Abstract: In the present paper, we study the saturation order in the space \(L^1({\mathbb {R}})\) for the sampling Kantorovich series based upon bandlimited kernels. The above study is based on the so-called Fourier transform method, introduced in 1960 by P. L. Butzer. As a first result, the saturation order is derived in a Bernstein class; here, it is crucial to derive the Fourier transform of the above sampling-type series, which can be expressed in a suitable closed form. Subsequently, the saturation is reached in the whole space \(L^1({\mathbb {R}})\) . At the end of the paper, several examples of bandlimited kernels, such as the Fejér’s and Bochner–Riesz’s kernel, have been recalled and the saturation order of the corresponding sampling Kantorovich series has been stated. PubDate: 2019-07-13

Abstract: We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it is to date the only extension to the bilinear setting of the recent Carleson embedding theorem by Culiuc and Treil that features a matrix Carleson measure and a matrix weight. It is well known that a Carleson embedding theorem implies a Doob’s maximal inequality and this holds true in the matrix weighted setting with an appropriately defined maximal operator. It is also known that a dimensional growth must occur in the Carleson embedding theorem with matrix Carleson measure, even with trivial weight. We give a definition of a maximal type function whose norm in the matrix weighted setting does not grow with dimension. PubDate: 2019-06-27

Abstract: We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearities. In this paper the concave term will be parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter \(\lambda \) varies. PubDate: 2019-06-24

Abstract: The original version of the article unfortunately contained few errors under Preliminaries section. The corrected text is given below. PubDate: 2019-06-18

Abstract: The boundary value problem for the first-order integro-differential equation is considered with the periodic boundary condition, polynomially dependent on the spectral parameter. The inverse problem is studied, which consists in reconstruction of the convolution kernel and the polynomial in the boundary condition, by using the spectrum. We obtain (1) uniqueness, (2) a constructive procedure for solution, (3) necessary and sufficient conditions for solvability of the inverse problem. PubDate: 2019-06-10

Abstract: We study non-regularity of growth of the fractional Cauchy transform $$\begin{aligned} f(z)=\int _{-\pi }^{\pi } \frac{d\psi (t)}{(1-ze^{-it})^\alpha }, \quad \alpha >0, \psi \in BV[-\pi ,\pi ], \end{aligned}$$ in terms of the modulus of continuity of the function \(\psi \) . Sharp estimates of the lower logarithmic order of f are found. In the case \(\alpha \in (0,1)\) the estimates are of different form than that for the logarithmic order. PubDate: 2019-06-06

Abstract: In this article we present a unified treatment for obtaining bounds on the potential energy of codes in the general context of polynomial metric spaces (PM-spaces). The lower bounds we derive via the linear programming techniques of Delsarte and Levenshtein are universally optimal in the sense that they apply to a broad class of energy functionals and, in general, cannot be improved for the specific subspace. Tests are presented for determining whether these universal lower bounds (ULB) can be improved on larger spaces. Our ULBs are applicable on the Euclidean sphere, infinite projective spaces, as well as Hamming and Johnson spaces. Asymptotic results for the ULB for the Euclidean spheres and the binary Hamming space are derived for the case when the cardinality and dimension of the space grow large in a related way. Our results emphasize the common features of the Levenshtein’s universal upper bounds for the cardinality of codes with given separation and our ULBs for energy. We also introduce upper bounds for the energy of designs in PM-spaces and the energy of codes with given separation. PubDate: 2019-06-06

Abstract: We discuss the notion of an inner function for spaces of analytic functions in multiply connected domains in \({\mathbb {C}}\) , giving a historical overview and comparing several possible definitions. We explore connections between inner functions, zero-divisors for Hardy spaces and Bergman spaces, and weighted reproducing kernels. After recording some obstructions and negative results, we suggest avenues for further research and point out several open problems. PubDate: 2019-06-06

Abstract: In this paper we prove two properties of the weighted Hardy space for the unit disc with the weight function satisfying the Muckenhoupt condition. PubDate: 2019-05-25

Abstract: Given a system of functions \({\mathbf {f}}=(f_1,\ldots ,f_d)\) analytic on a neighborhood of some compact subset E of the complex plane with simply connected complement in the extended complex plane, we give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of row sequences of multipoint Hermite–Padé approximants under a general extremal condition on the table of interpolation points. The exact rate of convergence of these denominators is provided and the rate of convergence of the simultaneous approximants is estimated. These results allow us to detect the location of the poles of the system of functions which are in some sense closest to E. PubDate: 2019-05-25

