Jürgen Prestin | University of Lübeck (original) (raw)

Papers by Jürgen Prestin

Journal of Applied Mathematics and Mechanics, 1996

. In this paper, localization properties of trigonometric polynomial Hermite operators are discus... more . In this paper, localization properties of trigonometric polynomial Hermite operators are discussed. In particular, time frequency uncertainty and operator norms are compared for the different types of fundamental interpolants which serve as scaling functions for a trigonometric multiresolution analysis. x1. Introduction Recently, several different approaches to periodic multiresolution analyses have been presented. For example, periodic scaling functions and wavelets are discussed by Narcowich and Ward [3] who investigate their time frequency behaviour in terms of an uncertainty principle for periodic functions due to Breitenberger [1]. The periodic basis functions in [3] possess an uncertainty product of O( p n) for increasing dimension n of the corresponding spaces. On the other hand, uniformly bounded uncertainty products are computed by Selig [6] for trigonometric fundamental Lagrange interpolants based on de la Vall'ee Poussin means. A multiresolution analysis generated b...

Applied and numerical harmonic analysis, 2017

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) ... more Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) n−l−1 (r 2)r l Y lm (ϑ , ϕ), |m| ≤ l < n ∈ N, L (l+1/2) n−l−1 being a generalized Laguerre polynomial, Y lm a spherical harmonic, constitute an orthonormal basis of the space L 2 on R 3 with Gaussian weight exp(−r 2). These basis functions are used extensively, e.g., in biomolecular dynamic simulations. However, to the present, there is no reliable algorithm available to compute the Fourier coefficients of a function with respect to the SGL basis functions in a fast way. This paper presents such generalized FFTs. We start out from an SGL sampling theorem that permits an exact computation of the SGL Fourier expansion of bandlimited functions. By a separation-of-variables approach and the employment of a fast spherical Fourier transform, we then unveil a general class of fast SGL Fourier transforms. All of these algorithms have an asymptotic complexity of O(B 4), B being the respective bandlimit, while the number of sample points on R 3 scales with B 3. This clearly improves the naive bound of O(B 7). At the same time, our approach results in fast inverse transforms with the same asymptotic complexity as the forward transforms. We demonstrate the practical suitability of our algorithms in a numerical experiment. Notably, this is one of the first performances of generalized FFTs on a non-compact domain. We conclude with a discussion, including the layout of a true O(B 3 log 2 B) fast SGL Fourier transform and inverse, and an outlook on future developments.

Citeseer

Ich erkläre hiermit, dass ich die vorliegende Arbeit selbstständig verfasst und keine anderen als... more Ich erkläre hiermit, dass ich die vorliegende Arbeit selbstständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. ... 2.3 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Spherical Radial Functions . . . . . . . . . . . . . . . . . . . . . . . ...

arXiv (Cornell University), Dec 17, 2017

We obtain estimates of the Lp-error of the bivariate polynomial interpolation on the Lissajous-Ch... more We obtain estimates of the Lp-error of the bivariate polynomial interpolation on the Lissajous-Chebyshev node points for wide classes of functions including non-smooth functions of bounded variation in the sense of Hardy-Krause. The results show that Lperrors of polynomial interpolation on the Lissajous-Chebyshev nodes have almost the same behavior as the polynomial interpolation in the case of the tensor product Chebyshev grid.

arXiv (Cornell University), Apr 6, 2017

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) ... more Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) n−l−1 (r 2)r l Y lm (ϑ, ϕ), |m| ≤ l < n ∈ N, constitute an orthonormal polynomial basis of the space L 2 on R 3 with radial Gaussian weight exp(−r 2). We have recently

arXiv (Cornell University), Feb 16, 2016

We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding t... more We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding to the generalized Jacobi translation and the Ditzian-Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer-Bernstein polynomial operators.

