arXiv subjects
Publications and source records attributed to Bálint Farkas.
We investigate Lie-Trotter product formulae for abstract nonlinear evolution equations with delay. Using results from the theory of nonlinear contraction semigroups in Hilbert spaces, we explain the convergence of the splitting procedure. The order of convergence is also investigated in detail, and some numerical illustrations are presented.
It is not hard to prove that a uniformly continuous real function, whose integral up to infinity exists, vanishes at infinity, and it is probably little known that this statement runs under the name "Barbalat's Lemma." In fact, the latter name is frequently used in control theory, where the lemma is used to obtain Lyapunov-like stability theorems for non-linear and non-autonomous systems. Barbalat's Lemma is qualitative in the sense that it asserts that a function has certain properties, here convergence to zero. Such qualitative statements can typically be proved by "soft analysis", such as indirect proofs. Indeed, in the original 1959 paper by Barbalat, the lemma was proved by contradiction and this proof prevails in the control theory textbooks. In this short note we first give a direct, "hard analyis" proof of the lemma, yielding quantitative results, i.e. rates of convergence to zero. This proof allows also for immediate generalizations. Finally, we unify three different versions which recently appeared and discuss their relation to the original lemma.
Given $T_1,\dots, T_n$ commuting power-bounded operators on a Banach space we study under which conditions the equality $\ker (T_1-\mathrm{I})\cdots (T_n-\mathrm{I})=\ker(T_1-\mathrm{I})+\cdots +\ker (T_n-\mathrm{I})$ holds true. This problem, known as the periodic decomposition problem, goes back to I. Z. Ruzsa. In this short note we consider the case when $T_j=T(t_j)$, $t_j>0$, $j=1,\dots, n$ for some one-parameter semigroup $(T(t))_{t\geq 0}$. We also look at a generalization of the periodic decomposition problem when instead of the cyclic semigroups $\{T_j^n:n \in \mathbb{N}\}$ more general semigroups of bounded linear operators are considered.
If a function $f:\mathbb{R}\to\mathbb{R}$ can be represented as the sum of $n$ periodic functions as $f=f_1+\dots+f_n$ with $f(x+α_j)=f(x)$ ($j=1,\dots,n$), then it also satisfies a corresponding $n$-order difference equation $Δ_{α_1}\dotsΔ_{α_n} f=0$. The periodic decomposition problem asks for the converse implication, which may hold or fail depending on the context (on the system of periods, on the function class in which the problem is considered, etc.). The problem has natural extensions and ramifications in various directions, and is related to several other problems in real analysis, Fourier and functional analysis. We give a survey about the available methods and results, and present a number of intriguing open problems.
We work out the optimization problem, initiated by K. Soundararajan, for the choice of the underlying polynomial P used in the construction of the weight function in the Goldston--Pintz--Yildirim method for finding small gaps between primes. First we reformulate to a maximization problem on L^2[0,1] for a self-adjoint operator T, the norm of which is then the maximal eigenvalue of T. To find eigenfunctions and eigenvalues, we derive a differential equation which can be explicitly solved. The aimed maximal value is S(k)=4/(k+ck^{1/3}), achieved by the (k-1)st integral of x^{1-k/2}J_{k-2}(a_1\sqrt{x}), where a_1 is the first positive root of the (k-2)nd Bessel function J_{k-2} and as such, is asymptotically ck^{1/3} with a well-known constant c. As this naturally gives rise to a number of technical problems in the application of the GPY method, we also construct a polynomial P which is a simpler function yet it furnishes an approximately optimal extremal quantity, 4/(k+Ck^{1/3}) with some other constant C. In the forthcoming paper of J. Pintz (also to appear in the Turán-100 Memorial Volume by de Gruyter), it is indeed shown how this quasi-optimal choice of the polynomial in the weight finally can exploit the GPY method to its theoretical limits.
