Uniform Approximation on the Graph of a Smooth Map in Cn

1980 ◽  
Vol 32 (6) ◽  
pp. 1390-1396 ◽  
Author(s):  
Barnet M. Weinstock
Keyword(s):  
Uniform Approximation ◽  
Compact Set ◽  
Continuous Complex ◽  
Smooth Map ◽  
Image Position ◽  
Complex Valued

Let X be a compact set in Cn, and let ƒ1, …, ƒm, m ≧ n, be continuous, complex-valued functions on X which have C1 extensions to some neighborhood of X. We wish to describe the algebra A of continuous complex-valued functions on X which can be approximated uniformly by polynomials in the functions z1 …, zn, ƒ1 …, ƒm. For this purpose we introduce the setsand

Note on Best Approximation of |x|

1979 ◽  
Vol 22 (3) ◽  
pp. 363-366
Author(s):  
Colin Bennett ◽  
Karl Rudnick ◽  
Jeffrey D. Vaaler
Keyword(s):  
General Problem ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Best Uniform Approximation ◽  
Linear Fractional Transformations ◽  
Symmetric Complex ◽  
Image Position ◽  
Complex Valued ◽  
Special Case

In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximationand the extremal transformations U whenever they exist.

scholarly journals Some Results On The Asymptotic Behavior Of Linear Systems

1955 ◽  
Vol 7 ◽  
pp. 531-538 ◽  
Author(s):  
M. Marcus
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Matrix Differential Equation ◽  
Complex Matrix ◽  
Continuous Complex ◽  
Matrix Differential ◽  
Image Position ◽  
Vector Matrix ◽  
Complex Valued ◽  
Square Complex

1. Introduction. We consider first in §2 the asymptotic behavior as t → ∞ of the solutions of the vector-matrix differential equation(1.1) ,where A is a constant n-square complex matrix, B{t) a continuous complex valued n-square matrix defined on [0, ∞ ), and x a complex n-vector.

scholarly journals Some Fourier division problems

1971 ◽  
Vol 12 (2) ◽  
pp. 167-186
Author(s):  
R. E. Edwards
Keyword(s):  
Fourier Series ◽  
Convergent Fourier Series ◽  
Circle Group ◽  
Division Problems ◽  
Continuous Complex ◽  
Image Position ◽  
Complex Valued

Let T denote the circle group, C the set of continuous complex-valued functions on T, and A the set of f ∈ C having absolutely convergent Fourier series:I standing for the set of integers.

Fubini and martingale theorems in C* -algebras

1979 ◽  
Vol 86 (1) ◽  
pp. 57-69
Author(s):  
C. J. K. Batty
Keyword(s):  
Measure Theory ◽  
Integration Theory ◽  
Affine Function ◽  
Radon Measures ◽  
C Algebras ◽  
Topological Measure Theory ◽  
Continuous Complex ◽  
Second Dual ◽  
Image Position ◽  
Complex Valued

The basic integration theory of Radon measures on locally compact spaces, as described in (5), has been developed in various directions both in commutative and non-commutative analysis. Thus if Ω is a compact Hausdorff space, and C(Ω) denotes the space of continuous complex-valued functions on Ω, Kaplan (16) showed how topological measure theory can be performed in the second dual C(Ω)** of C(Ω). Here as usual C(Ω)* is identified with the space of Radon measures on Ω by associating with a measure μ the linear functional φμ whereThus if f is a bounded real-valued function on Ω, there is an affine function f* defined on the set P(Ω) of Radon probability measures μ on Ω, byand f* extends by linearity to a functional in C(Ω)**. Conversely any function x on P(Ω) determines a function x0 on Ω. bywehere εω is the unit point of mass at ω.

Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups

1986 ◽  
Vol 99 (2) ◽  
pp. 273-283 ◽  
Author(s):  
Anthony To-Ming Lau
Keyword(s):  
Continuous Functions ◽  
Supremum Norm ◽  
Hausdorff Topology ◽  
Closed Subalgebra ◽  
Continuous Complex ◽  
Arens Multiplication ◽  
Image Position ◽  
Bounded Continuous Functions ◽  
Complex Valued ◽  
Uniformly Continuous

Let G be a topological semigroup, i.e. G is a semigroup with a Hausdorff topology such that the map (a, b) → a.b from G × G into G is continuous when G × G has the product topology. Let C(G) denote the space of complex-valued bounded continuous functions on G with the supremum norm. Let LUC (G) denote the space of bounded left uniformly continuous complex-valued functions on G i.e. all f ε C(G) such that the map a → laf of G into C(G) is continuous when C(G) has a norm topology, where (laf )(x) = f (ax) (a, x ε G). Then LUC (G) is a closed subalgebra of C(G) invariant under translations. Furthermore, if m ε LUC (G)*, f ε LUC (G), then the functionis also in LUC (G). Hence we may define a productfor n, m ε LUC(G)*. LUC (G)* with this product is a Banach algebra. Furthermore, ʘ is precisely the restriction of the Arens product defined on the second conjugate algebra l∞(G)* = l1(G)** to LUC (G)*. We refer the reader to [1] and [10] for more details.

