asymptotics and the Riemann Hypothesis.

Abstract: We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number m generates a complete or incomplete Fourier basis for a Cantor-type measure with scale g.

Abstract: In the present paper an orthogonal system of Chebyshev–Markov rational fractions is considered. We introduce the corresponding Fourier series and find the Dirichlet integral. We obtain the decomposition of the function |x| into Fourier series with respect to the considered system in explicit form and an asymptotic estimate of the uniform approximation of this function by partial sums of the rational Fourier–Chebyshev series.

Abstract: We consider the partial theta function $$\theta (q,z): = \sum olimits_{j = 0} \infty {{q {j(j + 1)/2}}{z j}} $$ , where z ∈ ℂ is a variable and q ∈ ℂ, 0 < |q| < 1, is a parameter. Set $$\alpha 0: = \sqrt 3 /2\pi = 0.2756644477....$$ We show that, for n ≥ 5, for |q| ≤ 1 − 1/(α0n) and for k ≥ n there exists a unique zero ξk of θ(q,.) satisfying the inequalities |q|−k+1/2 < |ξk| < |q|−k−1/2; all these zeros are simple ones. The moduli of the remaining n−1 zeros are ≤ |q|−n+1/2. A spectral value of q is a value for which θ(q,.) has a multiple zero. We prove the existence of the spectral values 0.4353184958... ± i 0.1230440086... for which θ has double zeros −5.963... ± i 6.104....

Abstract: Let S N , N = 1, 2,... be a random walk on the integers, let α be an irrational number and let Z N = {S N α>}, where {·} denotes fractional part. Then Z N , N = 1, 2,... is a random walk on the circle, and from classical results of probability theory it follows that the distribution of Z N converges weakly to the uniform distribution. We determine the precise speed of convergence, which, in addition to the distribution of the elementary step X of the random walk S N , depends sensitively on the rational approximation properties of α.

Abstract: The paper gives a survey on various concepts of negligible sets in infinite-dimensional linear spaces, in particular, related to the research of Jean- Pierre Kahane Some open problems are also mentioned

Abstract: In this paper the Riemann–Hilbert problem of the theory of analytic functions in weighted Smirnov classes is considered. Under certain conditions on the coefficients, the Noetherness of this problem is proved. In the case of solvability general solutions of homogeneous and also non-homogeneous Riemann–Hilbert problem are constructed. As sufficient condition on the weight function for the solvability of the corresponding problems is obtained.

Abstract: Let $$\{\lambda_{n}\}_{n=1}^\infty$$ be a strictly increasing sequence of positive real numbers diverging to infinity and let $$\{\mu_{n}\}_{n=1}^\infty$$ be a sequence of positive integers. Consider the exponential system $${E_{\Lambda \{ {t k}{e {{\lambda _n}t}}:k = 0,1,2,3,...,{\mu _n} - 1\} _{n = 1} \infty }}$$ Assuming the density condition $$\mathop {\lim }\limits_{t \to \infty } \frac{{\sum {_{\lambda n \leqslant {t {{\mu _n}}}}} }}{t} = d < \infty $$ and some other restrictions, we prove that every function in the closure of the linear span of EΛ in some weighted Banach spaces on the real line R is extended to an entire function represented by a Taylor–Dirichlet series $$g(z) = \sum\limits_{n = 1} \infty {(\sum\limits_{k = 0} {{\mu _n} - 1} {{c_n},{k {{z k}}}} )} {e {{\lambda _n}z}},{c_n},k \in C$$ We also consider a problem in a weighted L2(ℝ) Hilbert space as well as a moment problem on the real line.

Abstract: We give the solution of the Turan extremal problem for compact supported functions on the half-line with nonnegative Jacobi transform and the dual Fejer extremal problem for even nonnegative entire functions of exponential type that are Jacobi transforms. We prove the uniqueness of the extremal functions. The Markov quadrature formula on the half-line at zeros of the modified Jacobi function is used for the proof of these results.

Abstract: We continue the study of several related extremal problems for functions analytic in a simply connected domain G with a rectifiable Jordan boundary Γ. In particular, the problem of optimal recovery of a derivative at a point z0 ∈ G from limit boundary values given with an error on a measurable part γ1 of the boundary Γ for the class Q of functions with limit boundary values bounded by 1 on γ0 = Γ γ1 as well as the problem of the best approximation of the derivative at a point z0 ∈ G by bounded linear functionals in L∞(γ1) on the class Q. Complete exact solutions of the considered problems are obtained.