We prove that a separable Banach space $E$ does not contain a copy of the space $\co$ of null-sequences if and only if for every doubly power-bounded operator $T$ on $E$ and for every vector $x\in E$ the relative compactness of the sets $\{T^{n+m}x-T^nx: n\in \NN\}$ (for some/all $m\in\NN$, $m\geq 1$) and $\{T^nx:n\in \NN\}$ are equivalent. With the help of the Jacobs--de Leeuw--Glicksberg decomposition of strongly compact semigroups the case of (not necessarily invertible) power-bounded operators is also handled.
We present easy to verify conditions implying stability estimates for operator matrix splittings which ensure convergence of the associated Trotter, Strang and weighted product formulas. The results are applied to inhomogeneous abstract Cauchy problems and to boundary feedback systems.
We provide a general product formula for the solution of nonautonomous abstract delay equations. After having shown the convergence we obtain estimates on the order of convergence for differentiable history functions. Finally, the theoretical results are demonstrated on some typical numerical examples.
Continuing earlier investigations, we analyze the convergence of operator splitting procedures combined with spatial discretization and rational approximations.
We provide general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tool is the application of evolution semigroup methods, allowing the direct application of existing results on autonomous problems. The results are then illustrated by the example of a imaginary time Schrödinger equation with time dependent potential. We also obtain convergence rates for the Strang-splitting applied to this problem.
For a given bi-continuous semigroup T on a Banach space X we define its adjoint on an appropriate closed subspace X^o of the norm dual X'. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology (X^o,X). An application is the following: For K a Polish space we consider operator semigroups on the space C(K) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(K) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(K) are precisely those that are adjoints of a bi-continuous semigroups on C(K). We also prove that the class of bi-continuous semigroups on C(K) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if K is not Polish space this is not the case.
For a nonempty compact set D of R we determine the maximal possible dimension of a subspace X of polynomial functions of degree at most m which possesses a positive bases (where positivity is understood on D). The exact value of this maximal dimension depends on topological features of the base set D. We show that at many cases dimension m can be achieved.Whereas only for low m or finite sets D it is possible to have m+1 (the full dimension of the space of polynomials of degree at most m) dimensional subspace X with positive basis. Therefore, it turns out that for no D it is possible to have a positive basis of the polynomial space of degree at most m for all m in N.
Consider a_1,a_2,...,a_n, arbitrary elements of R. We characterize those real functions f that decompose into the sum of a_j-periodic functions, i.e., f=f_1+...+f_n with D_{a_j}f(x):=f(x+a_j)-f(x)=0. We show that f has such a decomposition if and only if for all partitions to B_1, B_2,... B_N of {a_1,a_2,...,a_n} with B_j consisting of commensurable elements with least common multiples b_j, one has D_{b_1}... D_{b_N}f=0. Actually, we prove a more general result for periodic decompositions of real functions f defined on an Abelian group A, and, in fact, we even consider invariant decompositions of functions f defined on some abstract set A, with respect to commuting, invertible self-mappings of the set A. We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real valued periodic decomposition of an integer valued function implies the existence of an integer valued periodic decomposition with the same periods.
Recent methods developed by Tao \cite{tao}, Kolountzakis and Matolcsi \cite{nspec} have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in $\RR^5$ Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an example of a non-spectral tile. In search of lower dimensional non-spectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias and Wang \cite{lagwang}. In particular, we prove here that the USC and the ``tile $\to$ spectral'' direction of Fuglede's conjecture are equivalent in any dimensions. Also, we show by an example that the sufficient condition of Lagarias and Szabó \cite{lagszab} for the existence of universal spectra is not necessary. This fact causes considerable difficulties in producing lower dimensional examples of tiles which have no spectra. We overcome these difficulties by invoking some ideas of Révész and Farkas \cite{revfark}, and obtain non-spectral tiles in $\RR^3$. Fuglede's conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The 1 dimensional case is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz \cite{covmey}.