scholarly journals A note on rectifiable curves

1956 ◽  
Vol 40 ◽  
pp. 12-14
Author(s):  
W. F. Newns
Keyword(s):  
Bounded Variation ◽  
Plane Curve ◽  
Arc Length ◽  
Absolutely Continuous ◽  
Rectifiable Curve ◽  
Continuous Complex ◽  
Image Position ◽  
Rectifiable Curves ◽  
Complex Valued ◽  
Equality Holding

Let f be a continuous complex-valued function of a real parameter whose real and imaginary parts are of bounded variation in the range (a, b) of the parameter, so that the range of f is a rectifiable plane curve. The main results connecting the arc-length s with the parametrization are as follows:Theorem 1 (Tonelli). For any rectifiable curve,equality holding for all α, β (a ≤ α < β ≤ b) if and only if f is absolutely continuous in (a, b).

Central Double Centralizers on Quasi-Central Banach Algebras with Bounded Approximate Identity

1983 ◽  
Vol 35 (2) ◽  
pp. 373-384
Author(s):  
Sin-Ei Takahasi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Central Complex ◽  
Double Centralizer ◽  
Continuous Complex ◽  
Image Position ◽  
Centralizer Algebra ◽  
Complex Valued ◽  
Canonical Image

We assume throughout this paper that A is a semi-simple, quasi-central, complex Banach algebra with a bounded approximate identity {eα}. The author [6] has shown that every central double centralizer T on A can be, under suitable conditions, represented as a bounded continuous complex-valued function ΦT on Prim A, the structure space of A with the hull-kernel topology, such thatHere x + P for P ∊ Prim A denotes the canonical image of x in A/P. This map Φ is called Dixmier's representation of Z(M(A)), the central double centralizer algebra of A. We denote by τ the canonical isomorphism of A into the Banach algebra D(A) with the restricted Arens product as defined in [6]. Also denote by μ Davenport's representation of Z(M(A)). In fact, this map μ is given byfor each T ∊ Z(M(A)).

14.—A Generalised Walsh-Lebesgue Theorem

1975 ◽  
Vol 73 ◽  
pp. 231-234 ◽  
Author(s):  
A. G. O'Farrell
Keyword(s):  
Compact Set ◽  
Uniform Limit ◽  
Continuous Complex ◽  
Complex Valued ◽  
Lebesgue Theorem

SynopsisLet X be the boundary of a compact set which does not separate the plane, C. Let Φ and Ψ be homeomorphisms of C to C with opposite orientations. Then every continuous complex-valued function on X is the uniform limit on X of sums p(Φ)+q(ψ), where p and q are analytic polynomials.

scholarly journals Approximation with norms defined by derivations

1975 ◽  
Vol 20 (1) ◽  
pp. 18-24
Author(s):  
J. M. Briggs
Keyword(s):  
Continuous Function ◽  
Linear Mapping ◽  
Coordinate Function ◽  
Supremum Norm ◽  
Polynomial Functions ◽  
Continuous Complex ◽  
Image Position ◽  
Complex Valued ◽  
Ordinary Derivative

A linear mapping D of the algebra of polynomial functions P[0, 1] into the algebra of all continuous complex-valued functions C[0,1] is called a derivation provided D(fg) = fD(g) + gD(f) for all polynomials f and g. The derivations of P[0, 1] into C[0,1] are easily seen to be all mappings of the form Dw where w is a continuous function on [0, 1] and Dw (f) = wf' (f' denotes the ordinary derivative of f). In fact, w = D(x) where x is the coordinate function. Let Dw be such a derivation, and let ∥ · ∥ denote the supremum norm on C[0,1]. Then Dw gives rise to an algebra norm ∥ · ∥w on P[0,1] denned by .

Isometries of self-adjoint complex function spaces

1989 ◽  
Vol 105 (1) ◽  
pp. 133-138 ◽  
Author(s):  
A. J. Ellis
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Complex Function ◽  
Supremum Norm ◽  
Affine Functions ◽  
Continuous Complex ◽  
Image Position ◽  
Complex Valued ◽  
Uniformly Closed

By a complex function space A we will mean a uniformly closed linear space of continuous complex-valued functions on a compact Hausdorff space X, such that A contains constants and separates the points of X. We denote by S the state-spaceendowed with the w*-topology. If A is self-adjoint then it is well known (cf. [1]) that A is naturally isometrically isomorphic to , and re A is naturally isometrically isomorphic to A(S), where (respectively A(S)) denotes the Banach space of all complex-valued (respectively real-valued) continuous affine functions on S with the supremum norm.

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