Abstract: In the first part of the paper, it is proved that for 1 < p < ∞ the couple (K , K ∞ ) of coinvariant subspaces of the shift operator on the unit circle is K-closed in the couple (L p (T),L∞ (T)). This property underlies basically all problems of real interpolation for the first couple. Also, a weighted analog of the above statement is established. In the second part it is shown that, given two closed ideals I and J in a uniform algebra such that the complex conjugate of I ∩ J is not included in some of them, the sum I + J is not closed. Though the methods of study in the two parts are quite different, the topics are related by the fact that the question treated in the second part emerged during the work on the first.

Abstract: Let A and B be standard operator algebras on Banach spaces X and Y, respectively. In this paper, we show that every map completely preserving fixed points property from A onto B is either an isomorphism or (in the complex case) a conjugate isomorphism. Also we show that every map completely preserving kernel of operators from A onto B is a scalar multiple of either an isomorphism or (in the complex case) a conjugate isomorphism. B be standard operator algebras on Banach spaces X and Y, respectively. In this paper, we show that every map completely preserving the fixed points property from A onto B is either an isomorphism or (in the complex case) a conjugate isomorphism. Also we show that every map completely preserving the kernel of operators from A onto B is a scalar multiple of either an isomorphism or (in the complex case) a conjugate isomorphism.

Abstract: In this paper, we study the concept of a Segal Frechet algebra and investigate and generalize many known results about abstract Segal algebras, for Segal Frechet algebras. Moreover, we characterize closed ideals of Segal Frechet algebras, and show that the ideal theorem is also valid for Frechet algebras.

Abstract: We study the problem of tiling and packing in vector spaces over finite fields and its connections with zeroes of classical exponential sums. In particular, we study tilings mostly in two and three dimensions and packings in dimension two. A combination of Fourier analytic and algebraic methods is employed.

Abstract: Suppose Λ is a discrete infinite set of nonnegative real numbers. We say that Λ is of type 1 if the series $$s(x) = \sum olimits_{\lambda \in \wedge } {f(x + \lambda )} $$ satisfies a zero-one law. This means that for any non-negative measurable f: ℝ → [0,+∞) either the convergence set C(f, Λ) = {x: s(x) < +∞} = ℝ modulo sets of Lebesgue zero, or its complement the divergence set D(f, Λ) = {x: s(x) = +∞} = ℝ modulo sets of measure zero. If Λ is not of type 1 we say that Λ is of type 2. We show that there is a universal Λ with gaps monotone decreasingly converging to zero such that for any open subset G ⊂ ℝ one can find a characteristic function f G such that G ⊂ D(f G , Λ) and C(f G , Λ) = ℝ\G modulo sets of measure zero. We also consider the question whether C(f,Λ) can contain non-degenerate intervals for continuous functions when D(f, Λ) is of positive measure. The above results answer some questions raised in a paper of Z. Buczolich, J-P. Kahane, and D. Mauldin.

Abstract: We apply an improvement of the Delsarte LP-bound to give a new proof of the non-existence of finite projective planes of order 6, and uniqueness of finite projective planes of order 7 The proof is computer aided, and it is also feasible to apply to higher orders like 8, 9 and, with further improvements, possibly 10 and 12

Abstract: We investigate the Plancherel–Polya inequality $$\sum {_{k \in \mathbb{Z}}} |f(k){|^2} \leqslant {c_2}\left( \sigma \right)||f||_{{L^2}\left( \mathbb{R} \right)}^2$$ on the set of entire functions f of exponential type at most σ whose restrictions to the real line belong to the space L2(ℝ). We prove that c2(σ) = [σ/π] for σ > 0 and describe the extremal functions.

Abstract: Given a measure μ on a bounded domain Ω ⊂ ℝn with C1 boundary, we investigate the following problem: when is a weighted harmonic Bergman space $$A_\alpha^p(\Omega)$$ continuously embedded in weighted space Lp(Ω) = Lp(μ, Ω)? We give a sufficient Carleson type condition for all α > −1 and 0 < p < ∞ which is also necessary for $$p > 1 + \frac{{\alpha + 2}}{{n - 2}}$$ .

