23 citations · 40 across the 30 of their papers we have counts for
33 papers
Representer Theorem in Complex Reproducing Kernel Hilbert Spaces with Applications to Fock and Hardy Spaces and Superoscillations
Natanael Alpay, Antonino De Martino, Kamal Diki
We introduce a complex-valued counterpart of the representer theorem in machine learning. We study several learning and minimization problems in reproducing kernel Hilbert spaces (…
Polyanalytic Gaussian Radial Basis Function Kernel and Itô-Hermite Polynomials
Hendrik De Bie, Antonino De Martino, Kamal Diki
We introduce a polyanalytic extension of the Gaussian radial basis function (RBF) kernel by computing the action of the convolution operator on normalized Hermite functions. In par…
Superoscillations and the Klein-Gordon equation via the Fourier method
Kamal Diki, Simon Verbruggen
We investigate the time-evolution problem associated with the Klein-Gordon equation, using superoscillations as initial data. Additionally, the Segal-Bargmann transform is used to…
Towards Fock Spaces in Hypercomplex Analysis
Kamal Diki
The Bargmann-Fock space(or Fock space for short) is a fundamental example of reproducing kernel Hilbert spaces that has found fascinating applications across multiple fields of cur…
Short-time Fourier transform and superoscillations
Daniel Alpay, Antonino De Martino, Kamal Diki +1
In this paper we investigate new results on the theory of superoscillations using time-frequency analysis tools and techniques such as the short-time Fourier transform (STFT) and t…
New aspects of Bargmann transform using Touchard polynomials and hypergeometric functions
Daniel Alpay, Antonino De Martino, Kamal Diki
In this paper we study the ranges of the Schwartz space and its dual (space of tempered distributions) under the Segal-Bargmann transform. The char…
Gaussian RBF-kernels via Fock spaces: quaternionic and several complex variables settings
Antonino De Martino, Kamal Diki
In this paper we study two extensions of the complex-valued Gaussian radial basis function (RBF) kernel and discuss their connections with Fock spaces in two different settings. Fi…
The Bicomplex-Real Calculus and Applications to Bicomplex Hermite-Itô Polynomials
Daniel Alpay, Kamal Diki, Mihaela Vajiac
In this paper we extend notions of complex C-R-calculus and complex Hermite polynomials to the bicomplex setting and compare the bicomplex polyanalytic function theory to the class…
The Bicomplex Tensor Product, a Bicomplex Choi Theorem and Applications
Daniel Alpay, Antonino De Martino, Kamal Diki +1
In this paper we extend the concept of tensor product to the bicomplex case and use it to prove the bicomplex counterpart of the classical Choi theorem in the theory of complex mat…
The Fueter-Sce mapping and the Clifford-Appell polynomials
Antonino De Martino, Kamal Diki, Ali Guzmán Adán
The Fueter-Sce theorem provides a procedure to obtain axially monogenic functions, which are in the kernel of generalized Cauchy-Riemann operator in . This resul…
Superoscillations and Fock spaces
Daniel Alpay, Fabrizio Colombo, Kamal Diki +2
In this paper we use techniques in Fock spaces theory and compute how the Segal-Bargmann transform acts on special wave functions obtained by multiplying superoscillating sequences…
An approach to the Gaussian RBF kernels via Fock spaces
Daniel Alpay, Fabrizio Colombo, Kamal Diki +1
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 i…
A Hörmander-Fock space
Daniel Alpay, Fabrizio Colombo, Kamal Diki +2
In a recent paper we used a basic decomposition property of polyanalytic functions of order in one complex variable to characterize solutions of the classical $\overline{\parti…
New Fueter Variables Associated to the Global Operator in the Quaternionic Case
Daniel Alpay, Kamal Diki, Mihaela Vajiac
The purpose of this paper is to develop a new theory of three non-commuting quaternionic variables and its related Schur analysis theory for a modified version of the quaternionic…
Hörmander's -method, -problem and polyanalytic function theory in one complex variable
Daniel Alpay, Fabrizio Colombo, Kamal Diki +2
In this paper we consider the classical -problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of…
Two Bicomplex and One Multicomplex Least Mean Square algorithms
Daniel Alpay, Kamal Diki, Mihaela Vajiac
We study and introduce new gradient operators in the complex and bicomplex settings, inspired from the well-known Least Mean Square (LMS) algorithm invented in 1960 by Widrow and H…
On the Mittag Leffler Bargmann (MLB) transform
Natanael Alpay, Kamal Diki
We introduce the Segal-Bargmann transform associated to the Mittag Leffler Fock space and study how it will be connected to the Fourier transform. We will discuss also the counterp…
Discrete analytic functions, structured matrices and a new family of moment problems
Daniel Alpay, Fabrizio Colombo, Kamal Diki +2
Using Zeilberger generating function formula for the values of a discrete analytic function in a quadrant we make connections with the theory of structured reproducing kernel space…
On the connection between the Fueter-Sce-Qian theorem and the generalized CK-extension
Antonino De Martino, Kamal Diki, Alí Guzmán Adán
The Fueter-Sce-Qian theorem provides a way of inducing axial monogenic functions in from holomorphic intrinsic functions of one complex variable. This result was…
A note on the complex and bicomplex valued neural networks
Daniel Alpay, Kamal Diki, Mihaela Vajiac
In this paper we first write a proof of the perceptron convergence algorithm for the complex multivalued neural networks (CMVNNs). Our primary goal is to formulate and prove the pe…
Generalized Appell polynomials and Fueter-Bargmann transforms in the polyanalytic setting
Antonino De Martino, Kamal Diki
Tnis paper deals with some special integral transforms in the setting of quaternionic valued slice polyanalytic functions. In particular, using the polyanalytic Fueter mappings it…
Reproducing kernel Hilbert spaces of polyanalytic functions of infinite order
Daniel Alpay, Fabrizio Colombo, Kamal Diki +1
In this paper we introduce reproducing kernel Hilbert spaces of polyanalytic functions of infinite order. First we study in details the counterpart of the Fock space and related re…
On the polyanalytic short-time Fourier transform in the quaternionic setting
Antonino De Martino, Kamal Diki
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 rece…
Poly slice monogenic functions, Cauchy formulas and the PS-functional calculus
Daniel Alpay, Fabrizio Colombo, Kamal Diki +1
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…