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

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 δE 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

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

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 C1 in an open set U ⊂ ℝN+m, \({\cal M}\) be a N-dimensional C1 submanifold of U and define \({\cal T}: = \left\{{z \in {\cal M}:H\left(z \right),K\left(z \right) \in {T_z}{\cal M}} \right\}\). Then the obvious property If z0 ∈ \({\cal M}\) is an interior point (relative to \({\cal M}\))of \({\cal T}\) then \(\left[{H,K} \right]\left({{z_0}} \right) \in

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 \({V_n}\left(x \right) = \sum\nolimits_{j = 0}^n {{a_j}\cos \left({jx} \right)} \), x ∈ (0, 2π), with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables (asymptotically) has \(2n/\sqrt 3 \) expected real roots. On the other hand, out of many ways to construct a dependent random polynomial

In this paper, BSE properties of some Banach function algebras are studied. We show that Lipschitz algebras Lipα (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 Lipα(X, d)suchas LipA (X, α), Lipn (X, α) and Lip(X, M, α). BSE properties of Bloch type spaces \({{\cal B}_\alpha}\) and Zygmund

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.

We investigate 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 spectral characteristics

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 (π,Hπ) be an irreducible, square-integrable representation modulo the center Z(N) of N on a Hilbert space Hπ of formal dimension dπ. If g ∈ Hπ is an integrable vector and the set

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, Dégot proved that, for each a ∈ (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 lie inside the unit

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 Fréchet spaces.

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.

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 \(A, B \in \mathbb{B}(\mathcal{H})\) be two strictly positive operators such that A ≤ B and \(m1_{\mathcal{H}}\leq{B}\leq{M1_{\mathcal{H}}}\) for some scalars 0 < m < M. Then $$B^p\leq{\rm{exp}}(\frac{M1_{\mathcal{H}}-B}{M-m}{\rm{ln}} \;m^p+\frac{B-m1_{\mathcal{H}}}{M-m}{\rm{ln}} \;M^p)\leq{K(m, M, p, q)}A^q$$ for p ≤ 0, −1 ≤ q ≤ 0 where K(m, M, p, q) is the generalized

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 analogue 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 in [1] in the setting of pluripotential theory arising from polynomials associated to a convex body C in (ℝ+)d. Here we discuss C-Robin functions and their applications. In the particular case where C is a triangle in (ℝ+)2 with vertices (0, 0), (b, 0), (0, a), a, b > 0, we generalize results of T. Bloom to construct families of polynomials which recover the C-extremal function

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

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

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.

In this paper, a new characterization is provided for the boundedness, compactness and essential norm of the difference of two weighted composition operators on weighted-type spaces in the unit ball of ℂn.

In this paper we study analogues of the perfect splines for weighted Sobolev classes of functions defined on the half-line. Maximally oscillating splines play important role in the solution of certain extremal problems. In particular, using these splines, we characterize the modulus of continuity of the differential operator.

Let ℋ and \({\cal K}\) be complex infinite dimensional separable Hilbert spaces. We denote by \({M_C} = \left( {\begin{array}{*{20}{c}} A&C \\ 0&B \end{array}} \right)\) a 2 × 2 upper triangular operator matrix acting on \({\cal H} \oplus {\cal K}\), where \(A \in {\cal B}\left({\cal H} \right),\,B \in {\cal B}\left({\cal K} \right)\) and \(C \in {\cal B}\left({{\cal K},{\cal H}} \right)\). In this

This paper is a follow up of a recent work of the author and Y. Sawano [6], in 2019. We investigate the boundedness of the bilinear fractional integral operator introduced by Grafakos in [3], 1992. When the local integrability index s falls 1 with weights and t exceeds 1, He and Yan obtained some results on this operator was worked on Morrey spaces earlier in [7] 2018. Later in the paper [6] we considered

Steinmetz [16] considered the first order non-linear differential equations$$C(z, f)(f^\prime)^2+B(z, f)f^\prime+A(z, f)=0,$$where A(z, f), B(z, f), C(z, f) are polynomials in f with rational coefficients in z and pointed out that the above equation must reduce into some certain types when it admits transcendental meromorphic solutions. In this paper, we will consider its difference version$$C(z, f)f(z+c)^2+B(z

We consider the Hankel operator with symbols of forms and study the relationship between the compactness of this operator and the compactness of ∂̄-Neumann operator.

In this paper, B-Fredholm linear relations in Hilbert spaces are introduced and some of their properties are given. In particular, it is shown that they are completely characterized in terms of an algebraic decomposition with a Fredholm linear relation and a bounded nilpotent operator. The behaviour of a polynomial in them is also investigated.

We provide sufficient conditions on the wavelets on a local field K for the wavelet system to form an unconditional basis for the Hardy space H1(K) and the Lebesgue spaces Lp(K), 1 < p < ∞.

Let \(\mathcal{B}(\mathcal{H})\) be the algebra of all bounded linear operators on a separable complex Hilbert space \(\mathcal{H}\). We introduce the J-decomposition property for projections in \(\mathcal{B}(\mathcal{H})\), and prove that the projection E in \(\mathcal{B}(\mathcal{H})\) has J-decomposition property with respect to a particular space decomposition, which is related to Hal-mos’ two

We give an exposition of some simple but applicable cases of worst-case growth of a polynomial in terms of its uniform norm on a given compact set K ⊂ ℂd. Included is a direct verification of the formula for the pluripotential extremal function for a real simplex. Throughout we attempt to make the exposition as accessible to a general (analytic) audience as possible, avoiding wherever possible the

We prove a Hille–Yosida type theorem for relatively uniformly continuous positive semigroups on vector lattices. We introduce the notions of relatively uniformly continuous, differentiable, and integrable functions on ℝ+. These notions allow us to study the generators of relatively uniformly continuous semigroups. Our main result provides sufficient and necessary conditions for an operator to be the

In this paper we first introduce the Heron and Heinz means of two convex functionals. Afterwards, some inequalities involving these functional means are investigated. The operator versions of our theoretical functional results are immediately deduced. We also obtain new refinements of some known operator inequalities via our functional approach in a fast and nice way.

The paper addresses questions on existence and structure of universal functions for spaces L1(E), E ⊂ [0, 1)2, with respect to the doubleWalsh system in the sense of signs of Fourier coefficients. It is shown that for each ε > 0 one can find a measurable set Eε ⊂ [0, 1)2 with measure |Eε| > 1−ε, such that by a proper modification of any integrable function f ∈ L1[0, 1)2 outside Eε one can get an integrable

For a locally compact group G, let \(\mathcal{L}^{\Phi}\)(G) and \(\mathcal{L}_\omega^{\Phi}\)(G) be Orlicz and weighted Orlicz spaces, respectively, where Φ is a Young function and ω is a weight on G. We study the harmonic and convolution operators on Orlicz and weighted Orlicz spaces. We prove that under some conditions the harmonic operators on \(\mathcal{L}^{\Phi}\)(G) and \(\mathcal{L}_\omega^{\Phi}\)(G)