Generalized Functions, Volume 4

Gel′fand, I.; Vilenkin, N. Ya.

doi:10.1090/chel/380

Cited by 24 publications

(38 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the domain of generalized functions, any function is differentiable of any times. The Fourier transform of any function exists (Gelfand and Vilenkin [67], Griffel [68]). …”

Section: Spectra Of Three Excitationsmentioning

confidence: 99%

Three Classes of Fractional Oscillators

2018

Symmetry

View full text Add to dashboard Cite

This article addresses three classes of fractional oscillators named Class I, II and III. It is known that the solutions to fractional oscillators of Class I type are represented by the Mittag-Leffler functions. However, closed form solutions to fractional oscillators in Classes II and III are unknown. In this article, we present a theory of equivalent systems with respect to three classes of fractional oscillators. In methodology, we first transform fractional oscillators with constant coefficients to be linear 2-order oscillators with variable coefficients (variable mass and damping). Then, we derive the closed form solutions to three classes of fractional oscillators using elementary functions. The present theory of equivalent oscillators consists of the main highlights as follows. (1) Proposing three equivalent 2-order oscillation equations corresponding to three classes of fractional oscillators; (2) Presenting the closed form expressions of equivalent mass, equivalent damping, equivalent natural frequencies, equivalent damping ratio for each class of fractional oscillators; (3) Putting forward the closed form formulas of responses (free, impulse, unit step, frequency, sinusoidal) to each class of fractional oscillators; (4) Revealing the power laws of equivalent mass and equivalent damping for each class of fractional oscillators in terms of oscillation frequency; (5) Giving analytic expressions of the logarithmic decrements of three classes of fractional oscillators; (6) Representing the closed form representations of some of the generalized Mittag-Leffler functions with elementary functions. The present results suggest a novel theory of fractional oscillators. This may facilitate the application of the theory of fractional oscillators to practice.

show abstract

“…In the domain of generalized functions, any function is differentiable of any times. The Fourier transform of any function exists (Gelfand and Vilenkin [67], Griffel [68]). …”

Section: Spectra Of Three Excitationsmentioning

confidence: 99%

Three Classes of Fractional Oscillators

2018

Symmetry

View full text Add to dashboard Cite

show abstract

“…Analogously to the situation in finite dimensions, one can expect that a smoother Fourier transform will produce a measure supported in proper subspaces of E a . The support of the measure is indeed related to continuity properties of the Fourier transform [1][2][3]22,23]. As an extreme example of this relation, consider the weakest possible topology in E, having only the empty set and E itself as open sets.…”

Section: Measures On Linear Spacesmentioning

confidence: 99%

“…In this section we consider Gaussian measures on infinite dimensional real linear spaces, following [1][2][3]. In this approach, and following the lines of Section 5, we start with a characteristic function-determined in this case by an inner product-in a linear space E, thus defining the measure initially on the algebraic dual E a .…”

Section: Gaussian Measuresmentioning

confidence: 99%

“…On the other hand, as is typically the case in infinite dimensions, there are no Borel measures on S (R d ) which remain quasi-invariant under the transitive action of all translations, i.e., with respect to the natural action of S (R d ) (seen as a group) on itself [1,2]. [Just like in the discussion at the end of the previous section, this immediately leads to the existence of non-equivalent S-quasi-invariant measures.…”

Section: Gaussian Measures On S (R D )mentioning

confidence: 99%

“…The author's background is mathematical physics and quantum field theory, and that is likely to be reflected in the text, but an hopefully successful effort was made to produce a review of interest to a broader audience. We have references [1][2][3] as our main inspiration. Obviously, some important topics are not dealt with, and others are discussed from a particular perspective, instead of another.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Topics of Measure Theory on Infinite Dimensional Spaces

Velhinho

2017

Mathematics

View full text Add to dashboard Cite

show abstract

Towards analytical model optimization in atmospheric tomography

Helin

Kindermann²,

Saxenhuber³

2016

Math Methods in App Sciences

View full text Add to dashboard Cite

Modern ground‐based telescopes rely on a technology called adaptive optics in order to compensate for the loss of angular resolution caused by atmospheric turbulence. Next‐generation adaptive optics systems designed for a wide field of view require a stable and high‐resolution reconstruction of the turbulent atmosphere. By introducing a novel Bayesian method, we address the problem via reconstructing the atmospheric turbulence strength profile and the turbulent layers simultaneously, where we only use wavefront measurements of incoming light from guide stars. Most importantly, we demonstrate how this method can be used for model optimization as well. We propose two different algorithms for solving the maximum a posteriori estimate: the first approach is based on alternating minimization and has the advantage of integrability into existing atmospheric tomography methods. In the second approach, we formulate a convex non‐differentiable optimization problem, which is solved by an iterative thresholding method. This approach clearly illustrates the underlying sparsity‐enforcing mechanism for the strength profile. By introducing a tuning/regularization parameter, an automated model reduction of the layer structure of the atmosphere is achieved. Using numerical simulations, we demonstrate the performance of our method in practice. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

Generalized Functions, Volume 4

Cited by 24 publications

References 0 publications

Three Classes of Fractional Oscillators

Three Classes of Fractional Oscillators

Topics of Measure Theory on Infinite Dimensional Spaces

Towards analytical model optimization in atmospheric tomography

Contact Info

Product

Resources

About