Right order Turan-type converse Markov inequalities for convex domains on the plane

For a convex domain $K$ in the complex plane, the well-known general Bernstein-Markov inequality holds asserting that a polynomial $p$ of degree $n$ must have $||p'|| < c(K) n^2 ||p||$. On the other hand for polynomials in general, $||p'||$ can be arbitrarily small as compared to $||p||$. The situation changes when we assume that the polynomials in question have all their zeroes in the convex body $K$. This was first investigated by Tur\'an, who showed the lower bounds $||p'|| \ge (n/2) ||p||$ for the unit disk $D$ and $||p'|| > c \sqrt{n} ||p||$ for the unit interval $I:=[-1,1]$. Although partial results provided general lower estimates of lower order, as well as certain classes of domains with lower bounds of order $n$, it was not clear what order of magnitude the general convex domains may admit here. Here we show that for all compact and convex domains $K$ with nonempty interior and polynomials $p$ with all their zeroes in $K$ $||p'|| > c(K) n ||p||$ holds true, while $||p'|| < C(K) n ||p||$ occurs for any $K$. Actually, we determine $c(K)$ and $C(K)$ within a factor of absolute numerical constant

A comparative analysis of Bernstein type estimates for the derivative of multivariate polynomials

We compare the yields of two methods to obtain Bernstein type pointwise estimates for the derivative of a multivariate polynomial in points of some domain, where the polynomial is assumed to have sup norm at most 1. One method, due to Sarantopoulos, relies on inscribing ellipses into the convex domain K. The other, pluripotential theoretic approach, mainly due to Baran, works for even more general sets, and yields estimates through the use of the pluricomplex Green function (the Zaharjuta -Siciak extremal function). Using the inscribed ellipse method on non-symmetric convex domains, a key role was played by the generalized Minkowski functional a(K,x). With the aid of this functional, our current knowledge is precise within a constant (squareroot 2) factor. Recently L. Milev and the author derived the exact yield of this method in the case of the simplex, and a number of numerical improvements were obtained compared to the general estimates known. Here we compare the yields of this real, geometric method and the results of the complex, pluripotential theoretical approaches on the case of the simplex. In conclusion we can observe a few remarkable facts, comment on the existing conjectures, and formulate a number of new hypothesis

On uniform asymptotic upper density in locally compact abelian groups

Starting out from results known for the most classical cases of N, Z^d, R^d or for sigma-finite abelian groups, here we define the notion of asymptotic uniform upper density in general locally compact abelian groups. Even if a bit surprising, the new notion proves to be the right extension of the classical cases of Z^d, R^d. The new notion is used to extend some analogous results previously obtained only for classical cases or sigma-finite abelian groups. In particular, we show the following extension of a well-known result for Z of Furstenberg: if in a general locally compact Abelian group G a subset S of G has positive uniform asymptotic upper density, then S-S is syndetic

Decomposition as the sum of invariant functions with respect to commuting transformations

Let A be an arbitrary set. For any transformation T (self-map of A) let T(f)(x):=f(T(x)) (for all x in A) be the usual shift operator. A function g is called periodic, i.e., invariant mod T, if Tg=g (=Ig, where I is the identity operator). As a natural generalization of various earlier investigations in different function spaces, we study the following problem. Let T_j (j=1,...,n) be arbitrary commuting mappings -- transformations -- from A into A. Under what conditions can we state that a function f from A to A is the sum of "periodic", that is, T_j-invariant functions f_j? An obvious necessary condition is that the corresponding multiple difference operator annihilates f, i.e., D_1 ... D_n f= 0, where D_j:=T_j-I. However, in general this condition is not sufficient, and our goal is to complement this basic condition with others, so that the set of conditions will be both necessary and sufficient

Rendezvous numbers in normed spaces

In previous papers, we used abstract potential theory, as developed by Fuglede and Ohtsuka, to a systematic treatment of rendezvous numbers. We introduced energies, Chebyshev constants as two variable set functions, and the modified notion of rendezvous intervals which proved to be rather nicely behaved even for only lower semicontinuous kernels or for not necessarily compact metric spaces. Here we study the rendezvous and average numbers of possibly infinite dimensional normed spaces. It turns out that very general existence and uniqueness results hold for the modified rendezvous numbers in all Banach spaces. We also observe the connections of these "magical numbers" to Chebyshev constants, Chebyshev radius and entropy. Applying the developed notions with the available methods we calculate the rendezvous numbers or rendezvous intervals of certain concrete Banach spaces. In particular, a satisfactory description of the case of L_p spaces is obtained for all p>0

