CHARACTERIZATIONS OF UNIFORM CONVEXITY FOR DIFFERENTIABLE FUNCTIONS - Alabdali, Osama; Guessab, Allal; Schmeißer, Gerhard (Applicable Analysis and Discrete Mathematics, 2019)
We consider convex functions in d real variables. For applications, for example in optimization, various strengthened forms of convexity have been introduced. Among them, uniform convexity is one of the most general, defined by involving a so-called modulus phi. Inspired by three classical characterizations of ordinary convexity, we aim at characterizations of uniform convexity by conditions in terms of the gradient or the Hessian matrix of the considered function for certain classes of moduli phi.
CRIS-Team WoS Importer:2020-01-07

Classical and Approximate Exponential Sampling Formula: Their Interconnections in Uniform and Mellin–Lebesgue Norms - Bardaro, C.; Butzer, P. L.; Mantellini, I.; Schmeißer, Gerhard; Stens, R. L. (Birkhauser, 2023)
We show that the exponential sampling theorem and its approximate version for functions belonging to a Mellin inversion class are equivalent in the sense that, within the setting of Mellin analysis, each can be obtained from the other as a corollary. The approximate version is considered for both, convergence in the uniform norm and in the Mellin–Lebesgue norm. An important tool is the introduction of a Mellin version of the mixed Hilbert transform and its continuity properties. Our chapter extends the analogous equivalence between the classical and the approximate sampling theorem of Fourier analysis.
CRIS-Team Scopus Importer:2024-12-16
Keywords: Approximate exponential sampling formula; Exponential sampling formula; Mellin bandlimited functions; Mellin inversion classes; Mixed Mellin–Hilbert transform

Development of a new concept of polar analytic functions useful in Mellin analysis - Bardaro, Carlo; Butzer, Paul L.; Mantellini, Ilaria; Schmeißer, Gerhard (Complex Variables and Elliptic Equations, 2019)
In this paper, we develop the concept of polar analyticity introduced in Bardaro C, et al. [A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces. Math Nachr. 2017;290:2759–2774] and successfully applied in Mellin analysis and in quadrature of functions defined on the positive real axis (see Bardaro C, et al. [Quadrature formulae for the positive real axis in the setting of Mellin analysis: sharp error estimates in terms of the Mellin distance]. Calcolo. 2018;55(3):26. Available from: https://doi.org/10.1007/s10092-018-0268-1]). This appears as a simple way to describe functions which are analytic on a part of the Riemann surface of the logarithm. We study analogues of Cauchy's integral theorems for polar-analytic functions and obtain two series expansions in terms of polar-derivatives and Mellin polar-derivatives, respectively. We also describe some geometric properties of polar-analytic functions related to conformality. By these studies, we launch the proposal to develop a complete complex function theory, independent of the classical function theory, which is built upon the new notion of polar analyticity.
CRIS-Team Scopus Importer:2019-08-06
Keywords: Cauchy's integral formula; conformality; D. Khavinson; Morera's theorem; Polar-analytic functions; Taylor's expansion

Exponential Sampling Series: Convergence in Mellin-Lebesgue Spaces - Bardaro, Carlo; Mantellini, Ilaria; Schmeißer, Gerhard (Results in Mathematics, 2019)
In this paper we study norm-convergence to a function f of its generalized exponential sampling series in weighted Lebesgue spaces. Key roles are taken by a result on the norm-density of the test functions and the notion of bounded coarse variation. Some examples are described.
CRIS-Team WoS Importer:2019-06-18

Exponential sampling with a multiplier - Bardaro, Carlo; Mantellini, Ilaria; Schmeißer, Gerhard (Sampling Theory, Signal Processing, and Data Analysis, 2023)
The exponential sampling formula has some limitations. By incorporating a Mellin bandlimited multiplier, we extend it to a wider class of functions with a series that converges faster. This series is a generalized exponential sampling series with some interesting properties. Moreover, under a side condition, any generalized exponential sampling series that is interpolating can be generated by a Mellin bandlimited multiplier. For an error analysis, we consider a truncated series with 2 N+ 1 terms and look for a highest speed of convergence as N→ ∞. We show by using a certain non-bandlimited multiplier, which introduces in addition an aliasing error, that we can achieve a higher rate of convergence to the function, namely O(e-αN) with α> 0 , than with the truncated series of an exact formula. The results are illustrated by three examples.
CRIS-Team Scopus Importer:2023-02-17
Keywords: Exponential sampling; Mellin–Bernstein spaces; Mellin–Paley–Wiener spaces; Multipliers; Polar-analytic functions

Integration of polar-analytic functions and applications to Boas’ differentiation formula and Bernstein’s inequality in Mellin setting - Bardaro, Carlo; Butzer, Paul L.; Mantellini, Ilaria; Schmeißer, Gerhard (Bolletino dell Unione Matematica Italiana, 2020)
We establish a general version of Cauchy’s integral formula and a residue theorem for polar-analytic functions, employing the new notion of logarithmic poles. As an application, a Boas-type differentiation formula in Mellin setting and a Bernstein-type inequality for polar Mellin derivatives are deduced.
CRIS-Team Scopus Importer:2020-06-09
Keywords: Boas’ differentiation formula; Cauchy’s integral formulae; Identity theorems; Logarithmic poles; Polar-analytic functions

Polar-Analytic Functions: Old and New Results, Applications - Bardaro, Carlo; Butzer, Paul L.; Mantellini, Ilaria; Schmeißer, Gerhard (Results in Mathematics, 2022)
Here we review the notion of polar analyticity introduced in a previous paper and successfully applied in Mellin analysis and for quadrature formulae over the positive real axis. This approach provides a simple way of describing functions which are analytic on a part of the Riemann surface of the logarithm. New results are also obtained.
CRIS-Team WoS Importer:2022-02-11

Valiron’s Interpolation Formula and a Derivative Sampling Formula in the Mellin Setting Acquired via Polar-Analytic Functions - Bardaro, Carlo; Butzer, Paul L.; Mantellini, Ilaria; Schmeißer, Gerhard (Computational Methods and Function Theory, 2020)
In this paper, we first recall some recent results on polar-analytic functions. Then we establish Mellin analogues of a classical interpolation of Valiron and of a derivative sampling formula. As consequences a new differentiation formula and an identity theorem in Mellin–Bernstein spaces are obtained. The main tool in the proofs is a residue theorem for polar-analytic functions.
CRIS-Team Scopus Importer:2020-08-21
Keywords: Cauchy’s integral formulae; Exponential formula for derivatives; Identity theorem; Logarithmic poles; Mellin–Bernstein spaces; Polar-analytic functions; Valiron’s interpolation formula