Given two different complex numbers a and b, a bounded linear operator T acting on an infinite-dimensional complex Hilbert space H is said to be {a, b}-quadratic if (T − a)(T − b) = 0. We provide in this paper a complete description of all surjective maps Φ (not necessarily additive) on the algebra \({\cal B}(H)\) of all bounded linear operators on H that satisfy S − λT is {a, b}-quadratic if and only

In this paper, we investigate general Lp-mixed brightness integrals which were defined by Yan and Wang. We give their extremal values and establish Brunn-Minkowski type inequalities.

We study several versions of the improved geometric Rellich inequalities in the setting of identities. Our results present an easy and explicit way to understand the sharp constants as well as the nonexistence of optimizers for many geometric Rellich’s type inequalities.

We study the compactness of composition operators on the Bergman spaces of certain bounded convex domains in ℂn with non-trivial analytic discs contained in the boundary. As a consequence we characterize that compactness of the composition operator with a continuous symbol (up to the closure) on the Bergman space of the polydisc.

Let P(x) be an arbitrary algebraic polynomial of degree n with all zeros in the unit interval −1 ≤ x ≤ 1. We establish the Turán-type inequality ‖P″‖0 ≥ n(e/4)‖P‖0, where \({\left\| f \right\|_0} = \exp \left( {{1 \over 2}\int_{ - 1}^1 {\ln \left| {f(x)} \right|\,dx} } \right)\) is the geometric mean of a function. This estimate is extremal for any even n. We also obtain the following Turán-type inequality

This paper has two purposes. One is to establish a version of Nevanlinna theory based on the historic so-called Jackson difference operator \({D_q}f(z) = {{f(qz) - f(z)} \over {qz - z}}\) for meromorphic functions of zero order in the complex plane ℂ. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Picard theorem, five-value theorem and Wiman-Valiron theorem

We discuss the relationship on the periodicity of a transcendental entire function with its differential polynomials. For example, we obtain that if f is a transcendental entire function, k is a non-negative integer and if (anfn + ⋯ + a1f)(k) is a periodic function, then f is also a periodic function, where a1, … an (≠ 0) are constants. Our results are related to Yang’s Conjecture on the periodicity

Let E be an arbitrary closed set on the unit circle ∂ⅅ and u be a harmonic function on the unit disk ⅅ satisfying |u(z)| ≾ (1 − |z|)γρ−q(z)where ρ(z) = dist (z, E), γ, q are some real constants, γ ≤ q. We establish an estimate of the conjugate \(\tilde u\) of the same type which is sharp in some sense, and in the case E = ∂ⅅ coincides with known estimates. As an application we describe growth classes

We are concerned with basic properties of the Helmholtz decomposition of the Muckenhoupt Ap-weighted Lp-space (\(L_w^p(\Omega )\))n for any domain Ω in ℝn (n ∈ ℤ, n ≥ 2), 1 < p < ∞ and Muckenhoupt Ap-weight w ∈ Ap. Set \({p^\prime}: = {p \over {p - 1}}\) and \({w^\prime}: = {w^{ - {1 \over {p - 1}}}}\). Then the consistency of the Helmholtz decomposition of \({(L_w^p(\Omega ))^n}\) and \({(L_{{w^\

We introduce and study the k-Hankel Gabor transform. We investigate the localization operators for this transform. In particular, we study their trace class properties and we prove that they belong to the Schatten-von Neumann class. We also study the eigenvalues and eigenfunctions of the time-frequency localization operator. Finally we give some results on the spectrograms for the k-Hankel Gabor transform

The classical Weyl Law says that if NM(λ) denotes the number of eigenvalues of the Laplace operator on a d-dimensional compact manifold M without a boundary that are less than or equal to λ, then $${N_M}(\lambda ) = c{\lambda ^d} + O({\lambda ^{d - 1}}).$$ This paper explores the prospects of improvements of Weyl remainders on products of manifolds. In particular we obtain a polynomial improvement

The aim of this work is to derive several analytical properties for the class of entire functions defined by $${{\cal I}_\phi}(t) = \sum\limits_{n = 0}^\infty {{1 \over {{W_\phi}(n + 1)}}{{{t^n}} \over {n!}}\,\,\;{\rm{with}}\,\;{W_\phi}(1) = 1,\;\;\,{W_\phi}(n + 1) = \prod\limits_{k = 1}^n {\phi (k),}} $$ where the parameter ϕ ranges over the set of Bernstein functions. This includes the expression

In this paper, we prove some difference analogue of second main theorems of meromorphic mapping from ℂm into an algebraic variety V intersecting a finite set of fixed hypersurfaces in subgeneral position. As an application, we prove a result on algebraic degeneracy of holomorphic curves on \({\cal P}_c^1\) intersecting hypersurfaces and difference analogue of Picard’s theorem on holomorphic curves

We give conditions for boundedness of Hausdorff operators on real Hardy spaces H1 over homogeneous spaces of locally compact groups with local doubling property. Special cases of the hyperbolic plane and 2-sphere are considered. In this cases Hausdorff operators look as orbital like integrals.

Szegő proved that the trigonometric inequality $$0 < \sum\limits_{k = 1}^n {\left({n - k + 1} \right)\left({n - k + 2} \right)\left({\cos \left({kx} \right) - \cos \left({ky} \right)} \right)}$$ holds for all integers n ≥ 1and real numbers x, y with 0 ≥ x ≥ π/2, α ≥ y ≥ π, where α = 1.98231 …. We show that the inequality is valid for a larger domain, namely for all n ≥ 1and x ∈ [0, π/2], y ∈ (π/2,

This paper is devoted to investigating the growth and zeros of meromorphic solutions of the generalized Fermat functional equation $${f^2}\left({z + 1} \right) + {A_1}\left(z \right)f\left({z + 1} \right)f\left({z + 1} \right)f\left(z \right) + {A_2}{f^2}\left(z \right) = {A_3}\left(z \right),$$ where A1(z), A2(z), A3(z) are polynomials with A3(z) ≢ 0. The corresponding homogeneous equation of the

In this article we discuss the behaviour of Θ-means of Walsh—Fourier series of a function in dyadic Hardy spaces Hp and dyadic homogeneous Banach spaces X. Namely, we estimate the rate of the approximation by Θ-means in terms of modulus of continuity in X and best approximation in Hp. Our main theorem is a generalization of a result of Fridli, Manchanda and Siddiqi [7]. Moreover, it extends a previous

Direct and inverse theorems are proved for approximation by bounded Vilenkin systems in weighted Orlicz spaces. The equivalence between the K-functional constructed with help of generalized derivative and three types of realizations functionals is established. We also obtain a simultaneous approximation result.

We introduce the Orlicz space corresponding to the Young function and, by virtue of the equivalence theorem between the modified K-functional and modulus of smoothness, establish the direct, inverse, and equivalence theorems for the Jacobi weighted Szász—Durrmeyer—Bézier operators in the Orlicz spaces.

The purpose of this paper is to prove a Hardy type inequality associated with the n-dimensional Hankel transform (n ≥ 1) for the exponent σ0 = n(2 − p) + n(2v + 1) by using the atomic decomposition. Moreover, we extend this inequality for “small” atoms to exponents σ satisfying σmin ≤ σ ≤ σmax.

In this paper, we discuss measure theoretic characterizations for Moore-Penrose inverse of Lambert conditional operators, denoted by (MwEMu)†, in some operator classes on L2(Σ) such as p-hyponormal, centered, n-normal, bi-normal, partial isometry, quasinilpotent and EP operator. Moreover, we prove some basic results on (MwEMu)†, for instance, triple reverse order law and lower and upper bounds for

In the present work, we give an answer to questions posed in [3]. We precisely study the mutual singularity of Hewitt-Stromberg measures on Bedford-McMullen carpets.

