Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- Adaptive decomposition (1)
- Blaschke-Potapov (1)
- Co-isometry (1)
- Convolution operators (1)
- Entire functions (1)
-
- Finite Impulse Response (1)
- Fourier transforms (1)
- Functions of the infinitesimal generator (1)
- Gaussian processes (1)
- Hankel operator (1)
- Hilbert space (1)
- Infinite order differential operators (1)
- Isometry (1)
- Laurent polynomials (1)
- Matrix-valued Blaschke products (1)
- Matrix-valued functions and Hardy spaces (1)
- Maximum selection principle (1)
- Quaternionic Laplace-Stieltjes transform (1)
- Quaternionic functional calculus (1)
- Quaternionic infinitesimal generators (1)
- Random noise (1)
- Realization (1)
- S-resolvent operator (1)
- S-spectrum (1)
- Superoscillations (1)
- Tensor methods (1)
- Topology (1)
Articles 1 - 6 of 6
Full-Text Articles in Other Mathematics
On A Class Of Quaternionic Positive Definite Functions And Their Derivatives, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
On A Class Of Quaternionic Positive Definite Functions And Their Derivatives, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we start the study of stochastic processes over the skew field of quaternions. We discuss the relation between positive definite functions and the covariance of centered Gaussian processes and the construction of stochastic processes and their derivatives. The use of perfect spaces and strong algebras and the notion of Fock space are crucial in this framework.
The Mathematics Of Superoscillations, Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, Jeff Tollaksen
The Mathematics Of Superoscillations, Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, Jeff Tollaksen
Mathematics, Physics, and Computer Science Faculty Articles and Research
In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the …
On A Class Of Quaternionic Positive Definite Functions And Their Derivatives, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
On A Class Of Quaternionic Positive Definite Functions And Their Derivatives, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we start the study of stochastic processes over the skew field of quaternions. We discuss the relation between positive definite functions and the covariance of centered Gaussian processes and the construction of stochastic processes and their derivatives. The use of perfect spaces and strong algebras and the notion of Fock space are crucial in this framework.
Characterizations Of Families Of Rectangular, Finite Impulse Response, Para-Unitary Systems, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz
Characterizations Of Families Of Rectangular, Finite Impulse Response, Para-Unitary Systems, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz
Mathematics, Physics, and Computer Science Faculty Articles and Research
We here study Finite Impulse Response (FIR) rectangular, not necessarily causal, systems which are (para)-unitary on the unit circle (=the class U). First, we offer three characterizations of these systems. Then, introduce a description of all FIRs in U, as copies of a real polytope, parametrized by the dimensions and the McMillan degree of the FIRs.
Finally, we present six simple ways (along with their combinations) to construct, from any FIR, a large family of FIRs, of various dimensions and McMillan degrees, so that whenever the original system is in U, so is the whole family.
A key role is …
Adaptive Orthonormal Systems For Matrix-Valued Functions, Daniel Alpay, Fabrizio Colombo, Tao Qian, Irene Sabadini, Tao Qian
Adaptive Orthonormal Systems For Matrix-Valued Functions, Daniel Alpay, Fabrizio Colombo, Tao Qian, Irene Sabadini, Tao Qian
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we consider functions in the Hardy space Hp×q2 defined in the unit disc of matrix-valued. We show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably modified matrix-valued Blaschke product, in an adaptive way. The procedure is based on a generalization to the matrix-valued case of the maximum selection principle which involves not only selections of suitable points in the unit disc but also suitable orthogonal projections. We show that the maximum selection principle gives rise to a convergent algorithm. Finally, we discuss the case of real-valued signals.
Functions Of The Infinitesimal Generator Of A Strongly Continuous Quaternionic Group, Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, David P. Kimsey
Functions Of The Infinitesimal Generator Of A Strongly Continuous Quaternionic Group, Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, David P. Kimsey
Mathematics, Physics, and Computer Science Faculty Articles and Research
The analogue of the Riesz-Dunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. In this paper we suppose that T is the infinitesimal generator of a strongly continuous group of operators (ZT (t))t2R and we show how we can define bounded operators f(T ), where f belongs to a class of functions which is larger than the class of slice regular functions, using the quaternionic Laplace-Stieltjes transform. This class will include functions that are slice regular on the S-spectrum of T but not necessarily at infinity. Moreover, …