Abstract: It is well known that the notions of domain of holomorphy and weak domain of holomorphy are equivalent. If \(X({\varOmega })\) is a space of holomorphic functions we extend these notions to \(X({\varOmega })\) -domain of holomorphy and weak \(X({\varOmega })\) -domain of holomorphy. For several function spaces \(X(\varOmega )\) , satisfying weak assumptions, we prove that the notions of \(X(\varOmega )\) -domain of holomorphy and weak \(X(\varOmega )\) -domain of holomorphy are equivalent and that in this case the set of non-extendable functions in \(X({\varOmega })\) is a dense \(G_\delta \) -subset of \(X({\varOmega })\) . Similar results are obtained for the stronger notion of total unboundedness. Finally we provide examples of new spaces \(X({\varOmega })\) , where all the above hold. Mainly they are localized versions of classical function spaces and combinations of them. PubDate: 2019-05-24

Abstract: Ever since Armitage proved that every nonnegative superharmonic function in a bounded domain of bounded curvature in \({{\mathbb {R}}^n}\) is \(L^p\) -integrable for \(0<p<n/(n-1)\) , the global integrability of nonnegative supersolutions has attracted many mathematicians. Illustrating related potential theoretic notions, we show how this problem has been settled and extended to the parabolic case. PubDate: 2019-05-24

Abstract: Brosamler’s formula gives a probabilistic representation of the solution of the Neumann problem for the Laplacian on a smooth bounded domain \(D\subset \mathbb {R}^n\) in terms of the reflecting Brownian motion in D. The original proof, as well as other proofs in the literature (e.g., in the case of Lipschitz domains), are based on potential theory (transition densities of the reflecting Brownian motion). We give new proofs of Brosamler’s formula using (path trajectories of) stochastic processes. More precisely, we use a connection between the Dirichlet and the Neumann boundary problems, and the explicit description of the reflecting Brownian motion and its boundary local time in terms of the free Brownian motion. The results are obtained in the case of the Euclidean unit ball in any dimension and in the case of smooth \(C^{1,\alpha }\) planar simply connected domains, for continuous boundary data, and then extended to the case of bounded measurable data, respectively integrable boundary data. A new Brosamler-type formula in terms of the free Brownian motion is also given. PubDate: 2019-05-23

Abstract: It is known that, generically in the space \(H({\mathbb {D}})\) of functions holomorphic in the unit disc \({\mathbb {D}}\) , the sequences \((S_n f)\) of partial sums of Taylor series behave extremely erratically on the unit circle \({\mathbb {T}}\) . According to a result of Gardiner and Manolaki, the situation changes in a significant way if \(f \in H({\mathbb {D}})\) has nontangential limits on subsets of \({\mathbb {T}}\) of positive arc length measure. In this case each convergent subsequence tends to the nontangential limit function almost everywhere. We consider the question to which extent in spaces of holomorphic functions where nontangential limits are guaranteed, “spurious” limit functions, that is, limit functions different than the nontangential limit may appear on small subsets of \({\mathbb {T}}\) . PubDate: 2019-05-23

Abstract: Let U be a bounded open subset of the complex plane. Let \(0<\alpha <1\) and let \(A_{\alpha }(U)\) denote the space of functions that satisfy a Lipschitz condition with exponent \(\alpha \) on the complex plane, are analytic on U and are such that for each \(\epsilon >0\) , there exists \(\delta >0\) such that for all z, \(w \in U\) , \( f(z)-f(w) \le \epsilon z-w ^{\alpha }\) whenever \( z-w < \delta \) . We show that if a boundary point \(x_0\) for U admits a bounded point derivation for \(A_{\alpha }(U)\) and U has an interior cone at \(x_0\) then one can evaluate the bounded point derivation by taking a limit of a difference quotient over a non-tangential ray to \(x_0\) . Notably our proofs are constructive in the sense that they make explicit use of the Cauchy integral formula. PubDate: 2019-05-23

Abstract: According to the Schwarz symmetry principle, every harmonic function vanishing on a real-analytic curve has an odd continuation, while a harmonic function satisfying homogeneous Neumann condition has an even continuation. Using a technique of Dirichlet to Neumann and Robin to Neumann operators, we derive reflection formulae for non-homogeneous Neumann and Robin conditions from a reflection formula subject to a non-homogeneous Dirichlet condition. PubDate: 2019-05-23

Abstract: Let X be a Banach holomorphic function space on the unit disk. A linear polynomial approximation scheme for X is a sequence of bounded linear operators \(T_n:X\rightarrow X\) with the property that, for each \(f\in X\) , the functions \(T_n(f)\) are polynomials converging to f in the norm of the space. We completely characterize those spaces X that admit a linear polynomial approximation scheme. In particular, we show that it is not sufficient merely that polynomials be dense in X. PubDate: 2019-05-22

Abstract: A logharmonic mapping f is a mapping that is a solution of the nonlinear elliptic partial differential equation \(\dfrac{\overline{f_{ \overline{z}}}}{\overline{f}}=a\dfrac{f_{z}}{f}\) . In this paper we investigate the univalence of logharmonic mappings of the form \( f=zH\overline{G},\) where H and G are analytic on a linearly connected domain. We discuss the relation with the univalence of its analytic counterparts. Stable Univalence and its consequences are also considered. PubDate: 2019-05-22