Rendezvous numbers of metric spaces - a potential theoretic approach

The present work draws on the understanding how notions of general potential theory - as set up, e.g., by Fuglede - explain existence and some basic results on the "magical" rendezvous numbers. We aim at a fairly general description of rendezvous numbers in a metric space by using systematically the potential theoretic approach. In particular, we generalize and explain results on invariant measures, hypermetric spaces and maximal energy measures, when showing how more general proofs can be found to them.Comment: 10 page

Potential theoretic approach to rendezvous numbers

We analyze relations between various forms of energies (reciprocal capacities), the transfinite diameter, various Chebyshev constants and the so-called rendezvous or average number. The latter is originally defined for compact connected metric spaces (X,d) as the (in this case unique) nonnegative real number r with the property that for arbitrary finite point systems {x1,...,xn} in X, there exists some point x in X with the average of the distances d(x,xj) being exactly r. Existence of such a miraculous number has fascinated many people; its normalized version was even named "the magic number" of the metric space. Exploring related notions of general potential theory, as set up, e.g., in the fundamental works of Fuglede and Ohtsuka, we present an alternative, potential theoretic approach to rendezvous numbers.Comment: 21 page

On pointwise estimates of positive definite functions with given support

The following problem originated from a question due to Paul Turan. Suppose $\Omega$ is a convex body in Euclidean space \RR^d or in \TT^d, which is symmetric about the origin. Over all positive definite functions supported in $\Omega$, and with normalized value 1 at the origin, what is the largest possible value of their integral? From this Arestov, Berdysheva and Berens arrived to pose the analogous pointwise extremal problem for intervals in \RR. That is, under the same conditions and normalizations, and for any particular point $z\in\Omega$, the supremum of possible function values at $z$ is to be found. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to \RR^d and non-convex domains as well. We present another approach to the problem, giving the solution in \RR^d and for several cases in \TT^d. In fact, we elaborate on the fact that the problem is essentially one-dimensional, and investigate non-convex open domains as well. We show that the extremal problems are equivalent to more familiar ones over trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relation of the problem for the space \RR^d to that for the torus \TT^d is given, showing that the former case is just the limiting case of the latter. Thus the hiearachy of difficulty is established, so that trigonometric polynomial extremal problems gain recognition again.Comment: 19 page

On a problem of Turan about positive definite functions

We study the following question posed by Turan. Suppose K is a convex body in Euclidean space which is symmetric with respect to the origin. Of all positive definite functions supported in K, and with value 1 at the origin, which one has the largest integral? It is probably the case that the extremal function is the indicator of the half-body convolved with itself and properly scaled, but this has been proved only for a small class of domains so far. We add to this class of known "Turan domains" the class of all spectral convex domains. These are all convex domains which have an orthogonal basis of complex exponentials. As a corollary we obtain that all convex domains which tile space by translation are Turan domains. We also give a new proof that the Euclidean ball is a Turan domain.Comment: 8 pages, 1 figur

Tur\'an's extremal problem for positive definite functions on groups

We study the following question: Given an open set $\Omega$, symmetric about 0, and a continuous, integrable, positive definite function $f$, supported in $\Omega$ and with $f(0)=1$, how large can $\int f$ be? This problem has been studied so far mostly for convex domains $\Omega$ in Euclidean space. In this paper we study the question in arbitrary locally compact abelian groups and for more general domains. Our emphasis is on finite groups as well as Euclidean spaces and \ZZ^d. We exhibit upper bounds for $\int f$ assuming geometric properties of $\Omega$ of two types: (a) packing properties of $\Omega$ and (b) spectral properties of $\Omega$. Several examples and applications of the main theorems are shown. In particular we recover and extend several known results concerning convex domains in Euclidean space. Also, we investigate the question of estimating $\int_{\Omega}f$ over possibly dispersed sets solely in dependence of the given measure $m:=|\Omega|$ of $\Omega$. In this respect we show that in \RR and \ZZ the integral is maximal for intervals.Comment: 18 pages, 1 figur