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Papers by kamal diki

Research paper thumbnail of An approach to the Gaussian RBF kernels via Fock spaces

Journal of Mathematical Physics

We use methods from the Fock space and Segal–Bargmann theories to prove several results on the Ga... more We use methods from the Fock space and Segal–Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods and in support vector machine classification algorithms. Complex analysis techniques allow us to consider several notions linked to the radial basis function (RBF) kernels, such as the feature space and the feature map, using the so-called Segal–Bargmann transform. We also show how the RBF kernels can be related to some of the most used operators in quantum mechanics and time frequency analysis; specifically, we prove the connections of such kernels with creation, annihilation, Fourier, translation, modulation, and Weyl operators. For the Weyl operators, we also study a semigroup property in this case.

Research paper thumbnail of On the Quaternionic Short-Time Fourier and Segal–Bargmann Transforms

Mediterranean Journal of Mathematics, 2021

In this paper, we study a special one-dimensional quaternion short-time Fourier transform (QSTFT)... more In this paper, we study a special one-dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal–Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb’s uncertainty principle. We provide also the reproducing kernel associated with the Gabor space considered in this setting.

Research paper thumbnail of On the polyanalytic short-time Fourier transform in the quaternionic setting

In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized He... more In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized Hermite functions as windows. It turns out that such a transform is based on the recent theory of slice polyanalytic functions on quaternions. Indeed, we will use the notions of true and full slice polyanalytic Fock spaces and Segal-Bargmann transforms. We prove new properties of this QSTFT including a Moyal formula, a reconstruction formula and a Lieb’s uncertainty principle. ‘ese results extend a recent paper of the authors which studies a QSTFT having a Gaussian function as a window. AMS Classification: 44A15, 30G35, 42C15, 46E22

Research paper thumbnail of Correction to: On Slice Polyanalytic Functions of a Quaternionic Variable

Results in Mathematics, 2021

In [2] we initiated the theory of polyanalytic functions of a quaternionic variable in the slice ... more In [2] we initiated the theory of polyanalytic functions of a quaternionic variable in the slice hyperholomorphic setting.

Research paper thumbnail of Fock and Hardy spaces: Clifford-Appell case

arXiv: Complex Variables, 2020

In this paper, we study a specific system of Clifford-Appell polynomials and in particular their ... more In this paper, we study a specific system of Clifford-Appell polynomials and in particular their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows to obtain various function spaces, by specifying a suitable sequence of real numbers. We focus on the Fock and Hardy cases in this setting, and we study the action of the Fueter mapping and its range.

Research paper thumbnail of On the global operator and Fueter mapping theorem for slice polyanalytic functions

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as sol... more In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball.

Research paper thumbnail of On a Polyanalytic Approach to Noncommutative de Branges–Rovnyak Spaces and Schur Analysis

In this paper we begin the study of Schur analysis and de Branges-Rovnyak spaces in the framework... more In this paper we begin the study of Schur analysis and de Branges-Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, we consider the counterpart of the half-space case, and the corresponding Hardy space, Schur multipliers and Carath\'eodory multipliers.

Research paper thumbnail of Poly slice monogenic functions, Cauchy formulas and the PS-functional calculus

Since 2006 the theory of slice hyperholomorphic functions and the related spectral theory on the ... more Since 2006 the theory of slice hyperholomorphic functions and the related spectral theory on the S-spectrum have had a very fast development. This new spectral theory based on the S-spectrum has applications for example in the formulation of quaternionic quantum mechanics, in Schur analysis and in fractional diffusion problems. The notion of poly slice analytic function has been recently introduced for the quaternionic setting. In this paper we study the theory of poly slice monogenic functions and the associated functional calculus, called PS-functional calculus, which is the polyanalytic version of the S-functional calculus. Also for this poly monogenic functional calculus we use the notion of S-spectrum.

Research paper thumbnail of On the Bargmann-Fock-Fueter and Bergman-Fueter integral transforms

Journal of Mathematical Physics

Bargmann-Fock type in the se ing of quaternionic valued slice hyperholomorphic and Cauchy-Fueter ... more Bargmann-Fock type in the se ing of quaternionic valued slice hyperholomorphic and Cauchy-Fueter regular functions. e construction is based on the well-known Fueter mapping theorem. In particular, starting with the normalized Hermite functions we can construct an Appell system of quaternionic regular polynomials. e ranges of such integral transforms are quaternionic reproducing kernel Hilbert spaces of regular functions. New integral representations and generating functions in this quaternionic se ing are obtained in both the Fock and Bergman cases.