In this paper, we prove a mean ergodic theorem for nonexpansive mappings in Hadamard (nonpositive curvature complete metric) spaces, which extends the Baillon nonlinear ergodic theorem. The main result shows that the sequence given by the Karcher means of iterations of a nonexpansive mapping with a nonempty fixed point set converges weakly to a fixed point of the mapping. This result also remains true

We study continuity properties for commutators of Calderón-Zygmund and fractional integral operators between generalized Zygmund spaces of L log L type, in the variable exponent setting with different weights. In order to reach this goal we use two different approaches: the first one is related to generalized bump conditions on a pair of weights, allowing us to handle with a wide class of symbol involved

We prove asymptotic expansions of generalized partial theta functions with a nonprincipal Dirichlet character and relate these expansions to certain L-series.

We utilize a combination of integral transforms, including the Laplace transform, with some classical results in analytic number theory concerning the Riemann ξ-function, to obtain a new integral equation. This integral equation is generalized to self-dual principal automorphic L-functions. We also provide a new proof of known functional-type identities from analytic number theory, and recast some

In this paper, we give optimal embedding relations between local Hardy spaces and α-modulation spaces. By a different approach, we extend the main results obtained by Kobayashi, Miyachi and Tomita in [Studia Math., 192 (2009), 79–96].

In this paper, we are interested in estimates of the Dunkl kernels on some special sets. We improving results of M.F.E. de Jeu and M. Rösler [6].

The purpose of this paper is to study several different properties of the holomorphic \({\cal N}\left( {p,q,s} \right)\)-type functions in the unit ball \(\mathbb{B}\) of ℂn via Carleson measure techniques. More precisely, we establish an atomic decomposition on such spaces and then we solve the Gleason’s problem on them. We also study the distance problems between Bergman-type spaces \({A^{ - {q \over

In this paper we characterize the functions that map the positive general monotone sequences to themselves, i.e., we obtain a necessary and a sufficient condition for the functions φ to satisfy $${\sum\limits_{k = n}^{2n} {\left| {{a_{k + 1}} - {a_k}} \right| \le C{a_n} \Rightarrow \sum\limits_{k = n}^{2n} {\left| {\phi \left( {{a_{k + 1}}} \right)} \right| \le {C^\prime }\phi \left( {{a_n}} \right)

For a topological dynamical system we characterize the decomposition of the state space induced by the fixed space of the corresponding Koopman operator. For this purpose, we introduce a hierarchy of generalized orbits and obtain the finest decomposition of the state space into absolutely Lyapunov stable sets. Analogously to the measure-preserving case, this yields that the system is topologically

Let f(z) be a transcendental meromorphic function, whose zeros have multiplicity at least 3. Set α(z): = β(z)exp (γ(z), where β(z) is a nonconstant elliptic function and γ(z) is an entire function. If σ(f(z)) > σ(α(z)), then f′(z) = α(z) has infinitely many solutions in the complex plane.

We consider the classes of functions that are defined on a metric compact, take values in an L -space (i.e. semi-isotropic semi-linear metric space) and have a given majorant of modulus of continuity. For a wide class of operators Λ that act on such function classes, we solve the problem of the optimal recovery based on inaccurate values of the functions in a finite number of points. As an application

Let $$\left\{ {{d_k}} \right\}_{k = 1}^\infty $$ be an upper-bounded sequence of positive integers and let δ E be the uniformly discrete probability measure on the finite set E . For 0 < ρ < 1, the infinite convolution $${\mu _{\rho \left\{ {0,{d_k}} \right\}}}: = {\delta _{\rho \left\{ {0,{d_1}} \right\}}}*{\delta _{{\rho ^2}\left\{ {0,{d_2}} \right\}}}* \cdots $$ is called an infinite Bernoulli convolution

We show a characterization for the boundedness of the commutators for bilinear fractional integral operators B α (0 < α < n ) on Morrey spaces. Moreover, we obtain that if b ∈ CMO, then the commutators [ b, B α ] i ( i = 1, 2) are separately compact operators on Morrey spaces where CMO denotes the BMO-closure of C c ∞ (ℝ n ). A necessary condition for commutators [ b, B α ] i ( i = 1, 2) to be jointly

In this article, we present a new method to study relative Cuntz-Krieger algebras for higher-rank graphs. We only work with edges rather than paths of arbitrary degrees. We then use this method to simplify the existing results about relative Cuntz-Krieger algebras. We also give applications to study ideals and quotients of Toeplitz algebras.