Mitteilungen der Deutschen mathematiker-Vereinigung, Dec 30, 2022

arXiv (Cornell University), Apr 18, 2016

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) ... more Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) n−l−1 (r 2)r l Y lm (ϑ , ϕ), |m| ≤ l < n ∈ N, L (l+1/2) n−l−1 being a generalized Laguerre polynomial, Y lm a spherical harmonic, constitute an orthonormal basis of the space L 2 on R 3 with Gaussian weight exp(−r 2). These basis functions are used extensively, e.g., in biomolecular dynamic simulations. However, to the present, there is no reliable algorithm available to compute the Fourier coefficients of a function with respect to the SGL basis functions in a fast way. This paper presents such generalized FFTs. We start out from an SGL sampling theorem that permits an exact computation of the SGL Fourier expansion of bandlimited functions. By a separation-of-variables approach and the employment of a fast spherical Fourier transform, we then unveil a general class of fast SGL Fourier transforms. All of these algorithms have an asymptotic complexity of O(B 4), B being the respective bandlimit, while the number of sample points on R 3 scales with B 3. This clearly improves the naive bound of O(B 7). At the same time, our approach results in fast inverse transforms with the same asymptotic complexity as the forward transforms. We demonstrate the practical suitability of our algorithms in a numerical experiment. Notably, this is one of the first performances of generalized FFTs on a non-compact domain. We conclude with a discussion, including the layout of a true O(B 3 log 2 B) fast SGL Fourier transform and inverse, and an outlook on future developments.

Ukrainian Mathematical Journal, Oct 1, 2017

UDC 517.5 We prove direct and inverse theorems on the approximation of 2⇡-periodic functions by T... more UDC 517.5 We prove direct and inverse theorems on the approximation of 2⇡-periodic functions by Taylor-Abel-Poisson operators in the integral metric.

arXiv (Cornell University), Feb 11, 2020

We study approximation properties of general multivariate periodic quasiinterpolation operators, ... more We study approximation properties of general multivariate periodic quasiinterpolation operators, which are generated by distributions/functions ϕj and trigonometric polynomials ϕj. The class of such operators includes classical interpolation polynomials (ϕj is the Dirac delta function), Kantorovich-type operators (ϕj is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on ϕj and ϕj , we obtain upper and lower bound estimates for the Lp-error of approximation by quasi-interpolation operators in terms of the best and best one-sided approximation, classical and fractional moduli of smoothness, K-functionals, and other terms.

Gem - International Journal on Geomathematics, Mar 23, 2019

Journal of Mathematical Analysis and Applications

arXiv (Cornell University), Oct 30, 2022

Approximative properties of the Taylor-Abel-Poisson linear summation method of Fourier series are... more Approximative properties of the Taylor-Abel-Poisson linear summation method of Fourier series are considered for functions of several variables, periodic with respect to the hexagonal domain, in the integral metric. In particular, direct and inverse theorems are proved in terms of approximations of functions by the Taylor-Abel-Poisson means and K-functionals generated by radial derivatives. Bernstein type inequalities for L 1-norm of high-order radial derivatives of the Poisson kernel are also obtained.

Frontiers in Applied Mathematics and Statistics, 2020

Mathematics and Computers in Simulation, 2016

Different mathematical models are built of a selection of mechanisms, and they reproduce observat... more Different mathematical models are built of a selection of mechanisms, and they reproduce observations in a quantitatively different manner. A suitable error functional is used to compare the models and to detect mechanisms, which probably caused the observations. For this aim, parameter identification oftentimes is seen as the determination of a best approximation out of the set of feasible solutions of the model, which can be identified with the set. In this paper, the comparison of different model approaches is discussed with respect to observation data from a disease that occurred 1988 in Chernivtsi in Ukraine. The cause of the disease remained unclear. The quantitative measure of the error functional and selected qualitative properties are used to distinguish the models. Even though only a small set of data for the number of affected persons is available, a comparison of a contamination model and an epidemical model suggest that the cause of the disease is rather an infection than an exposure to environmental toxin.

Cybernetics and Systems Analysis, Jul 1, 2018

The properties of wavelets based on Jacobi polynomials are analyzed. The conditions are considere... more The properties of wavelets based on Jacobi polynomials are analyzed. The conditions are considered under which these wavelets are mutually orthogonal and under which the wavelet basis is characterized by a minimum Riesz ratio. These problems lead to the solution of systems of nonlinear equations by a method proposed earlier by the authors.

Reports of the National Academy of Sciences of Ukraine

Let ε be the set of all entire functions on the complex plane C. Let us consider the class XE of ... more Let ε be the set of all entire functions on the complex plane C. Let us consider the class XE of all complex Banach spaces X such that X ⊇ ε . For (X, ⎥⎥ ⋅ ⎥⎥)∈XE and g ∈X we write En, X (g ) = inf {⎥⎥ g − p⎥⎥: p∈Πn }, where Πn is the set of all polynomials with degree at most n. We describe all X ∈XE for which the relation lim n→∞ (En, X( g ))1/n = 0 holds if and only if g ∈ ε.

We obtain estimates of the Lp-error of the bivariate polynomial interpolation on the Lissajous-Ch... more We obtain estimates of the Lp-error of the bivariate polynomial interpolation on the Lissajous-Chebyshev node points for wide classes of functions including non-smooth functions of bounded variation in the sense of Hardy-Krause. The results show that Lperrors of polynomial interpolation on the Lissajous-Chebyshev nodes have almost the same behavior as the corresponding polynomial interpolation in the case of the tensor product Chebyshev grid.

