On a minimal property of trigonometric interpolation at equidistant nodes

Computing ◽  
1981 ◽  
Vol 27 (4) ◽  
pp. 371-372 ◽  
Author(s):  
B. Sündermann
Keyword(s):  
Trigonometric Interpolation ◽  
Equidistant Nodes ◽  
Minimal Property

SOME 2-PERIODIC TRIGONOMETRIC INTERPOLATION PROBLEMS ON EQUIDISTANT NODES

Analysis ◽  
1991 ◽  
Vol 11 (2-3) ◽  
Author(s):  
A. SHARMA ◽  
J . SZABADOS ◽  
R . S . VARGA
Keyword(s):  
Trigonometric Interpolation ◽  
Interpolation Problems ◽  
Equidistant Nodes

LACUNARY TRIGONOMETRIC INTERPOLATION ON EQUIDISTANT NODES

Analysis ◽  
1986 ◽  
Vol 6 (2-3) ◽  
Author(s):  
A. Jakimovski ◽  
A. Sharma
Keyword(s):  
Trigonometric Interpolation ◽  
Equidistant Nodes

Convergence of lacunary trigonometric interpolation on equidistant nodes

1982 ◽  
Vol 39 (1-3) ◽  
pp. 27-37 ◽  
Author(s):  
S. Riemenschneider ◽  
A. Sharma ◽  
P. W. Smith
Keyword(s):  
Trigonometric Interpolation ◽  
Equidistant Nodes

scholarly journals Trigonometric interpolation

1992 ◽  
Vol 35 (3) ◽  
pp. 457-472
Author(s):  
T. N. T. Goodman ◽  
A. Sharma
Keyword(s):  
Interpolation Problem ◽  
Arbitrary Sequence ◽  
Trigonometric Interpolation ◽  
Equidistant Nodes ◽  
Explicit Expressions

We consider interpolation at 2n equidistant nodes in [0,π) from the space ℱN spanned by sines and cosines of odd multiples of x. This interpolation problem is shown to be correct for an arbitrary sequence of derivatives specified at all the nodes. Explicit expressions for the fundamental polynomials are obtained and it is shown that under mild smoothness assumptions on the function f interpolant from ℱN converges uniformly to f as the node spacing goes to zero.

Minimal invariant spaces in formal topology

1997 ◽  
Vol 62 (3) ◽  
pp. 689-698 ◽  
Author(s):  
Thierry Coquand
Keyword(s):  
Direct Consequence ◽  
Arithmetical Progression ◽  
Standard Result ◽  
Continuous Map ◽  
Compact Topological Space ◽  
Combinatorial Theorem ◽  
Minimal Property ◽  
Special Instance ◽  
Invariant Spaces ◽  
Formal Topology

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.

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