Research paper thumbnail of Composition of Segal–Bargmann transforms

Complex Variables and Elliptic Equations

We introduce and discuss some basic properties of some integral transforms in the framework of sp... more We introduce and discuss some basic properties of some integral transforms in the framework of specific functional Hilbert spaces, the holomorphic Bargmann-Fock spaces on C and C 2 and the sliced hyperholomorphic Bargmann-Fock space on H. The first one is a natural integral transform mapping isometrically the standard Hilbert space on the real line into the two-dimensional Bargmann-Fock space. It is obtained as composition of the one and two dimensional Segal-Bargmann transforms and reduces further to an extremely integral operator that looks like a composition operator of the one-dimensional Segal-Bargmann transform with a specific symbol. We study its basic properties, including the identification of its image and the determination of a like-left inverse defined on the whole two-dimensional Bargmann-Fock space. We also examine their combination with the Fourier transform which lead to special integral transforms connecting the two-dimensional Bargmann-Fock space and its analogue on the complex plane. We also investigate the relationship between special subspaces of the twodimensional Bargmann-Fock space and the slice-hyperholomorphic one on the quaternions by introducing appropriate integral transforms. We identify their image and their action on the reproducing kernel.

Research paper thumbnail of The Cholewinski‐Fock space in the slice hyperholomorphic setting

Mathematical Methods in the Applied Sciences

Research paper thumbnail of On Slice Polyanalytic Functions of a Quaternionic Variable

Research paper thumbnail of Polynomial Approximation in Slice Regular Fock Spaces

Complex Analysis and Operator Theory

The main purpose of this paper is to prove some density results of polynomials in Fock spaces of ... more The main purpose of this paper is to prove some density results of polynomials in Fock spaces of slice regular functions. The spaces can be of two different kinds since they are equipped with different inner products and contain different functions. We treat both the cases, providing several results, some of them based on constructive methods which make use of the Taylor expansion and of the convolution polynomials. We also prove quantitative estimates in terms of higher order moduli of smoothness and in terms of the best approximation quantities.

Research paper thumbnail of An approach to the Gaussian RBF kernels via Fock spaces

Journal of Mathematical Physics

We use methods from the Fock space and Segal–Bargmann theories to prove several results on the Ga... more We use methods from the Fock space and Segal–Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods and in support vector machine classification algorithms. Complex analysis techniques allow us to consider several notions linked to the radial basis function (RBF) kernels, such as the feature space and the feature map, using the so-called Segal–Bargmann transform. We also show how the RBF kernels can be related to some of the most used operators in quantum mechanics and time frequency analysis; specifically, we prove the connections of such kernels with creation, annihilation, Fourier, translation, modulation, and Weyl operators. For the Weyl operators, we also study a semigroup property in this case.

Research paper thumbnail of On the Quaternionic Short-Time Fourier and Segal–Bargmann Transforms

Mediterranean Journal of Mathematics, 2021

In this paper, we study a special one-dimensional quaternion short-time Fourier transform (QSTFT)... more In this paper, we study a special one-dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal–Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb’s uncertainty principle. We provide also the reproducing kernel associated with the Gabor space considered in this setting.

Research paper thumbnail of On the polyanalytic short-time Fourier transform in the quaternionic setting

In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized He... more In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized Hermite functions as windows. It turns out that such a transform is based on the recent theory of slice polyanalytic functions on quaternions. Indeed, we will use the notions of true and full slice polyanalytic Fock spaces and Segal-Bargmann transforms. We prove new properties of this QSTFT including a Moyal formula, a reconstruction formula and a Lieb’s uncertainty principle. ‘ese results extend a recent paper of the authors which studies a QSTFT having a Gaussian function as a window. AMS Classification: 44A15, 30G35, 42C15, 46E22

Research paper thumbnail of Correction to: On Slice Polyanalytic Functions of a Quaternionic Variable

Results in Mathematics, 2021

In [2] we initiated the theory of polyanalytic functions of a quaternionic variable in the slice ... more In [2] we initiated the theory of polyanalytic functions of a quaternionic variable in the slice hyperholomorphic setting.

