Classical Approximation

1995 ◽  
Author(s):  
Marek A. Kowalski ◽  
Krzystof A. Sikorski ◽  
Frank Stenger
Keyword(s):  
20Th Century ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Inner Product ◽  
Normed Spaces ◽  
Remez Algorithm ◽  
Best Uniform Approximation ◽  
Converse Theorems ◽  
Inner Product Spaces ◽  
Finite Dimensional

In this chapter we acquaint the reader with the theory of approximation of elements of normed spaces by elements of their finite dimensional subspaces. The theory of best approximation was originated between 1850 and 1860 by Chebyshev. His results and ideas have been extended and complemented in the 20th century by other eminent mathematicians, such as Bernstein, Jackson, and Kolmogorov. Initially, we present the classical theory of best approximation in the setting of normed spaces. Next, we discuss best approximation in unitary (inner product) spaces, and we present several practically important examples. Finally, we give a reasonably complete presentation of best uniform approximation, along with examples, the Remez algorithm, and including converse theorems about best approximation. The goal of this section is to present some general results on approximation in normed spaces.

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.

Self-adjoint operators on inner product spaces over fields of power series

2008 ◽  
Vol 58 (4) ◽  
Author(s):  
Hans Keller ◽  
Ochsenius Herminia
Keyword(s):  
Power Series ◽  
Adjoint Operator ◽  
Inner Product ◽  
Closed Field ◽  
Product Spaces ◽  
Binary Forms ◽  
Inner Product Spaces ◽  
Finite Dimensional ◽  
Orthogonal Decompositions ◽  
Adjoint Operators

AbstractTheorems on orthogonal decompositions are a cornerstone in the classical theory of real (or complex) matrices and operators on ℝn. In the paper we consider finite dimensional inner product spaces (E, ϕ) over a field K = F((χ 1, ..., x m)) of generalized power series in m variables and with coefficients in a real closed field F. It turns out that for most of these spaces (E, ϕ) every self-adjoint operator gives rise to an orthogonal decomposition of E into invariant subspaces, but there are some salient exceptions. Our main theorem states that every self-adjoint operator T: (E, ϕ) → (E, ϕ) is decomposable except when dim E is a power of 2 with exponent at most m, and ϕ is a tensor product of pairwise inequivalent binary forms. In the exceptional cases we provide an explicit description of indecomposable operators.

scholarly journals Some equivalent characterizations of inner product spaces and their consequences

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1587-1599 ◽  
Author(s):  
Dan Marinescu ◽  
Mihai Monea ◽  
Mihai Opincariu ◽  
Marian Stroe
Keyword(s):  
Inner Product ◽  
Product Spaces ◽  
Normed Spaces ◽  
Inner Product Spaces

In this paper, first we prove the equivalence of the Frechet and the Jordan-Neumann relations in normed spaces. Next, we will present some characterizations of the inner product spaces and we will show their equivalence. Finally, we will prove some consequences of our main results.

Vector Spaces

2021 ◽  
pp. 47-102
Author(s):  
Adel N. Boules
Keyword(s):  
Invariant Subspaces ◽  
Extensive Study ◽  
Vector Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Direct Sums ◽  
Infinite Dimensional ◽  
Quotient Spaces ◽  
Inner Product Spaces ◽  
Finite Dimensional

The first three sections of this chapter provide a thorough presentation of the concepts of basis and dimension. The approach is unified in the sense that it does not treat finite and infinite-dimensional spaces separately. Important concepts such as algebraic complements, quotient spaces, direct sums, projections, linear functionals, and invariant subspaces make their first debut in section 3.4. Section 3.5 is a brief summary of matrix representations and diagonalization. Then the chapter introduces normed linear spaces followed by an extensive study of inner product spaces. The presentation of inner product spaces in this section and in section 4.10 is not limited to finite-dimensional spaces but rather to the properties of inner products that do not require completeness. The chapter concludes with the finite-dimensional spectral theory.

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