The property of unique continuation in certain spaces spanned by rational functions on compact nowhere dense sets

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It is shown that some set of all step functions (and the set of all uniform limits of such functions) allows an embedding into a compact subset (with respect to weak-star topology) of the set of all finitely additive measures of bounded variation in the form of an everywhere dense subset. In particular, we consider the set of all step functions (the set of all uniform limits of such functions) such that an integral of absolute value of the functions with respect to nonnegative finitely additive measure λ is equal to unity. For these sets, the possibility of embedding is proved without any additional assumptions on λ; this generalizes the previous results. Using the Sobczyk-Hammer decomposition theorem, we show that for λ with the finite range, the above-mentioned sets of functions allow an embedding into the unit sphere (in the strong norm-variation) of weakly absolutely continuous measures with respect to λ in the form of an everywhere dense subset. For λ with an infinite range, the above-mentioned sets of functions allow an embedding into the unit ball of weakly absolutely continuous measures with respect to λ in the form of an everywhere dense subset. The results can be helpful for an extension of linear impulse control problems in the class of finitely additive measures to obtain robust representations of reachable sets given by constraints of asymptotic character.

Orthogonal rational functions with complex poles: The Favard theorem

There are numerous kinds of sequences of rational functions, such as sequences of Pad6 approximants and sequences of rational functions of best approximation on a given point set, where the poles of the individual functions are not known even asymptotically, and no effective methods for their determination are at hand. Yet regions of convergence and degree of convergence depend heavily on the location of those poles. If assumptions on the location of poles are made, it may be possible to deduce results on regions and degree of convergence, as has been shown by Baker, Chisholm, Walsh, and others. The present note on these topics seems appropriate because a number of previous results, as set forth in a recent report by Baker [1], can be materially improved, especially with reference to degree of convergence. We emphasize Pad6 approximants and rational functions of best approximation. A rational function is said to be of type (m, n) if it can be expressed in the form

We study certain sequences of rational functions with poles outside the unit circle. Such kinds of sequences are recursively constructed based on sequences of complex numbers with norm less than one. In fact, such sequences are closely related to the Schur–Nevanlinna algorithm for Schur functions on the one hand, and to orthogonal rational functions on the unit circle on the other. We shall see that rational functions belonging to a Schur–Nevanlinna sequence can be used to parametrize the set of all solutions of an interpolation problem of Nevanlinna–Pick type for Schur functions.

In this paper, we show that if Fe Lv(k, a) on V where T denotes the border of a compact bordered Riemann surface R, then F can be uniquely written as the sum of a function in Hv(k, a) and a function in Gv(k, a) and moreover that F can be approximated on Y in V norm to within Ant+a by a sequence of rational functions on the union of R with its double.

Let K⊂C be compact and triangular schemes of nodes in K and poles in C\K be specified. We obtain necessary and sufficient conditions on the distribution of the poles and nodes such that every function f holomorphic on K may be approximated uniformly on K by a sequence of rational functions with the given poles which interpolate ∫ in the given nodes. We then show that certain point systems known to be well suited in other interpolation problems may be used successfully in our case. Suppose G⊂C is a multiply connected domain. We then show that those point systems may be employed similarly to define sets of poles in C\G and sets of nodes in G such that every function ∫ holomorphic in G is the limit of the associated rational interpolants locally uniformly in G.

For any rational functions with complex coefficients $A(z), B(z)$ and $C(z)$, where $A(z)$, $C(z)$ are not identically zero, we consider the sequence of rational functions $H_{m}(z)$ with generating function $\sum H_{m}(z)t^{m}=1/(A(z)t^{2}+B(z)t+C(z))$. We provide an explicit formula for the limiting pair correlation function of the roots of $\prod_{m=0}^{n}H_{m}(z)$, as $n\rightarrow\infty$, counting multiplicities, on certain closed subarcs $J$ of a curve $\mathcal{C}$ where the roots lie. We give an example where the limiting pair correlation function does not exist if $J$ contains the endpoints of $\mathcal{C}$.

Let μ be a positive bounded Borel measure on a subset I of the real line, and A = {α_1...,α_n} a sequence of arbitrary complex poles outside I. Suppose {φ_1,...,φ_n} is the sequence of rational functions with poles in A orthonormal on I with respect to μ. First, we are concerned with reducing the number of different coeffcients in the three term recurrence relation satisfied by these orthornormal rational functions. Next, we consider the case in which I = [-1, 1] and μ satisfies the Erdös-Turán condition μ' > 0 a.e. (where μ' is the Radon-Nikodym derivative of the measure μ with respect to the Lebesgue measure), to discuss the convergence of φ_{n+1}(x)/φ_n(x) as n tends to infinity and to derive asymptotic formulas for the recurrence coefficients in the three term recurrence relation. Finally, we give a strong convergence result for φ_n(x) under the more restrictive condition that μ satisfies the Szegö condition (1 - x^2)^{-½} log μ'(x) ∈ L¹[-1; 1].

The paper sets out a simple procedure of general application whereby any function represented by a power series may be replaced by a sequence of rational functions, the advantage of such representation being that in many cases the rational functions converge much more rapidly than does the series. It often happens that the rational-function form continues to represent the given function accurately for values of the argument which make the series form diverge.The procedure is illustrated with a worked example, and the method is then applied to determine a simple expression for the maximum electric intensity between two equal charged spheres, when the sphere-gap is small compared with the diameters.

We consider forn=0, 1,... the nested spaces ℒ n of rational functions of degreen at most with given poles\(1/\bar \alpha _i , |\alpha _i |< 1, i = 1,...,n\). Given a finite measure supported on the unit circle, we associate with it a nested orthogonal basis of rational functions Φ0,...,Φ n for ℒ n ,n=0, 1,.... These Φ n satisfy a recurrence relation that generalizes the recurrence for Szegő polynomials.In this paper we shall prove a Favard type theorem which says that if one has a sequence of rational functions Φ n ∈ ℒ n which are generated by such a recurrence, then there will be a measure μ supported on the unit circle to which they are orthogonal. We shall give a sufficient condition for the uniqueness of this measure.

In the theory of zero distribution of polynomials, there are two branches that are closely related to each other. One branch deals with the conditions under which the normalized zero counting measure of a sequence of polynomials converges in the weak-star sense to the equilibrium measure of a Jordan curve. The other consists of discrepancy theorems, where the maximum deviation of these two measures is estimated. There are also theorems on weak-star convergence of the zero distribution of special sequences of rational functions to a measure that depends on the poles of these functions. In this paper, we give appropriate discrepancy theorems for rational functions. These new results also imply theorems about the weak-star convergence of the zero distribution of more general sequences of rational functions.

When Newton's method, or Halley's method is used to approximate the $p${th} root of $1-z$, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).

Yu. Lyubarskii and K. Seip K. (Revista Matematica Iberoamericana, 1997 (13), № 2) found the criterion for the existence of a uniqe solution of the interpolation problem f(λk) = bk in the Paley-Wiener spaces in terms of Makenhoupt’s (Ap) condition of entire functions of the exponential type at most π, where p is a real number more than 1, whose restriction to the real line coincides with the class of functions which module degree number p is integrable on R whith p-norm. These results make it possible to obtain the criterion of the unconditional basisness of the system of exponentials in the space of functions which module degree number p is an integrable on (–π; π). At the same time the sequence (λk) with a unique limit point at the infinity, for which mentioned interpolation problem has a uniqe solution is called a complete interpolating sequence in the Paley-Wiener spaces. For p = 2 those descriptions coincides with those given by Pavlov (1979), Nikolsky (1980), and Minkin (1982). We generalize these results to the weighted spaces of entire functions of the exponential type at most σ with the p-norm (there is a power function with exponent ω as a weight), where σ is a nonnegative real number, ω – real number more then –1. That is, we find conditions for the completeness of the interpolating sequence (λk) in the Paley-Wiener weighted space. Different forms of these conditions are considered. Among them there are Mackenhoupt’s conditions (continuous and discrete (Ap) conditions). There is proved that if (λk) is a complete interpolation sequence in the Paley-Wiener weighted space, then it is a relatively dense set in the space C. Also there constructed an example of a complete interpolating sequence for σ = π

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.