In this paper, the properties of Lipschitz functions are considered and the convergence of the special type of series of Fourier coefficients with respect to general orthonormal systems (ONS) is investigated. For the functions of ONS we find a condition under which the special series of Fourier coefficients of Lipschitz functions with respect to general orthonormal systems are convergent. The obtained

Let H, K be a couple of vector fields of class C 1 in an open set U ⊂ ℝ N+m , $${\cal M}$$ ℳ be a N -dimensional C 1 submanifold of U and define $${\cal T}: = \left\{{z \in {\cal M}:H\left(z \right),K\left(z \right) \in {T_z}{\cal M}} \right\}$$ T : = { z ∈ ℳ : H ( z ) , K ( z ) ∈ T z ℳ } . Then the obvious property If z 0 ∈ $${\cal M}$$ ℳ is an interior point (relative to $${\cal M}$$ ℳ )of $${\cal

We point out that the proof of Theorem 1.6 in the paper [2] by B. A. Bhayo and J. Sándor contains a mistake. Correcting this mistake is the main aim of this note.

It is well known that a random cosine polynomial $${V_n}\left(x \right) = \sum\nolimits_{j = 0}^n {{a_j}\cos \left({jx} \right)} $$ V n ( x ) = ∑ j = 0 n a j cos ( j x ) , x ∈ (0, 2 π ), with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables (asymptotically) has $$2n/\sqrt 3 $$ 2 n / 3 expected real roots. On the other hand, out of

In this paper, BSE properties of some Banach function algebras are studied. We show that Lipschitz algebras Lip α ( X, d ) and Dales-Davie algebras D ( X, M ) are BSE-algebras for certain underlying plane sets X . Moreover, we investigate BSE properties of certain subalgebras of Lip α ( X, d )suchas Lip A ( X, α ), Lip n ( X, α ) and Lip( X, M, α ). BSE properties of Bloch type spaces $${{\cal B}_\alpha}$$

In this paper, we study q-difference analogues of several central results in value distribution theory of several complex variables such as q-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura–Clunie theorem concerning partial q-difference polynomials is also obtained. Finally, we apply

We investigate the weighted approximation of functions in Lp-norm by Kantorovich modifications of the classical Baskakov operator, with weights of type (1+x)α, α ∈ ℝ. By defining an appropriate K-functional we prove direct inequalities for them.

The paper is an investigation of the analytic properties of a new class of special functions that appear in the kernels of a class of integral operators underlying the dynamics of matter relaxation processes in attractive fields. These functions, recently introduced by the second author, generate the kernels of the principal parts of these operators and play an important role in understanding their

This paper is devoted to studying the boundedness and compactness of commutators for Marcinkiewicz integrals with rough kernels in weighted Lebesgue spaces. The characterized theorems on the boundedness and compactness for such commutators are established, respectively. We improve and extend the previous results. Meanwhile, the quantitative weighted bounds for the commutator of Marcinkiewicz integral

We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let $N$ be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let $(\pi, \mathcal{H}_{\pi})$ be an irreducible, square-integrable representation modulo the center $Z(N)$ of $N$ on a Hilbert space $\mathcal{H}_{\pi}$ of formal dimension $d_\pi

Sendov's conjecture states that if all the zeroes of a complex polynomial $P(z)$ of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of $P(z)$. In a paper that appeared in 2014, D\'{e}got proved that, for each $a\in (0,1)$, there exists an integer $N$ such that for any polynomial $P(z)$ with degree greater than $N$, if $P(a) = 0$ and all zeroes

In this paper, we study the problem of approximation to reconstruct the value of a function of several variables from the approximate values of the Fourier coefficients of a function at a given fixed point. A theorem on the reconstruction of the value of a function with several variables at any given fixed point with a small error is given.

