Strong Unicity Research Articles

Examples of functions whose sequence of strong unicity constants is unbounded

Examples of functions whose sequence of strong unicity constants is unbounded

On the Strong Unicity of Best Chebyshev Approximation of Differentiable Functions

On the Strong Unicity of Best Chebyshev Approximation of Differentiable Functions

Strong unicity of arbitrary rate

The concept is introduced of strong unicity with respect to a rate function u, i.e., ∥ f − p∥ ⩾ ∥ f − p f ∥ + γu(∥ p − p f ∥), in approximating (with constraints) f in a Banach space X from an n-dimensional subspace V ( p ϵ V, p f denotes the best approximation to f, and γ denotes a positive constant). Past work has demonstrated examples of monotone approximation in C[ a, b], where V is Haar and the best u has polynomial decay of arbitrary even degree (i.e., u( t) = t 2 m , m = 1,2,…,). In particular, in this same setting examples are demonstrated where the best u decays exponentially (e.g., exp(−c 2t −2 3 ) ⩽ u(t) ⩽ t −2 3 exp(−c 1t −2 3 ) for constants 0 < c 1 < c 2) and a general statement is provided relating the best u to h″ when V = [1, x, h′( x), h( x)] and h ϵ C 2 satisfies certain conditions.

On extremal sets and strong unicity constants for certain C∞ functions

For each f continuous on the interval I, let B n ( f) denote the best uniform polynomial approximation of degree less than or equal to n. Let M n ( f) denote the corresponding strong unicity constant. For a certain class of nonrational functions F, it is shown that there exist positive constants α and β and a natural number N such that αn ⩽ M n ( f) ⩽ βn for n ⩾ N. The results of the present paper also provide concise estimates to the location of the extreme points of f − B n ( f). The set F includes the functions f α ( x) = e αx , α ≠ 0.

On the strong unicity of best Chebyshev approximation of differentiable functions

Let X X be a normed linear space, U n {U_n} an n n -dimensional Chebyshev subspace of X X . For f ∈ X f \in X denote by p ( f ) ∈ U n p(f) \in {U_n} its best approximation in U n {U_n} . The problem of strong unicity consists in estimating how fast the nearly best approximants g ∈ U n g \in {U_n} satisfying ‖ f − g ‖ ⩽ ‖ f − p ( f ) ‖ + δ \left \| {f - g} \right \| \leqslant \left \| {f - p(f)} \right \| + \delta approach p ( f ) p(f) as δ → 0 \delta \to 0 . In the present note we study this problem in the case when X = C r [ a , b ] X = {C^r}[a,b] is the space of r r -times continuously differentiable functions endowed with the supremum norm ( 1 ⩽ r ⩽ ∞ ) (1 \leqslant r \leqslant \infty ) .

Strong unicity constants for spline functions

In the last years there has been a great deal of interest in strong unicity constants for the space of polynomials of degree at most n. We consider analogous questions for the space of spline functions of degree at most n with k fixed knots. In this case we give formulas and bounds for strong unicity constants and investigate strong unicity for generalized convex functions of order n, i.e. functions whose (n+1)–st derivative is positive.

Strong unicity of best approximations : A numerical aspect

The set of functions in C(T) which have a strongly unique best approximation from a given finite-dimensional subspace is denoted by SU(G). Since strong unicity plays an important role in numerical computations and since there the functions are only known up to some error, it is natural to ask what are the functions from the interior of SU(G). A complete characterization of those functions is given and the result is applied to weak Chebyshev and spline subspaces.

Lebesgue and strong unicity constants for Zolotareff polynomials

Lebesgue and strong unicity constants for Zolotareff polynomials

On strong unicity of best approximation in C(R)

In the present note we give a necessary and sufficient condition on a Haar subspace which guarantees the strong unicity of best approximation in the space C(R) endowed with a certain metric Let C(R) denote the space of real valued continuous functions on the real line R endowed with the metric where h e C(R) is a given positive even function such that Furthermore, let Un be an n-dimensional Haar subspace of C(R), that is each p eUn not identically zero has at most n – 1 zeros on R. The problem of best approximation in metric of this type was investigated by G.Albinus [1]. We say that p0 e Un is a best approximation of f e C(R) if d(f,p0)≤ d(f,p) for each p e Un The following alternation theorem gives a characterization of best approximation.