Abstract: In this paper I present my first proof regarding the existence of universal Taylor series on the disc where the universal approximation was required on the boundary as well. It is a modification of a construction giving a negative answer to a question of S. Pichorides, where the approximation was valid only on the boundary of the disc. There was no use of Baire’s theorem in the above proofs. J.-P. Kahane suggested to use Baire’s theorem which yields stronger results with simpler proofs. Later, Baire’s theorem was systematically used in order to establish new generic universalities.

Abstract: We discuss the optimality of Kahane’s multidimensional Ingham type theorem and we establish a relationship between the absolute continuity of Bernoulli convolutions and some results on expansions in non-integer bases.

Abstract: We present a unified approach to obtain new series representations for various classical constants Among others, we prove that $$\log (2) = \frac{{17}}{{24}} + \sum\limits_{k = 2}^\infty {{{( - 1)}^k}} \frac{{{k^2} + k - 1/2}}{{(k - 1)k(k + 1)(k + 2)}}({H_k} - {H_{{{\left[ {k/2} \right]}}})^2}$$ $$G = - \frac{1}{2} + 2\sum\limits_{k = 1}^\infty {{{( - 1)}^k}\frac{{k(4{k^2} - 5)}}{{(4{k^2} - 1)(4{k^2} - 9)}}{{(2{H_{2k}} - {H_k})}^2}} $$ , $$\zeta (3) = \frac{{149}}{{144}} + \frac{1}{8}\sum\limits_{k = 2}^\infty {\frac{{(2k + 1)({k^4} + 2{k^3} + 3{k^2} + 2k - 2)}}{{{{(k - 1)}^2}{k^2}{{(k + 1)}^2}(k + 2)}}{{(2H_k^{(2)} - H_{[k/2]}^{(2)})}^2}} $$ , where $${H_k} = \sum olimits_{j = 1}^k {1/j} $$ and $$H_k^{(2)} = {\sum olimits_{j = 1}^k {1/j} ^2}$$ denote the harmonic numbers and the generalized harmonic numbers of order 2, respectively, and G is the Catalan constant

Abstract: Erdős, Polya and Turan conjectured 70 years ago that a linear combination of consecutive differences of primes takes infinitely often both positive and negative values if and only if the (fixed) coefficients of the linear combination do not have all the same sign. In this work we prove this conjecture in a somewhat more general form. Our proof is based on a method of Banks, Freiberg and Maynard which is again based on the method of Maynard, Tao and the Polymath 8 project which showed the existence of infinitely many prime gaps not exceeding 246.

Abstract: Fourier multipliers of the space W1,1(ℝd) are bounded functions m such that the convolution by F−1m extends into a bounded operator on W1,1(ℝd) Poornima exhibited in the eighties a family of such Fourier multipliers among which some are not Fourier transforms of bounded measures for d > 1 Her counterexamples are based on celebrated non-inequalities of Ornstein We will give new examples of Fourier multipliers, which are not Fourier transforms of measures and do not belong to her family The examples are constructed on the d-dimensional torus and then transferred to the Euclidean space

Abstract: We show that the class of sets E with the property that there exists a measure singular with respect to the Lebesgue measure whose Fourier transform tends to 0 at infinity and vanishes on E contains the Helson sets.

Abstract: We give an extension of the Mazur–Ulam theorem for isometries from real metrizable topological vector spaces into real normed spaces.

Abstract: Let L(H) be the algebra of all bounded linear operators on a Hilbert space H into itself, and let K(H) denote the ideal of all compact operators. Given A,B ∈ L(H), define the generalized derivation δA,B: L(H) → L(H) by δA,B(X) = AX − XB. If A = B, then δA,A = δA is the inner derivation implemented by A ∈ L(H). In this paper we establish some properties concerning the class of operators A ∈ L(H) wich satisfy $${\overline {R({\delta _A})} ^W} \cap \{ A\} ' \cap K(H) = \{ 0\} $$ , where $${\overline {R({\delta _A})} ^W}$$ is the weak closure of the range of δA, as a consequence, we show that the set $$\{ A \in L(H)|{\overline {R({\delta _A})} ^W} \cap K(H) = \{ 0\} \} $$ is norm-dense. We also give a large class of operators A verifying $${\overline {R({\delta _A})} ^W} \cap \{ A*\} '$$ contains no nonzero compact operator, and we describe some classes of operators A, B for wich we have $${\overline {R({\delta _{A,B}})} ^W} \cap Ker({\delta _{A*,B*}}) \cap K(H) = \{ 0\} $$ , where ker(δA*,B*) is the kernel of δA*,B*.