38 research outputs found
On the positivity of symmetric polynomial functions. Part III: Extremal polynomials of degree 4
No full textAbstractIn this paper, which is a continuation of [V. Timofte, On the positivity of symmetric polynomial functions. Part I: General results, J. Math. Anal. Appl. 284 (2003) 174–190] and [V. Timofte, On the positivity of symmetric polynomial functions. Part II: Lattice general results and positivity criteria for degrees 4 and 5, J. Math. Anal. Appl., in press], we study properties of extremal polynomials of degree 4, and we give the construction of some of them. The main results are Theorems 9, 13, 15, 16, and 18
Special uniform approximations of continuous vector-valued functions. Part II: special approximations in CX(T)⊗CY(S)
No full textAbstractIn this paper, which is a continuation of Timofte (J. Approx. Theory 119 (2002) 291–299, we give special uniform approximations of functions from CX⊗Y(T×S) and C∞(T×S,X⊗Y) by elements of the tensor products CX(T)⊗CY(S), respectively C0(T,X)⊗C0(S,Y), for topological spaces T,S and Γ-locally convex spaces X,Y (all four being Hausdorff)
An Isomorphic Characterization of L₁-Spaces
No full textThe talk will discuss a recent (2008) representation theorem for Hausdorff topological ordered vector spaces with the Riesz decomposition property, locally solid and sequentially complete topology, and the positive cone generated by a closed bounded set not containing the origin. Every such space is isomorphic to some L₁-space. The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces
Integral estimates for convergent positive series
No full textAbstractIt is shown that for every α>1, we have ∑k=n+1∞1kα=1(α−1)(n+θn)α−1 for some strictly decreasing sequence (θn)n⩾1 such that 12<θn<14[1+(1+12n+1)α], hence with limn→∞θn=12. This is only a particular case of more general new results on series defined by convex functions
Remainder Maps
No full textWe study remainder maps R_f(z) = Σₖ₌₁^∞ f(z + k), associated to convergent series defined by complex rational maps f ∈ ��(X), with coefficients in an arbitrary subfield �� ⊂ ℂ. All such maps are meromorphic on ℂ, form a vector space over ��, and generate what we call the remainder field ��(R_F) of ground ��. For remainder maps, we prove a criterion of algebraic independence over ��(X), stated in terms of ��-linear combinations. It turns out that any R_f is either a member of ��(X) or is transcendental, and so the field extension ��(R_F)/��(X) is purely transcendental. We show that any nonzero ϕ from the remainder field has a degree d(ϕ) ∈ ℤ, for which ϕ(x) / x^(d(ϕ)) has a nonzero limit (as x → ∞) in ��. This yields a natural asymptotic expansion of ϕ. Consequently, the remainder field can be extended to the field ��((X⁻¹)) of reverse formal Laurent series, and rational approximants of any order exist. For remainder maps, finding rational approximants reduces to the same problem for the fundamental case of f = 1 / X². We thus find the explicit formal series of any remainder map. In the general case, the difference 1 / R_f(x) - W_f(x) tends to 0 for some unique W_f ∈ ��[X], called here the inverse polynomial of R_f. In the real case, the convergence is monotonic and leads to sharp estimates. Iterating the construction leads to higher order inverse polynomials, with exponentially increasing degrees. In our example, the error made by replacing the sum of the series Σₙ≥1 1/n² (the fundamental case) by a corrected nth partial sum, is less than 10⁻⁴ for n = 1, less than 10⁻¹⁵ for n = 5, and less than 10⁻³² for n = 20. Our theory also includes the alternating remainder maps R̂_f(z) := Σₖ₌₁^∞ (-1)^(k-1) f(2z + k). Students are welcome
Representation of a class of locally convex (M)-lattices
No full textAbstractWe prove a representation theorem for Hausdorff locally convex (M)-lattices which are Dedekind σ-complete, and whose topologies are order σ-continuous and monotonically complete. These turn out to be the weighted spaces c0(T, H), defined in the paper for T ≠ ∅ and H ⊂ ℝT+. We also characterize the dual of c0(T, H), as the space l1 (T, H) defined in the last section. The known representation (on c0(T)) of Banach (M)-lattices with order continuous norm follows as a particular case. We obtain these results by first proving a new general isomorphism theorem, which seems to be of independent interest. Our notion of “monotonic topological completeness” is weaker than the usual completeness and seems to be very convenient in the framework of topological ordered vector spaces
On the positivity of symmetric polynomial functions. Part I: General results
No full textAbstractWe prove that a real symmetric polynomial inequality of degree d⩾2 holds on R+n if and only if it holds for elements with at most ⌊d/2⌋ distinct non-zero components, which may have multiplicities. We establish this result by solving a Cauchy problem for ordinary differential equations involving the symmetric power sums; this implies the existence of a special kind of paths in the minimizer of some restriction of the considered polynomial function. In the final section, extensions of our results to the whole space Rn are outlined. The main results are Theorems 5.1 and 5.2 with Corollaries 2.1 and 5.2, and the corresponding results for Rn from the last subsection. Part II will contain a discussion on the ordered vector space Hd[n] in general, as well as on the particular cases of degrees d=4 and d=5 (finite test sets for positivity in the homogeneous case and other sufficient criteria)
Uniform Approximation Under Constrains for Continuous Vector-Valued Functions
No full textThe talk will discuss approximations with constraints on the range and on the support. For any ε \u3e 0, the approximant ��ε of �� (continuous maps defined on a topological space ��) is required to take locally (on neighborhoods) values into finite-dimensional subspaces, and to satisfy the restrictions supₜ∈�� ||��(��) − ��ε(��)|| \u3c ε, ��ε(��) ⊆ co(��(��)), supp(��ε) ⊆ int(supp(��)). The result obtained has very distinct applications: a generalization of the Tietze-Dugundji extension theorem, a new proof of the fixed-point theorem of Schauder-Tikhonov, and a density result with respect to the inductive limit topology
On the positivity of symmetric polynomial functions. Part II: Lattice general results and positivity criteria for degrees 4 and 5
No full textAbstractWe prove that homogeneous symmetric polynomial inequalities of degree p∈{4,5} in n positive11Which will mean ⩾0, according to the terminology of ordered vector spaces. variables can be algorithmically tested, on a finite set depending on the given inequality (Theorem 13); the test-set can be obtained by solving a finite number of equations of degree not exceeding p−2. Section 3 discusses the structure of the ordered vector spaces (Hp[n],⪯) and (Hp[n],⋞). In Section 4, positivity criteria for degrees 4 and 5 are stated and proved. The main results are Theorems 10–14. Part III of this work will be concerned with the construction of extremal homogeneous symmetric polynomials (best inequalities) of degree 4 in n positive variables
An isomorphic characterization of L1-spaces
No full textAbstractWe show that a sequentially (τ)-complete topological vector lattice Xτ is isomorphic to some L1(μ), if and only if the positive cone can be written as X+ = ℝ+B for some convex, (τ)-bounded, and (τ)-closed set B ⊂ X+ \ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y+ = ℝ+Y+=ℝ+B¯(u,1) (for ‖u‖ >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials
