Displaying similar documents to “ L -Convergence of Finite Element Approximation”

Convergence of greedy approximation I. General systems

S. V. Konyagin, V. N. Temlyakov (2003)

Studia Mathematica

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We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as T ε ( x ) : = j D ε ( x ) e * j ( x ) e j , where D ε ( x ) : = j : | e * j ( x ) | ε . We study a generalized version of T ε that we call the weak thresholding approximation. We modify the T ε ( x ) in the following way. For ε > 0, t ∈ (0,1) we set D t , ε ( x ) : = j : t ε | e * j ( x ) | < ε and consider...

Convergence of greedy approximation II. The trigonometric system

S. V. Konyagin, V. N. Temlyakov (2003)

Studia Mathematica

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We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form G ( f ) : = k Λ f ̂ ( k ) e i ( k , x ) , where Λ d is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in...

On the approximation of real continuous functions by series of solutions of a single system of partial differential equations

Carsten Elsner (2006)

Colloquium Mathematicae

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We prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function f : s can be approximated with arbitrary accuracy by an infinite sum r = 1 H r ( x , . . . , x s ) C ( s ) of analytic functions H r , each solving the same system of universal partial differential equations, namely P ( x σ ; H r , H r / x σ , . . . , H r / x σ ) = 0 (σ = 1,..., s).

Some duality results on bounded approximation properties of pairs

Eve Oja, Silja Treialt (2013)

Studia Mathematica

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The main result is as follows. Let X be a Banach space and let Y be a closed subspace of X. Assume that the pair ( X * , Y ) has the λ-bounded approximation property. Then there exists a net ( S α ) of finite-rank operators on X such that S α ( Y ) Y and | | S α | | λ for all α, and ( S α ) and ( S * α ) converge pointwise to the identity operators on X and X*, respectively. This means that the pair (X,Y) has the λ-bounded duality approximation property.

On the multiples of a badly approximable vector

Yann Bugeaud (2015)

Acta Arithmetica

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Let d be a positive integer and α a real algebraic number of degree d + 1. Set α ̲ : = ( α , α ² , . . . , α d ) . It is well-known that c ( α ̲ ) : = l i m i n f q q 1 / d · | | q α ̲ | | > 0 , where ||·|| denotes the distance to the nearest integer. Furthermore, c ( α ̲ ) n - 1 / d c ( n α ̲ ) n c ( α ̲ ) for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that c ( n α ̲ ) C n - 1 / d for any integer n ≥ 1.

Three-space problems for the approximation property

A. Szankowski (2009)

Journal of the European Mathematical Society

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It is shown that there is a subspace Z q of q for 1 < q < 2 which is isomorphic to q such that q / Z q does not have the approximation property. On the other hand, for 2 < p < there is a subspace Y p of p such that Y p does not have the approximation property (AP) but the quotient space p / Y p is isomorphic to p . The result is obtained by defining random “Enflo-Davie spaces” Y p which with full probability fail AP for all 2 < p and have AP for all 1 p 2 . For 1 < p 2 , Y p are isomorphic to p .

On C * -spaces

P. Srivastava, K. K. Azad (1981)

Matematički Vesnik

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