Alternating Chebyshev Approximation with A Non-Continuous Weight Function

1976 ◽  
Vol 19 (2) ◽  
pp. 155-157 ◽  
Author(s):  
Charles B. Dunham
Keyword(s):  
Weight Function ◽  
Best Approximation ◽  
Closed Interval ◽  
Chebyshev Approximation ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Negative Function ◽  
Chebyshev Problem ◽  
Image Position ◽  
Continuous Weight

Let [α, β] be a closed interval and C[α, β] be the space of continuous functions on [α, β], For g a function on [α, β] defineLet s be a non-negative function on [α, β]. Let F be an approximating function with parameter space P such that F(A, .)∊ C[α, β] for all A∊P. The Chebyshev problem with weight s is given f ∊ C[α, β], to find a parameter A* ∊ P to minimize e(A) = ||s * (f - F(A, .))|| over A∊P. Such a parameter A* is called best and F(A*,.) is called a best approximation to f.

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

scholarly journals The Borel structure of iterates of continuous functions

1989 ◽  
Vol 32 (3) ◽  
pp. 483-494 ◽  
Author(s):  
Paul D. Humke ◽  
M. Laczkovich
Keyword(s):  
Banach Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Space Of Continuous Functions ◽  
Borel Structure ◽  
Image Position ◽  
Set Of Points ◽  
Positive Integers

Let C[0,1] be the Banach space of continuous functions defined on [0,1] and let C be the set of functions f∈C[0,1] mapping [0,1] into itself. If f∈C, fk will denote the kth iterate of f and we put Ck = {fk:f∈C;}. The set of increasing (≡ nondecreasing) and decreasing (≡ nonincreasing) functions in C will be denoted by ℐ and D, respectively. If a function f is defined on an interval I, we let C(f) denote the set of points at which f is locally constant, i.e.We let N denote the set of positive integers and NN denote the Baire space of sequences of positive integers.

Limit Point Criteria for Differential Equations, II

1974 ◽  
Vol 26 (02) ◽  
pp. 340-351 ◽  
Author(s):  
Don Hinton
Keyword(s):  
Differential Equations ◽  
Weight Function ◽  
Limit Point ◽  
Differential Operators ◽  
Linearly Independent ◽  
Integrable Functions ◽  
Image Position ◽  
Continuous Weight ◽  
Finite Singularity

We consider here singular differential operators, and for convenience the finite singularity is taken to be zero. One operator discussed is the operator L defined by where q 0 > 0 and the coefficients q t are real, locally Lebesgue integrable functions defined on an interval (a, b). For a given positive, continuous weight function h, conditions are given on the functions qi for which the number of linearly independent solutions y of L(y) = λhy (Re λ = 0) satisfying.

scholarly journals Averaging distances in certain Banach spaces

1997 ◽  
Vol 55 (1) ◽  
pp. 147-160 ◽  
Author(s):  
Reinhard Wolf
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Finite Dimension ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Empty Interval ◽  
Positive Real ◽  
Complete Discussion ◽  
Image Position

Let E be a Banach space. The averaging interval AI(E) is defined as the set of positive real numbers α, with the following property: For each n ∈ ℕ and for all (not necessarily distinct) x1, x2, … xn ∈ E with ∥x1∥ = ∥x2∥ = … = ∥xn∥ = 1, there is an x ∈ E, ∥x∥ = 1, such thatIt follows immediately, that AI(E) is a (perhaps empty) interval included in the closed interval [1,2]. For example in this paper it is shown that AI(E) = {α} for some 1 < α < 2, if E has finite dimension. Furthermore a complete discussion of AI(C(X)) is given, where C(X) denotes the Banach space of real valued continuous functions on a compact Hausdorff space X. Also a Banach space E is found, such that AI(E) = [1,2].

scholarly journals Numerical Solutions of a Class of Nonlinear Volterra Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
H. S. Malindzisa ◽  
M. Khumalo
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Closed Interval ◽  
Continuous Functions ◽  
Volterra Integral Equations ◽  
Collocation Methods ◽  
Uniform Mesh ◽  
Space Of Continuous Functions ◽  
Point Collocation ◽  
Uniqueness Of The Solution

We consider numerical solutions of a class of nonlinear (nonstandard) Volterra integral equations. We first prove the existence and uniqueness of the solution of the Volterra integral equation in the context of the space of continuous functions over a closed interval. We then use one-point collocation methods with a uniform mesh to construct solutions of the nonlinear (nonstandard) VIE and quadrature rules. We conclude that the repeated Simpson's rule gives better solutions when a reasonably large value of the stepsize is used.

scholarly journals Integro-differential equations of Volterra type

1970 ◽  
Vol 3 (1) ◽  
pp. 9-22 ◽  
Author(s):  
M. Rama Mohana Rao ◽  
Chris P. Tsokos
Keyword(s):  
Differential System ◽  
Basic Result ◽  
Sufficient Conditions ◽  
Continuous Operator ◽  
Continuous Functions ◽  
Uniform Asymptotic Stability ◽  
Space Of Continuous Functions ◽  
Volterra Type ◽  
The Stability ◽  
Image Position

The aim of this paper is concerned with studying the stability properties of an integro-differential system by reducing it into a scalar integro-differential equation. A theorem is stated about the existence of a maximal solution of such systems and a basic result on integro-differential inequalities. Utilizing these results we obtain sufficient conditions for uniform asymptotic stability of the trivial solution of the integro-differential system of the form where , with , , C(J) denotes the space of continuous functions, A a continuous operator such that A maps C(J) into C(J). The fruitfulness of the results of the paper are illustrated with two applications.

Best Trigonometric Approximation, Fractional Order Derivatives and Lipschitz Classes

1977 ◽  
Vol 29 (4) ◽  
pp. 781-793 ◽  
Author(s):  
P. L. Butzer ◽  
H. Dyckhoff ◽  
E. Görlich ◽  
R. L. Stens
Keyword(s):  
Fractional Order ◽  
Best Approximation ◽  
Continuous Functions ◽  
Trigonometric Approximation ◽  
Trigonometric Polynomials ◽  
Lipschitz Classes ◽  
The Best Approximation ◽  
Image Position ◽  
Best Trigonometric Approximation

Let C2π denote the space of 2π-periodic continuous functions and πn the set of trigonometric polynomials of degree ≦ n, where n ϵ P = {0, 1, … } . Given θ > 0, the well-known theorem of Stečkin and its converse state that the best approximation of an ƒ ϵ C2π with respect to the max-norm satisfies

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