Difference and (Δ)(\Delta) properties for some new classes

\begin{abstract} {In this paper we study difference and $(\Delta)$ properties for the classes of the form $C_0(J,X)$, \frak {g} \U, \U+\frak {g} \V, where \U, \V\in \{BUC(J,X), UC(J,X)\} and $\frak{g} (t)=e^{it^2}$, $t\in \mathbb{R}$. For functions whose differences belong to \F \in\{C_0(J,X), \frak {g} \U\}, we prove a new stronger $(\Delta)$ property (S$\Delta$P): If $f: J\to X$ and \Delta_h f \in \F, $h > 0$, then $f\in C(J,X)$ and (f-M_hf)\in \F, $h > 0$. \noindent See (Lemma 2.5, Theorems 3.1, 3.2, Lemma 3.4, Theorem 3.7). These results enabled us to prove $(\Delta)$ for \U+\frak {g} \V even when \U, \V\in UC(J,X) (Theorem 4.2). We give a new proof of a theorem of De Bruijn [10] stating: if J\in \{\Rdb_+, \Rdb\}, \phi\in \Cdb^J and \Delta_s \phi \in C(J,\Cdb) for each $s >0$, then $\phi= G+H$, where G \in C(J,\Cdb) and $H(t+s)=H(t)+H(s)$, $t,s\in J$, for functions $\phi: X^J$.} \end{abstract}Comment: 10 page

Recurrent solutions of neutral differential-difference systems

Results of Bohr-Neugebauer type are obtained for recurrent functions : If $y$ is a bounded uniformly continuous solution of a linear neutral difference-differential system with recurrent right-hand side, then $y$ is recurrent if $c_0 \not \subset X$ ; also analogues and extensions to half lines are given. For this, various subclasses "$rec$" are introduced which are linear (the set REC of all recurrent functions is not), invariant, closed etc. Also, analogues of the Bohl-Bohr-Amerio-Kadets and Esclangon- Landau results for REC are obtained.Comment: 20 page

Heat equation for weighted Banach space valued function spaces

We study the homogeneous equation (*) $u' = \Delta u$, $t > 0$, $u(0)=f\in wX$, where $wX$ is a weighted Banach space, $w(x)= (1+||x||)^k$, x\in \r^n with $k\ge 0$, $\Delta$ is the Laplacian, $Y$ a complex Banach space and $X$ one of the spaces BUC (\r^n,Y)\} , C_0 (\r^n,Y), L^p (\r^n,Y), $1 \le p < \infty$. It is shown that the mild solutions of (*) are still given by the classical Gauss-Poisson formula, a holomorphic $C_0$-semigroup

Existence of bounded uniformly continuous mild solutions on R\Bbb{R} of evolution equations and some applications

We prove that there is $x_{\phi}\in X$ for which (*)$\frac{d u(t)}{dt}= A u(t) + \phi (t)$, $u(0)=x$ has on \r a mild solution $u\in C_{ub} (\r,X)$ (that is bounded and uniformly continuous) with $u(0)=x_{\phi}$, where $A$ is the generator of a holomorphic $C_0$-semigroup $(T(t))_{t\ge 0}$ on ${X}$ with sup $_{t\ge 0} \,||T(t)|| < \infty$, $\phi\in L^{\infty} (\r,{X})$ and $i\,sp (\phi)\cap \sigma (A)=\emptyset$. As a consequence it is shown that if \n is the space of almost periodic $AP$, almost automorphic $AA$, bounded Levitan almost periodic $LAP_b$, certain classes of recurrent functions $REC_b$ and $\phi \in L^{\infty} (\r,{X})$ such that M_h \phi:=(1/h)\int_0^h \phi (\cdot+s)\, ds \in \n for each $h >0$, then u\in \n\cap C_{ub}. These results seem new and generalize and strengthen several recent Theorems.Comment: 16 page

Spectral criteria for solutions of evolution equations and comments on reduced spectra

