Some best-approximation theorems in tensor-product spaces

1981 ◽  
Vol 89 (3) ◽  
pp. 385-390 ◽  
Author(s):  
W. A. Light ◽  
E. W. Cheney
Keyword(s):  
Tensor Product ◽  
Best Approximation ◽  
Product Form ◽  
Product Spaces ◽  
Approximation Theorems ◽  
Univariate Functions ◽  
Bivariate Function ◽  
Image Position

We begin by describing a concrete example from the class of problems to be considered. A continuous bivariate function f defined on the square |t| ≤ 1, |s| ≤ 1 is to be approximated by a tensor-product form involving univariate functions. For example, the approximation may be prescribed to have the formin which the Ti are the Tchebycheff polynomials, and the coefficient functions xi(t) and yi(s) are to be chosen to achieve a good or best approximation. Will a best approximation exist? If so, how can it be obtained?

scholarly journals Best approximation problems in tensor-product spaces

1982 ◽  
Vol 102 (2) ◽  
pp. 437-446 ◽  
Author(s):  
J. R. Respess ◽  
Elliott Cheney
Keyword(s):  
Tensor Product ◽  
Best Approximation ◽  
Product Spaces ◽  
Approximation Problems

scholarly journals Classification of tensor product immersions which are of 1-type

1994 ◽  
Vol 36 (2) ◽  
pp. 255-264 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Tensor Product ◽  
Binary Operation ◽  
Inner Product ◽  
Dimensional Euclidean Space ◽  
Product Spaces ◽  
Image Position ◽  
Product Map ◽  
Tensor Product Immersion

Let V and W be two vector spaces over the field of real numbers R. Then we have the notion of the tensor product V ⊗ W. If V and W are inner product spaces with their inner products given respectively by «,»v and «,» w, then V ⊗ W is also an inner product space with inner product denned byLet Em denote the m-dimensional Euclidean space with the canonical Euclidean inner product. Then, with respect to the inner product defined above, Em ⊗Em is isometric to Em. By applying this algebraic notion, we have the notion of tensor product mapf ⊗h: M→ E: M ⊗= Em; associated with any two maps f: M→Em and h:M→E of a given Riemannian manifold (M, g) defined as follows:Denote by R(M) the set of all transversal immersions from an n-dimensional Riemannian manifold (M, g) into Euclidean spaces; i. e., immersions f:M→Em with f(p) ∉T*(TPM) for p ∈ M. Then ⊗ is a binary operation on R(M). Hence, if f: Mm and h: M→Em are immersions belonging to R(M), then their tensor product map f ⊗ h: M→ Em ⊗ Em ≡ Emm is an immersion in R(M), called the tensor product immersionof f and h.

scholarly journals Approximation with monotone norms in tensor product spaces

1992 ◽  
Vol 68 (2) ◽  
pp. 183-205 ◽  
Author(s):  
W.A Light ◽  
M.v Golitschek ◽  
E.W Cheney
Keyword(s):  
Tensor Product ◽  
Product Spaces

ON THE HYPERSTABILITY OF A PEXIDERISED -QUADRATIC FUNCTIONAL EQUATION ON SEMIGROUPS

2018 ◽  
Vol 97 (3) ◽  
pp. 459-470 ◽  
Author(s):  
IZ-IDDINE EL-FASSI ◽  
JANUSZ BRZDĘK
Keyword(s):  
Functional Equation ◽  
Commutative Semigroup ◽  
Quadratic Functional Equation ◽  
Inner Product ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Image Position

Motivated by the notion of Ulam stability, we investigate some inequalities connected with the functional equation $$\begin{eqnarray}f(xy)+f(x\unicode[STIX]{x1D70E}(y))=2f(x)+h(y),\quad x,y\in G,\end{eqnarray}$$ for functions $f$ and $h$ mapping a semigroup $(G,\cdot )$ into a commutative semigroup $(E,+)$, where the map $\unicode[STIX]{x1D70E}:G\rightarrow G$ is an endomorphism of $G$ with $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D70E}(x))=x$ for all $x\in G$. We derive from these results some characterisations of inner product spaces. We also obtain a description of solutions to the equation and hyperstability results for the $\unicode[STIX]{x1D70E}$-quadratic and $\unicode[STIX]{x1D70E}$-Drygas equations.

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 KKM theorems in modular function spaces

Filomat ◽  
2014 ◽  
Vol 28 (7) ◽  
pp. 1307-1313
Author(s):  
Nasrin Karamikabir ◽  
Abdolrahman Razani
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Best Approximation ◽  
Modular Function ◽  
Function Spaces ◽  
Minimax Inequality ◽  
Approximation Theorems ◽  
Coincidence Theorem ◽  
Modular Function Spaces ◽  
Ky Fan

In this paper, a coincidence theorem is obtained which is generalization of Ky Fan?s fixed point theorem in modular function spaces. A modular version of Fan?s minimax inequality is proved. Moreover, some best approximation theorems are presented for multi-valued mappings.

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