Properties of translation operator and the solution of the eigenvalue and boundary value problems of arbitrary space–time periodic metamaterials

There is a recent interest in understanding and exploiting the intriguing properties of space–time metamaterials. In the current manuscript, the time periodic circuit theory is exploited to introduce an appropriate translation operator that fully describes arbitrary space–time metamaterials. It is shown that the underlying mathematical machinery is identical to the one used in the analysis of linear time invariant periodic structures, where time and space eigen-decompositions are successively employed. We prove some useful properties the translation operator exhibits. The wave propagation inside the space time periodic metamaterial and the terminal characteristics can be rigorously determined via the expansion in the operators eigenvectors (space–time Bloch waves). Two examples are provided that demonstrate how to apply the framework. In the first, a space time modulated composite right left handed transmission line is studied and results are verified via time domain computations. Furthermore, we apply the theory to explain the non-reciprocal behaviour observed on a nonlinear transmission line manufactured in our lab. Bloch-waves are computed from the extracted circuit parameters. Results predicted using the developed machinery agree with both measurements and time domain analysis. Although the analysis was carried out for electric circuits, the approach is valid for different domains such as acoustic and elastic media.

Necessary Conditions for Boundedness of Translation Operator in de Branges Spaces

The translation operator is bounded in the Paley–Wiener spaces and, more generally, in the Bernstein spaces. The goal of this paper is to find some necessary conditions for the boundedness of the translation operator in the de Branges spaces, of which the Paley–Wiener spaces are special cases. Indeed, if the vertical translation operator $$T_\tau $$ T τ defined on the de Branges space $${\mathcal H}(E)$$ H ( E ) is bounded, then a suitably defined measure $$d\mu (z)$$ d μ ( z ) is a Carleson measure for the associated model space $$K(\Theta )$$ K ( Θ ) . This relation allows us to state necessary conditions for the boundedness of the vertical translation $$T_\tau $$ T τ . Finally, similar results are also obtained for the horizontal translation $$T_\sigma $$ T σ .

The heat semigroup and equation related to a Bessel-type operators and the canonical Fourier Bessel transform

In this paper we study a translation operator associated with the canonical Fourier Bessel transform $\mathcal{F}_{\nu}^{\mathbf{m}}.$ We then use it to derive a convolution product and study some of its important properties. As a direct application, we introduce the heat semigroup generated by the Bessel-type operators $$\Delta_{\nu}^{\mathbf{m}^{-1}}=\frac{d^{2}}{dx^{2}}+\left( \frac{2\nu +1}{x}+2i \frac{a}{b} x\right) \frac{d}{dx}-\left( \frac{a^{2}}{b^{2}}x^{2}-2i\left( \nu +1\right) \frac{a}{b}\right) $$ and use it to solve the initial value problem for the heat equation governed by $\Delta_{\nu}^{\mathbf{m}^{-1}}.$

Path integral for quantum dynamics with position-dependent mass within the displacement operator approach

In the context of quantum mechanics reformulated in a modified Hilbert space, we can formulate the Feynman’s path integral approach for the quantum systems with position-dependent mass particle using the formulation of position-dependent infinitesimal translation operator. Which is similar a deformed quantum mechanics based on modified commutation relations. An illustration of calculation is given in the case of the harmonic oscillator, the infinite square well and the inverse square plus Coulomb potentials.

On the K-functional in learning theory

[Formula: see text]-functionals are used in learning theory literature to study approximation errors in kernel-based regularization schemes. In this paper, we study the approximation error and [Formula: see text]-functionals in [Formula: see text] spaces with [Formula: see text]. To this end, we give a new viewpoint for a reproducing kernel Hilbert space (RKHS) from a fractional derivative and treat powers of the induced integral operator as fractional derivatives of various orders. Then a generalized translation operator is defined by Fourier multipliers, with which a generalized modulus of smoothness is defined. Some general strong equivalent relations between the moduli of smoothness and the [Formula: see text]-functionals are established. As applications, some strong equivalent relations between these two families of quantities on the unit sphere and the unit ball are provided explicitly.

Generalized integral transforms with the translation operator on function space

Main goal of this paper is to establish various basic formulas for the generalized integral transform involving the generalized convolution product. In order to establish these formulas, we use the translation operator which was introduced in [9]. It was not easy to establish basic formulas for the generalized integral transforms because the generalized Brownian motion process used in this paper has the nonzero mean function. In this paper, we can easily establish various basic formulas for the generalized integral transform involving the generalized convolution product via the translation operator.