We show that, for each compact subset of the real line of infinite cardinality with an isolated point, the space of Whitney jets on the set does not possess a basis consisting only of polynomials. On the other hand, polynomials are dense in any Whitney space. Thus, there are no general results about stability of bases in Frechet spaces.

Let f(z) be a non-constant meromorphic (entire) function of hyper-order strictly less than 1, n ≥ 3 (n ≥ 2) be an integer. It is shown that if fn(z) and fn(z + c) share a(≠ 0) ∈ ℂ and ∞ CM, then f(z) = t1f(z + c) or f(z) = t2f(z + 2c), where t1 and t2 satisfy $$t_i^n=1$$ , (i = 1, 2). Some examples are provided to show the sharpness of our results. In addition, we mainly obtain two uniqueness results

We show the following result: Let A,B ∈ B (H) be two strictly positive operators such that A ≤ B and m1H ≤ B ≤ M1H for some scalars 0 < m < M . Then B ≤ exp ( M1H −B M −m lnm + B −m1H M −m lnM ) ≤ K (m,M, p, q)A for p ≤ 0,−1 ≤ q ≤ 0 where K (m,M, p, q) is the generalized Kantorovich constant with two parameters. In addition, we obtain Kantorovich type inequalities for the chaotic order.

In this article we study the probability that the maximum over a symmetric interval [−a, a] of a univariate polynomial of degree at most n is attained at an endpoint. We give explicit formulas for the degree n = 1, 2,3 cases. The formula for the degree 3 case leads to a lower bound for the true probability. Numerical experiments indicate that this lower bound is actually quite a good approximation

We carry out complete membership to Kaplan classes of polynomials with all zeros on unit circle. In this way we extend Sheil-Small’s and Jahangiri’s theorems.

Using a local analog of the Wiener-Levi theorem, we investigate the class of measures on Euclidean space with discrete support and spectrum. Also, we find a new sufficient conditions for a discrete set in Euclidean space to be a coherent set of frequencies.

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

We continue the study in the setting of pluripotential theory arising from polynomials associated to a convex body $C$ in $({\bf R}^+)^d$. Here we discuss $C-$Robin functions and their applications. In the particular case where $C$ is a simplex in $({\bf R}^+)^2$ with vertices $(0,0),(b,0),(a,0)$, $a,b>0$, we generalize results of T. Bloom to construct families of polynomials which recover the $C-$extremal

In this paper, we study a Trudinger-Moser inequality with a vanishing weight in the unit disk. Precisely, let $$B$$ B be the unit disk and β ≥ 0 be a real number. Denote $${\cal S} = \left\{{u \in W_0^{1,2}\left(B\right);u} \right.$$ S = { u ∈ W 0 1 , 2 ( B ) ; u is a radially symmetric function and ∥∇ u ∥ 2 ≤ 1 . Suppose a function h ( x ) is radially symmetric, nonnegative, continuous on $$\overline{\mathbb{B}}$$

We discuss the problem of the best uniform approximation on the axis, i.e., in the space C(−∞,∞), of the differentiation operator of order k on the class of functions with bounded derivative of order n, 0 < k < n, by bounded linear operators in the space Lr,1 ≤ r < ∞. We give an exact solution of the problem for odd n ≥ 3 and all k, 0 < k < n. For even n, a result close to the best one is obtained

The sum of a sine series $$g\left({b,x} \right) = \sum\nolimits_{k = 1}^\infty {}$$ bk sin kx with coefficients forming a convex sequence b is known to be positive on the interval (0,π). To estimate its values near zero Telyakovskiĭ used the piecewise-continuous function $$\sigma \left({{\bf{b}},x} \right) = \left({1/m\left(x \right)} \right)\sum\nolimits_{k = 1}^{m\left(x \right) - 1} {{k^2}\left({{b_k}

The well-known stationary phase formula gives us a way to find asymptotics of oscillating integrals so long as the symbol is regular enough (in comparison to the large parameter controlling the oscillation). However in a number of applications we find ourselves with symbols that are not suitably regular. In this paper we obtain decay bounds for such oscillatory integrals.