Research paper thumbnail of Fock and Hardy spaces: Clifford-Appell case

arXiv: Complex Variables, 2020

In this paper, we study a specific system of Clifford-Appell polynomials and in particular their ... more In this paper, we study a specific system of Clifford-Appell polynomials and in particular their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows to obtain various function spaces, by specifying a suitable sequence of real numbers. We focus on the Fock and Hardy cases in this setting, and we study the action of the Fueter mapping and its range.

Research paper thumbnail of On the global operator and Fueter mapping theorem for slice polyanalytic functions

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as sol... more In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball.

Research paper thumbnail of On a Polyanalytic Approach to Noncommutative de Branges–Rovnyak Spaces and Schur Analysis

In this paper we begin the study of Schur analysis and de Branges-Rovnyak spaces in the framework... more In this paper we begin the study of Schur analysis and de Branges-Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, we consider the counterpart of the half-space case, and the corresponding Hardy space, Schur multipliers and Carath\'eodory multipliers.

Research paper thumbnail of Poly slice monogenic functions, Cauchy formulas and the PS-functional calculus

Since 2006 the theory of slice hyperholomorphic functions and the related spectral theory on the ... more Since 2006 the theory of slice hyperholomorphic functions and the related spectral theory on the S-spectrum have had a very fast development. This new spectral theory based on the S-spectrum has applications for example in the formulation of quaternionic quantum mechanics, in Schur analysis and in fractional diffusion problems. The notion of poly slice analytic function has been recently introduced for the quaternionic setting. In this paper we study the theory of poly slice monogenic functions and the associated functional calculus, called PS-functional calculus, which is the polyanalytic version of the S-functional calculus. Also for this poly monogenic functional calculus we use the notion of S-spectrum.

Research paper thumbnail of On the Bargmann-Fock-Fueter and Bergman-Fueter integral transforms

Journal of Mathematical Physics

Bargmann-Fock type in the se ing of quaternionic valued slice hyperholomorphic and Cauchy-Fueter ... more Bargmann-Fock type in the se ing of quaternionic valued slice hyperholomorphic and Cauchy-Fueter regular functions. e construction is based on the well-known Fueter mapping theorem. In particular, starting with the normalized Hermite functions we can construct an Appell system of quaternionic regular polynomials. e ranges of such integral transforms are quaternionic reproducing kernel Hilbert spaces of regular functions. New integral representations and generating functions in this quaternionic se ing are obtained in both the Fock and Bergman cases.

Research paper thumbnail of Composition of Segal–Bargmann transforms

Complex Variables and Elliptic Equations

We introduce and discuss some basic properties of some integral transforms in the framework of sp... more We introduce and discuss some basic properties of some integral transforms in the framework of specific functional Hilbert spaces, the holomorphic Bargmann-Fock spaces on C and C 2 and the sliced hyperholomorphic Bargmann-Fock space on H. The first one is a natural integral transform mapping isometrically the standard Hilbert space on the real line into the two-dimensional Bargmann-Fock space. It is obtained as composition of the one and two dimensional Segal-Bargmann transforms and reduces further to an extremely integral operator that looks like a composition operator of the one-dimensional Segal-Bargmann transform with a specific symbol. We study its basic properties, including the identification of its image and the determination of a like-left inverse defined on the whole two-dimensional Bargmann-Fock space. We also examine their combination with the Fourier transform which lead to special integral transforms connecting the two-dimensional Bargmann-Fock space and its analogue on the complex plane. We also investigate the relationship between special subspaces of the twodimensional Bargmann-Fock space and the slice-hyperholomorphic one on the quaternions by introducing appropriate integral transforms. We identify their image and their action on the reproducing kernel.

Research paper thumbnail of The Cholewinski‐Fock space in the slice hyperholomorphic setting

Mathematical Methods in the Applied Sciences

Research paper thumbnail of On Slice Polyanalytic Functions of a Quaternionic Variable

Research paper thumbnail of Polynomial Approximation in Slice Regular Fock Spaces

Complex Analysis and Operator Theory

The main purpose of this paper is to prove some density results of polynomials in Fock spaces of ... more The main purpose of this paper is to prove some density results of polynomials in Fock spaces of slice regular functions. The spaces can be of two different kinds since they are equipped with different inner products and contain different functions. We treat both the cases, providing several results, some of them based on constructive methods which make use of the Taylor expansion and of the convolution polynomials. We also prove quantitative estimates in terms of higher order moduli of smoothness and in terms of the best approximation quantities.