Best approximation in inner product spaces

  title={Best approximation in inner product spaces},
  author={Frank Deutsch},
Inner Product Spaces.- Best Approximation.- Existence and Uniqueness of Best Approximations.- Characterization of Best Approximations.- The Metric Projection.- Bounded Linear Functionals and Best Approximation from Hyperplanes and Half-spaces.- Error of Approximation.- Generalized Solutions of Linear Equations.- The Method of Alternating Projections.- Constrained Interpolation from a Convex Set.- Interpolation and Approximation.- Convexity of Chebyshev Sets. 
Best Simultaneous Approximation of Finite Set in Inner Product Space
In this paper, we find a way to give best simultaneous approximation of n arbitrary points in convex sets. First, we introduce a special hyperplane which is based on those n points. Then by using
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We examine best approximation by closed sets in a class of normed spaces with star-shaped cones. It is assumed that the norm on the space X under consideration is generated by a star-shaped cone.
Best approximation in quotient spaces
We obtain a sufficient and necessary theorems simple for proximality and Chebyshevity of the best approximate sets in quotient spaces. Also we consider compactness of the set of best approximate in
A Deep Monotone Approximation Operator Based on the Best Quadratic Lower Bound of Convex Functions
  • M. Yamagishi, I. Yamada
  • Mathematics, Computer Science
    IEICE Trans. Fundam. Electron. Commun. Comput. Sci.
  • 2008
By using the proposed lower bound, this paper derives a computationally efficient deep monotone approximation operator to the level set of the function, which realizes better approximation than subgradient projection which has been utilized, as a monot one to level sets of differentiable convex functions as well as nonsmooth conveX functions.
Local Linear Convergence for Alternating and Averaged Nonconvex Projections
It is proved that von Neumann’s method of “alternating projections” converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity.
Convex Functions: Constructions, Characterizations and Counterexamples
Preface 1. Why convex? 2. Convex functions on Euclidean spaces 3. Finer structure of Euclidean spaces 4. Convex functions on Banach spaces 5. Duality between smoothness and strict convexity 6.
Stochastic Approximation on Riemannian Manifolds
  • S. Shah
  • Mathematics, Computer Science
    Applied Mathematics & Optimization
  • 2019
The standard theory of stochastic approximation (SA) is extended to the case when the constraint set is a Riemannian manifold and a framework is developed for a projected SA scheme with approximate retractions.