Set Of Analytic Functions Research Articles

Integrating n-unisolvent sets of functions

We prove that integrating an n-unisolvent set F of continuous real-valued functions on some open or half-open interval of \( \Bbb R \) gives an (n + 1 )-unisolvent set S (F) of functions. If F solves the Hermite interpolation problem and the interval is open, then S (F) also solves the Hermite interpolation problem.¶In a second section we verify to what extent these results can be generalized to n-unisolvent sets of analytic functions defined on domains in \( \Bbb C \). The n-unisolvent sets in this section are linear spans of Chebyshev systems that are associated with completely (n - 1 )-valent functions.

Remarks on the descriptive metric characterization of singular sets of analytic functions

This work presents two remarks on the structure of singular boundary sets of functions analytic in the unit diskD: ¦z¦<1. The first remark concerns the conversion of the Plessner theorem. We prove that three pairwise disjoint subsetsE1,E2, andE3 of the unit circle Γ: ¦z¦=1,\( \cup _{i = 1}^3 E_i\) = Γ, are the setsI(ƒ) of all Plessner points,F(ƒ) of all Fatou points, andE(ƒ) of all exceptional boundary points, respectively, for a function ƒ holomorphic inD if and only ifE1 is aGδ-set andE3 is a\(G_{\delta \sigma }\)-set of linear measure zero. In the second part of the paper it is shown that for any\(G_{\delta \sigma }\)-subsetE of the unit circle Γ with a zero logarithmic capacity there exists a one-sheeted function onD whose angular limits do not exist at the points ofE and do exist at all the other points of Γ.

The range of a continuous linear functional over a class of functions defined by subordination

Let Δ = {z ∈ ℂ ⃒ ⃓z⃓ &lt;1) and H(Δ) the set of analytic functions on Δ. We recall the definition of subordination between two functions, say ƒ and g, analytic on Δ: this means that f(0)= g(0) and there is a function ρ ∈ H (Δ) such that ρ(0) = 0, ⃒ ρ(z)⃒&lt;1 if z ∈ Δ, and f(z) ≡ g(ρ(z)). Subordination between f and g will be denoted byf&lt;g. The Hadamard product (or convolution) of two functions and in H(Δ)is the function f * g ∈ H(Δ)definedas f * g (z)= .

H(ϕ) Spaces

AbstractLet ψ be a non-decreasing continuous subadditive function defined on [0, ∞) and satisfy ψ(x) = 0 if and only if x = 0. The space H(ψ) is defined as the set of analytic functions in the unit disk which satisfyand the space H+ (ψ) is the space of a f ∊ H(ψ) for whichwhere almost everywhere.In this paper we study the H(ψ) spaces and characterize the continuous linear functionals on H+ (ψ).

On Dual Sets of Analytic Functions

On Dual Sets of Analytic Functions

A stochastic proof of an extension of a theorem of Rado

The purpose of this article is to illustrate how the theorem of Lévy about conformal invariance of Brownian motion can be used to obtain information about boundary behaviour and removable singularity sets of analytic functions. In particular, we prove a Frostman–Nevanlinna–Tsuji type result about the size of the set of asymptotic values of an analytic function at a subset of the boundary of its domain of definition (Theorem 1). Then this is used to prove the following extension of the classical Radó theorem: If φ is analytic in B\K, where B is the unit ball of ℂ;n and K is a relatively closed subset of B, and the cluster set of φ at K has zero harmonic measure w.r.t. φ(B\K)\≠Ø, then φ extends to a meromorphic function in B (Theorem 2).

On Extreme Points of Subordination Families

Let F be the set of analytic functions in U = {z: I 1 ) < I 3 subordinate to a univalent function f. Let D = f(U). For g(.7) = f( k(z)) E F, let AX() denote the distance between g( e'l) and aD (boundary of D). We obtain the following results. (1) Iff' is Nevanlinna then f(2, log X(0) dO = -oc if and only if 7log(] -t IP(ef9) 1) dlS -x 0 (2) If g is an extreme point of the closed convex hull of F then fJ 2?log(l t-(e') ) dO0 = -x, 0 for the case when D is a Jordan domain subset to a half-plane and f' is Nevanlinna.