Mitteilungen der Deutschen Mathematiker-Vereinigung, 2019

Die . Internationale Mathematik-Olympiade (IMO) fand vom . bis zum . Juli  in Bath stat... more Die . Internationale Mathematik-Olympiade (IMO) fand vom . bis zum . Juli  in Bath statt, einer zum Weltkulturerbe der UNESCO zählenden malerischen Universitätsstadt im Westen Englands.  Schülerinnen und  Schüler aus  Ländern nahmen an dieser Olympiade teil. Sie war damit die größte IMO in der Geschichte. Erfreulich ist das Wachstum der letzen  Jahre. Der damalige Rekord der . IMO  in Deutschland mit  Ländern und  Teilnehmerinnen und  Teilnehmern ist inzwischen weit übertroffen. Die deutsche Mannschaft bestand aus sechs Schülern, Dr. Eric Müller als stellvertretendem Delegationsleiter, Dr. Christian Reiher als Observer A und dem Berichterstatter als Delegationsleiter. Alle sechs Teilnehmer haben schon an vielen Mathematik-Wettbewerben teilgenommen. Für Lukas Finn Groß war es die zweite Teilnahme an einer IMO und für Jonas Walter sogar schon die dritte Teilnahme. Letzterer hat insgesamt viermal die IMO-Vorbereitungslehrgänge durchlaufen. Maximilian Göbel, Lukas Finn Groß und Maximilian Keßler waren schon seit  dabei. Maximilian Keßler lebt seit vier Jahren in Barcelona und musste daher zu den Vorbereitungslehrgängen jeweils aus Spanien anreisen. Berichtenswert ist auch, dass Jonas Walter in seiner Freizeit klassische Gitarre spielt und schon erfolgreich bei Jugend Musiziert teilgenommen hat. Lukas Finn Groß war von - Mitglied in einem Schwimmverein und sehr erfolgreich bei Wettbewerben in NRW, besonders auf der  m-Strecke. Alle fünf Abiturienten wollen im Wintersemester ein Mathematik-Studium beginnen, Maximilian Keßler an der LMU in München, die anderen vier in Bonn.

Journal of Applied Mathematics and Mechanics, 1996

. In this paper, localization properties of trigonometric polynomial Hermite operators are discus... more . In this paper, localization properties of trigonometric polynomial Hermite operators are discussed. In particular, time frequency uncertainty and operator norms are compared for the different types of fundamental interpolants which serve as scaling functions for a trigonometric multiresolution analysis. x1. Introduction Recently, several different approaches to periodic multiresolution analyses have been presented. For example, periodic scaling functions and wavelets are discussed by Narcowich and Ward [3] who investigate their time frequency behaviour in terms of an uncertainty principle for periodic functions due to Breitenberger [1]. The periodic basis functions in [3] possess an uncertainty product of O( p n) for increasing dimension n of the corresponding spaces. On the other hand, uniformly bounded uncertainty products are computed by Selig [6] for trigonometric fundamental Lagrange interpolants based on de la Vall'ee Poussin means. A multiresolution analysis generated b...

Applied and numerical harmonic analysis, 2017

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) ... more Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) n−l−1 (r 2)r l Y lm (ϑ , ϕ), |m| ≤ l < n ∈ N, L (l+1/2) n−l−1 being a generalized Laguerre polynomial, Y lm a spherical harmonic, constitute an orthonormal basis of the space L 2 on R 3 with Gaussian weight exp(−r 2). These basis functions are used extensively, e.g., in biomolecular dynamic simulations. However, to the present, there is no reliable algorithm available to compute the Fourier coefficients of a function with respect to the SGL basis functions in a fast way. This paper presents such generalized FFTs. We start out from an SGL sampling theorem that permits an exact computation of the SGL Fourier expansion of bandlimited functions. By a separation-of-variables approach and the employment of a fast spherical Fourier transform, we then unveil a general class of fast SGL Fourier transforms. All of these algorithms have an asymptotic complexity of O(B 4), B being the respective bandlimit, while the number of sample points on R 3 scales with B 3. This clearly improves the naive bound of O(B 7). At the same time, our approach results in fast inverse transforms with the same asymptotic complexity as the forward transforms. We demonstrate the practical suitability of our algorithms in a numerical experiment. Notably, this is one of the first performances of generalized FFTs on a non-compact domain. We conclude with a discussion, including the layout of a true O(B 3 log 2 B) fast SGL Fourier transform and inverse, and an outlook on future developments.

Citeseer

Ich erkläre hiermit, dass ich die vorliegende Arbeit selbstständig verfasst und keine anderen als... more Ich erkläre hiermit, dass ich die vorliegende Arbeit selbstständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. ... 2.3 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Spherical Radial Functions . . . . . . . . . . . . . . . . . . . . . . . ...

arXiv (Cornell University), Dec 17, 2017

We obtain estimates of the Lp-error of the bivariate polynomial interpolation on the Lissajous-Ch... more We obtain estimates of the Lp-error of the bivariate polynomial interpolation on the Lissajous-Chebyshev node points for wide classes of functions including non-smooth functions of bounded variation in the sense of Hardy-Krause. The results show that Lperrors of polynomial interpolation on the Lissajous-Chebyshev nodes have almost the same behavior as the polynomial interpolation in the case of the tensor product Chebyshev grid.

arXiv (Cornell University), Apr 6, 2017

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) ... more Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) n−l−1 (r 2)r l Y lm (ϑ, ϕ), |m| ≤ l < n ∈ N, constitute an orthonormal polynomial basis of the space L 2 on R 3 with radial Gaussian weight exp(−r 2). We have recently

arXiv (Cornell University), Feb 16, 2016

We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding t... more We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding to the generalized Jacobi translation and the Ditzian-Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer-Bernstein polynomial operators.

Mitteilungen der Deutschen mathematiker-Vereinigung, Dec 30, 2022

arXiv (Cornell University), Apr 18, 2016

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) ... more Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2) n−l−1 (r 2)r l Y lm (ϑ , ϕ), |m| ≤ l < n ∈ N, L (l+1/2) n−l−1 being a generalized Laguerre polynomial, Y lm a spherical harmonic, constitute an orthonormal basis of the space L 2 on R 3 with Gaussian weight exp(−r 2). These basis functions are used extensively, e.g., in biomolecular dynamic simulations. However, to the present, there is no reliable algorithm available to compute the Fourier coefficients of a function with respect to the SGL basis functions in a fast way. This paper presents such generalized FFTs. We start out from an SGL sampling theorem that permits an exact computation of the SGL Fourier expansion of bandlimited functions. By a separation-of-variables approach and the employment of a fast spherical Fourier transform, we then unveil a general class of fast SGL Fourier transforms. All of these algorithms have an asymptotic complexity of O(B 4), B being the respective bandlimit, while the number of sample points on R 3 scales with B 3. This clearly improves the naive bound of O(B 7). At the same time, our approach results in fast inverse transforms with the same asymptotic complexity as the forward transforms. We demonstrate the practical suitability of our algorithms in a numerical experiment. Notably, this is one of the first performances of generalized FFTs on a non-compact domain. We conclude with a discussion, including the layout of a true O(B 3 log 2 B) fast SGL Fourier transform and inverse, and an outlook on future developments.

Ukrainian Mathematical Journal, Oct 1, 2017

UDC 517.5 We prove direct and inverse theorems on the approximation of 2⇡-periodic functions by T... more UDC 517.5 We prove direct and inverse theorems on the approximation of 2⇡-periodic functions by Taylor-Abel-Poisson operators in the integral metric.

arXiv (Cornell University), Feb 11, 2020

We study approximation properties of general multivariate periodic quasiinterpolation operators, ... more We study approximation properties of general multivariate periodic quasiinterpolation operators, which are generated by distributions/functions ϕj and trigonometric polynomials ϕj. The class of such operators includes classical interpolation polynomials (ϕj is the Dirac delta function), Kantorovich-type operators (ϕj is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on ϕj and ϕj , we obtain upper and lower bound estimates for the Lp-error of approximation by quasi-interpolation operators in terms of the best and best one-sided approximation, classical and fractional moduli of smoothness, K-functionals, and other terms.

Gem - International Journal on Geomathematics, Mar 23, 2019

Journal of Mathematical Analysis and Applications

arXiv (Cornell University), Oct 30, 2022

Approximative properties of the Taylor-Abel-Poisson linear summation method of Fourier series are... more Approximative properties of the Taylor-Abel-Poisson linear summation method of Fourier series are considered for functions of several variables, periodic with respect to the hexagonal domain, in the integral metric. In particular, direct and inverse theorems are proved in terms of approximations of functions by the Taylor-Abel-Poisson means and K-functionals generated by radial derivatives. Bernstein type inequalities for L 1-norm of high-order radial derivatives of the Poisson kernel are also obtained.

