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Polynomial Approximation on Polytopes
 
Vilmos Totik Bolyai Institute, University of Szeged, Hungary
Front Cover for Polynomial Approximation on Polytopes
Available Formats:
Electronic ISBN: 978-1-4704-1894-6
Product Code: MEMO/232/1091.E
112 pp 
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $45.00
Front Cover for Polynomial Approximation on Polytopes
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  • Front Cover for Polynomial Approximation on Polytopes
  • Back Cover for Polynomial Approximation on Polytopes
Polynomial Approximation on Polytopes
Vilmos Totik Bolyai Institute, University of Szeged, Hungary
Available Formats:
Electronic ISBN:  978-1-4704-1894-6
Product Code:  MEMO/232/1091.E
112 pp 
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $45.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2322014
    MSC: Primary 41;

    Polynomial approximation on convex polytopes in \(\mathbf{R}^d\) is considered in uniform and \(L^p\)-norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the \(L^p\)-case so called strong direct and converse results are also verified. The equivalence of the moduli of smoothness with an appropriate \(K\)-functional follows as a consequence.

    The results solve a problem that was left open since the mid 1980s when some of the present findings were established for special, so-called simple polytopes.

  • Table of Contents
     
     
    • 1. The continuous case
    • 1. The result
    • 2. Outline of the proof
    • 3. Fast decreasing polynomials
    • 4. Approximation on simple polytopes
    • 5. Polynomial approximants on rhombi
    • 6. Pyramids and local moduli on them
    • 7. Local approximation on the sets $K_a$
    • 8. Global approximation of $F=F_n$ on $S_{1/32}$ excluding a neighborhood of the apex
    • 9. Global approximation of $f$ on $S_{1/64}$
    • 10. Completion of the proof of Theorem
    • 11. Approximation in ${\mathbf R}^d$
    • 12. A $K$-functional and the equivalence theorem
    • 2. The $L^p$-case
    • 13. The $L^p$ result
    • 14. Proof of the $L^p$ result
    • 15. The dyadic decomposition
    • 16. Some properties of $L^p$ moduli of smoothness
    • 17. Local $L^p$ moduli of smoothness
    • 18. Local approximation
    • 19. Global $L^p$ approximation excluding a neighborhood of the apex
    • 20. Strong direct and converse inequalities
    • 21. The $K$-functional in $L^p$ and the equivalence theorem
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Volume: 2322014
MSC: Primary 41;

Polynomial approximation on convex polytopes in \(\mathbf{R}^d\) is considered in uniform and \(L^p\)-norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the \(L^p\)-case so called strong direct and converse results are also verified. The equivalence of the moduli of smoothness with an appropriate \(K\)-functional follows as a consequence.

The results solve a problem that was left open since the mid 1980s when some of the present findings were established for special, so-called simple polytopes.

  • 1. The continuous case
  • 1. The result
  • 2. Outline of the proof
  • 3. Fast decreasing polynomials
  • 4. Approximation on simple polytopes
  • 5. Polynomial approximants on rhombi
  • 6. Pyramids and local moduli on them
  • 7. Local approximation on the sets $K_a$
  • 8. Global approximation of $F=F_n$ on $S_{1/32}$ excluding a neighborhood of the apex
  • 9. Global approximation of $f$ on $S_{1/64}$
  • 10. Completion of the proof of Theorem
  • 11. Approximation in ${\mathbf R}^d$
  • 12. A $K$-functional and the equivalence theorem
  • 2. The $L^p$-case
  • 13. The $L^p$ result
  • 14. Proof of the $L^p$ result
  • 15. The dyadic decomposition
  • 16. Some properties of $L^p$ moduli of smoothness
  • 17. Local $L^p$ moduli of smoothness
  • 18. Local approximation
  • 19. Global $L^p$ approximation excluding a neighborhood of the apex
  • 20. Strong direct and converse inequalities
  • 21. The $K$-functional in $L^p$ and the equivalence theorem
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