New Support Points of S and Extreme Points of HS

Let S be the usual class of univalent analytic functionsf on (zIz I < 1) normalized byf(z) = z + a2z2 + . We prove that the functions z (X +y)z2 f,((z) = 1_)2 IxI = IyI=l, x #Y, which are support points of C(, the subclass of S of close-to-convex functions, and extreme points of 9C C, are support points of S and extreme points of 9CCS whenever 0 < larg(-x/y)l < 7r/4. We observe that the known bound of ir/4 for the acute angle between the omitted arc of a support point of S and the radius vector is achieved by the functions fx, with Jarg(-x/y)J = ir/4. Introduction. Let d be the set of analytic functions on the open unit disk. With the usual topology of uniform convergence on compacta C is a locally convex linear topological space. Suppose 93 c 6d. A function b in % is called a support point of 93 if b maximizes Re J over i for some continuous linear functional J on i? such that Re J is not constant on . Let 5C % denote the closed convex hull of %. A function b in 9C% is called an extreme point of 9C% if b = tbl + (1t)b2 implies b = b = b2 whenever 0 < t < 1 and b,, b2 E SC . Let S be the usual class of univalent functions f in 6! normalized by f(z) = z + 2 1 . 2A L.Bika D. R. a2Z + A. Pfluger [10] and L. Brickman and D. R. Wilken (3] have shown that if f is a support point of 5, then f maps the open unit disk to the complement of an analytic arc IF, which tends to xo with increasing modulus. Furthermore, r satisfies the 7T/4-property, i.e., if r is oriented so that r is (positively) traversed from the finite tip to oo, then the angle between the oriented tangent vector to r and the radius vector to r at any point is less than or equal to sr/4, with strict inequality at each point of r except possibly at the finite tip. In an early paper [1] L. Brickman proved that if f in S is an extreme point of S5, then f maps the open unit disk to the complement of an arc which tends to so with increasing modulus. Later W. E. Kirwan and R. W. Pell [9] improved Brickman's result. A special case of their result states that if f in S is an extreme point of 'CS and if the omitted arc of f is smooth, then the omitted arc of f satisfies the v/4-property, albeit, not necessarily with strict inequality. Since S and SCS are compact a lemma in Dunford and Schwartz [5, p. 440] implies that if f is an extreme point of CX 5, then f ES. The following lemma shows that in certain cases we can identify support points of S as extreme points of Received by the editors February 10, 1980; presented to the Society, August 19, 1980. 1980 Mathematics Subject Classification. Primary 30C75; Secondary 30C45.

The n-width of sets of analytic functions

The n-width of sets of analytic functions

A new method for deriving rigorous results on ππ scattering

A new approach to the problem of constraining the pi pi scattering amplitudes by means of the axiomatically proved properties of unitarity, analyticity, and crossing symmetry is developed. The method is based on the solution of an extremal problem on a convex set of analytic functions and provides a global description of the domain of values taken by any finite number of partial waves at an arbitrary set of unphysical energies, compatible with unitarity, the bounds at complex energies derived from generalised dispersion relations and the crossing integral relations. From this domain the authors obtain new absolute bounds for the amplitudes as well as rigorous correlations between the values of various partial waves.

Optimal analytic extrapolation for the scattering amplitude from cuts to interior points

Given a data function together with an error corridor for the scattering amplitude along some finite part of the cuts, one can construct effectively the whole set of analytic functions (``admissible amplitudes''), compatible with these conditions and bounded by a certain number M on the remaining part of the cuts. Depending on the actual value of an important constant ε0 computed from the data function and the bound M, this set may be void. If not, in every point of the cut plane the set of values of the admissible amplitudes fills densely a circle; explicit formulas are given for its radius η̂(z) and center f̂(z), the latter being the best possible estimate for the whole set. In contrast to the linear extrapolation obtained by Poisson weighted dispersion relations, here nonlinear functional methods were used. This paper contains an appendix written by Professor C. Foias, on some functional analytical methods used in connection with the computation of the numerical value of the constant ε0.

A characterization of the extreme points of some convex sets of analytic functions

A characterization of the extreme points of some convex sets of analytic functions

Lorentz expansion of a multiparticle amplitude

An asymptotic expansion of a multiparticle amplitude, written in terms of group-theoretical variables and free of kinematical singularities and constraints, is derived in terms of Lorentz pole contributions. The covariance relations of the amplitude are explicitly taken into account in order to express the Lorentz residues in terms of a set of analytic functions with explicitly known kinematical factors. The connection with the BCP amplitude is derived. Massive particles of arbitrary spin and parity are considered. The zeros of the residue functions at integral values of the Lorentz trajectory are briefly discussed.

Products of basic sets of analytic functions

Theorems are proved on the effectiveness of the product of two basic sets showing that in certain cases the effectiveness of the factors is necessary for that of the product. Certain corrections to the author’s previous paper ‘On the representation of analytic functions by infinite series’ are made.