Frontiers in Applied Mathematics and Statistics, 2020

Mathematics and Computers in Simulation, 2016

Different mathematical models are built of a selection of mechanisms, and they reproduce observat... more Different mathematical models are built of a selection of mechanisms, and they reproduce observations in a quantitatively different manner. A suitable error functional is used to compare the models and to detect mechanisms, which probably caused the observations. For this aim, parameter identification oftentimes is seen as the determination of a best approximation out of the set of feasible solutions of the model, which can be identified with the set. In this paper, the comparison of different model approaches is discussed with respect to observation data from a disease that occurred 1988 in Chernivtsi in Ukraine. The cause of the disease remained unclear. The quantitative measure of the error functional and selected qualitative properties are used to distinguish the models. Even though only a small set of data for the number of affected persons is available, a comparison of a contamination model and an epidemical model suggest that the cause of the disease is rather an infection than an exposure to environmental toxin.

Cybernetics and Systems Analysis, Jul 1, 2018

The properties of wavelets based on Jacobi polynomials are analyzed. The conditions are considere... more The properties of wavelets based on Jacobi polynomials are analyzed. The conditions are considered under which these wavelets are mutually orthogonal and under which the wavelet basis is characterized by a minimum Riesz ratio. These problems lead to the solution of systems of nonlinear equations by a method proposed earlier by the authors.

Reports of the National Academy of Sciences of Ukraine

Let ε be the set of all entire functions on the complex plane C. Let us consider the class XE of ... more Let ε be the set of all entire functions on the complex plane C. Let us consider the class XE of all complex Banach spaces X such that X ⊇ ε . For (X, ⎥⎥ ⋅ ⎥⎥)∈XE and g ∈X we write En, X (g ) = inf {⎥⎥ g − p⎥⎥: p∈Πn }, where Πn is the set of all polynomials with degree at most n. We describe all X ∈XE for which the relation lim n→∞ (En, X( g ))1/n = 0 holds if and only if g ∈ ε.

We obtain estimates of the Lp-error of the bivariate polynomial interpolation on the Lissajous-Ch... more We obtain estimates of the Lp-error of the bivariate polynomial interpolation on the Lissajous-Chebyshev node points for wide classes of functions including non-smooth functions of bounded variation in the sense of Hardy-Krause. The results show that Lperrors of polynomial interpolation on the Lissajous-Chebyshev nodes have almost the same behavior as the corresponding polynomial interpolation in the case of the tensor product Chebyshev grid.

Mitteilungen der Deutschen Mathematiker-Vereinigung, 2019

Die . Internationale Mathematik-Olympiade (IMO) fand vom . bis zum . Juli  in Bath stat... more Die . Internationale Mathematik-Olympiade (IMO) fand vom . bis zum . Juli  in Bath statt, einer zum Weltkulturerbe der UNESCO zählenden malerischen Universitätsstadt im Westen Englands.  Schülerinnen und  Schüler aus  Ländern nahmen an dieser Olympiade teil. Sie war damit die größte IMO in der Geschichte. Erfreulich ist das Wachstum der letzen  Jahre. Der damalige Rekord der . IMO  in Deutschland mit  Ländern und  Teilnehmerinnen und  Teilnehmern ist inzwischen weit übertroffen. Die deutsche Mannschaft bestand aus sechs Schülern, Dr. Eric Müller als stellvertretendem Delegationsleiter, Dr. Christian Reiher als Observer A und dem Berichterstatter als Delegationsleiter. Alle sechs Teilnehmer haben schon an vielen Mathematik-Wettbewerben teilgenommen. Für Lukas Finn Groß war es die zweite Teilnahme an einer IMO und für Jonas Walter sogar schon die dritte Teilnahme. Letzterer hat insgesamt viermal die IMO-Vorbereitungslehrgänge durchlaufen. Maximilian Göbel, Lukas Finn Groß und Maximilian Keßler waren schon seit  dabei. Maximilian Keßler lebt seit vier Jahren in Barcelona und musste daher zu den Vorbereitungslehrgängen jeweils aus Spanien anreisen. Berichtenswert ist auch, dass Jonas Walter in seiner Freizeit klassische Gitarre spielt und schon erfolgreich bei Jugend Musiziert teilgenommen hat. Lukas Finn Groß war von - Mitglied in einem Schwimmverein und sehr erfolgreich bei Wettbewerben in NRW, besonders auf der  m-Strecke. Alle fünf Abiturienten wollen im Wintersemester ein Mathematik-Studium beginnen, Maximilian Keßler an der LMU in München, die anderen vier in Bonn.