We revisit the notion of reduced spectra sp_{\Cal {F}} (\phi) for bounded measurable functions $\phi \in L^{\infty} (J,\Bbb{X})$, {\Cal {F}}\subset L^1_{loc}(J,\Bbb{X}). We show that it can not be obtained via Carleman spectra unless $\phi\in BUC(J,\Bbb{X})$ and {\Cal {F}} \subset BUC(J,\Bbb{X}). In section 3, we give two examples which seem to be of independent interest for spectral theory. In section 4, we prove a spectral criteria for bounded mild solutions of evolution equation (*) $\frac{d u(t)}{dt}= A u(t) + \phi (t)$, $u(0)=x\in \Bbb{X}$, $t\in {J}$, where $A$ is a closed linear operator on $\Bbb{X}$ and $\phi\in L^{\infty} ({J}, \Bbb{X})$ where {J} \in\{\r_+,\r\}.Comment: 16 page

Eberlein almost periodic functions that are not pseudo almost periodic

We construct Eberlein almost periodic functions $f_j : J \to H$ so that $||f_1(\cdot)||$ is not ergodic and thus not Eberlein almost periodic and $||f_2(.)||$ is Eberlein almost periodic, but $f_1$ and $f_2$ are not pseudo almost periodic, the Parseval equation for them fails, where J=\r_+ or \r and $H$ is a Hilbert space. This answers several questions posed by Zhang and Liu [18].Comment: 6 page

Equality of uniform and Carleman spectra for bounded measurable functions

In this paper we study various types of spectra of functions \phi:\jj\to X, where \jj\in\{\r_+,\r\} and $X$ is a complex Banach space. We show that uniform spectrum defined in [15] coincides with Carleman spectrum for $\phi\in L^{\infty}(\r,X)$. This result holds true also for Laplace (half-line) spectrum for \phi\in L^{\infty}(\r_+,X). We also indicate a class of bounded measurable functions for which Laplace spectrum and Carleman spectrum are equalComment: 20 page

Comparison of spectra of absolutely regular distributions and applications

We study the reduced Beurling spectra sp_{\Cal {A},V} (F) of functions F \in L^1_{loc} (\jj,X) relative to certain function spaces \Cal{A}\st L^{\infty}(\jj,X) and V\st L^1 (\r) and compare them with other spectra including the weak Laplace spectrum. Here \jj is \r_+ or \r and $X$ is a Banach space. If $F$ belongs to the space \f'_{ar}(\jj,X) of absolutely regular distributions and has uniformly continuous indefinite integral with 0\not\in sp_{\A,\f(\r)} (F) (for example if F is slowly oscillating and \A is $\{0\}$ or C_0 (\jj,X)), then $F$ is ergodic. If F\in \f'_{ar}(\r,X) and $M_h F (\cdot)= \int_0^h F(\cdot+s)\,ds$ is bounded for all $h > 0$ (for example if $F$ is ergodic) and if sp_{C_0(\r,X),\f} (F)=\emptyset, then ${F}*\psi \in C_0(\r,X)$ for all \psi\in \f(\r). We show that tauberian theorems for Laplace transforms follow from results about reduced spectra. Our results are more general than previous ones and we demonstrate this through examplesComment: 24 page

Dimension of spaces of polynomials on abelian topological semigroups

In this paper we study (continuous) polynomials $p: J\to X$, where $J$ is an abelian topological semigroup and $X$ is a topological vector space. If $J$ is a subsemigroup with non-empty interior of a locally compact abelian group $G$ and $G=J-J$, then every polynomial $p$ on $J$ extends uniquely to a polynomial on $G$. It is of particular interest to know when the spaces $P^n (J,X)$ of polynomials of order at most $n$ are finite dimensional. For example we show that for some semigroups the subspace $P^n_{R} (J,\mathbf{C})$ of Riss polynomials (those generated by a finite number of homomorphisms $\alpha: J\to \mathbf{R}$) is properly contained in $P^n (G,\mathbf{C})$. However, if $P^1 (J,\mathbf{C})$ is finite dimensional then $P^n_{R} (J,\mathbf{C})= P^n (J,\mathbf{C})$. Finally we exhibit a large family of groups for which $P^n (G,\mathbf{C})$ is finite dimensional.Comment: 9 page

Non-Gaussianity from Entanglement During Inflation

We compute the bi-spectrum of CMB temperature fluctuations for a state where the metric perturbation $\zeta$ is entangled with a spectator scalar field $\chi$. Novel terms in the cubic $\zeta$ action coupled to the scalar can be the dominant contribution to the bi-spectrum for such states and we highlight the differences between this result and the no-entanglement bi-spectrum. New shapes can be important in the bi-spectra leading to distinctive observational signatures.Comment: 17 pages, 7 figure