An Example on Strong Unicity Constants in Trigonometric Approximation

An Example on Strong Unicity Constants in Trigonometric Approximation

An example on strong unicity constants in trigonometric approximation

In this paper an explicit example is constructed to illustrate some interesting properties of the strong unicity constant in the case of approximations by trigonometric polynomials.

On Strong Unicity of L 1 -Approximation

On Strong Unicity of L 1 -Approximation

Uniform strong unicity for rational approximation

Let R n m denote the class of rational functions defined on a closed interval I with numerators in the class of polynomials of degree at most n and positive valued denominators in the class of polynomials of degree at most m. If f ϵ C( I) is normal, the well-known strong unicity theorem asserts that there is a smallest positive constant γ n, m ( f) such that ∥ f − R∥ ⩾ ∥ f − R f ∥ + γ n, m ( f)∥ R − R f ∥ for all R ϵ R n m , where R f is the best uniform approximation to f from R n m . In this paper, the dependence of γ n, m ( f) on f is investigated. Sufficient conditions are given to insure that inf fϵΓ γ n, m ( f) > 0, where Γ is a subset of C( I). Necessity of these conditions is investigated and examples are given to show that known results for R n 0 do not directly extend to R n m for m > 0.

On Poreda's problem on the strong unicity constants

Let c‘(I) denote the space of continuous. real~valued functions on the inter\,al I ICI. 6) endowed with the uniform norm /I 11. Let U,, denote the set of all algebraic polynomials of degree II or less. and let II denote the set of all algebraic polynomials. For a given SF C(I). with best uniform approximation r,,(.f) from n,,. Newman and Shapiro 1 IO] showed that there is a constant ;’ > 0 such that

Precise orders of strong unicity constants for a class of rational functions

Let R ̂ ⊆ C[∞-1, 1] denote a certain class of rational functions. For each f ϵ R̂, consider the polynomial of degree at most n that best approximates f in the uniform norm. The corresponding strong unicity constant is denoted by M n ( f). Then there exist positive constants α and β, not depending on n, such that an ⩽ M n ( f)⩽ βn, n = 1,2,….

Continuity and strong unicity of the best approximation operator on subintervals

Continuity and strong unicity of the best approximation operator on subintervals

Orders of strong unicity constants

Given f ϵ C( I), the growth of the strong unicity constant M n ( f) for changing dimension is considered. Under appropriate hypotheses it is shown that 2 n + 1 ⩽ M n ( f) ⩽ βn 2. Furthermore, relationships between certain Lebesgue constants and M n ( f) are established.

On strong unicity of 𝐿₁-approximation

Let C 1 {C_1} be the space of continuous functions on [ 0 , 1 ] [0,1] with norm ‖ f ‖ = ∫ 0 1 | f ( x ) | d x \left \| f \right \| = \int _0^1 {|f(x)|} dx , and let G ⊂ C 1 G \subset {C_1} be a finite dimensional unicity subspace, i.e. any f ∈ C 1 f \in {C_1} possesses a unique best approximation out of G G . Consider an arbitrary f ∈ C 1 f \in {C_1} such that 0 is its best approximation. Then for any 0 &gt; δ &gt; δ 0 0 &gt; \delta &gt; {\delta _0} and g ~ ∈ G \tilde g \in G with ‖ f − g ~ ‖ ⩽ ‖ f ‖ + δ \left \| {f - \tilde g} \right \| \leqslant \left \| f \right \| + \delta it follows that ‖ g ~ ‖ ⩽ K w f ∗ ( δ ) \left \| {\tilde g} \right \| \leqslant Kw_f^* (\delta ) , where w f ∗ ( h ) = ( w f − 1 ( h ) h ) − 1 w_f^*(h) = {(w_f^{ - 1}(h)h)^{ - 1}} and w f ( h ) {w_f}(h) denotes the modulus of continuity of f f . ( − 1 - 1 is used to denote the inverse function.)

Continuity of the strong unicity constant on C( X) for changing X

Continuity of the strong unicity constant on C( X) for changing X

Strong unicity and lipschitz conditions of order [formula omitted] for monotone approximation

Strong unicity and lipschitz conditions of order [formula omitted] for monotone approximation