ABSTRACT We construct polynomial approximations of Dzjadyk type (in terms of the k-th modulus of ... more ABSTRACT We construct polynomial approximations of Dzjadyk type (in terms of the k-th modulus of continuity, k⩾1) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E. Furthermore, the error in this approximation is decaying as e−cnα strictly inside E, where c and α are positive constants independent of the degree n of the approximating polynomial.
This is a survey (without proofs except for verifying a few new facts) of the slowest possible ra... more This is a survey (without proofs except for verifying a few new facts) of the slowest possible rate of convergence of a sequence of linear operators that converges pointwise to a linear operator. A sequence of linear operators (L n ) is said to converge to a linear operator Larbitrarily slowly (resp., almost arbitrarily slowly) provided that (L n ) converges to L pointwise, and for each sequence of real numbers (ϕ(n)) converging to 0, there exists a point x = x ϕ such that ||</font >Ln(x) -</font > L(x)||</font > ³</font > f</font >(n)\|{L}_{n}(x) - L(x)\| \geq \phi (n) for all n (resp., for infinitely many n). Two main “lethargy” theorems are prominent in this study, and they have numerous applications. The first lethargy theorem (Theorem 11.16) characterizes almost arbitrarily slow convergence. Applications of this lethargy theorem include the fact that a large class of polynomial operators (e.g., Bernstein, Hermite–Fejer, Landau, Fejer, and Jackson operators) all converge almost arbitrarily slowly to the identity operator. Also all the classical quadrature rules (e.g., the composite Trapezoidal Rule, composite Simpson’s Rule, and Gaussian quadrature) converge almost arbitrarily slowly to the integration functional. The second lethargy theorem (Theorem 11.21) gives useful sufficient conditions that guarantee arbitrarily slow convergence. In the particular case when the sequence of linear operators is generated by the powers of a single linear operator, there is a “dichotomy” theorem (Theorem 11.27) which states that either there is linear (fast) convergence or arbitrarily slow convergence; no other type of convergence is possible. Some applications of the dichotomy theorem include generalizations and sharpening of (1) the von Neumann-Halperin cyclic projections theorem, (2) the rate of convergence for intermittently (i.e., “almost” randomly) ordered projections, and (3) a theorem of Xu and Zikatanov.
The concept of a visible point of a convex set relative to a given point is introduced. A number ... more The concept of a visible point of a convex set relative to a given point is introduced. A number of basic properties of such visible point sets is developed. In particular, it is shown that this concept is useful in the study of best approximation, and it also seems to have potential value in the study of robotics.
Transactions of the American Mathematical Society, 1994
A general theory of best comonotone approximation in C[a, b] by elements of an «-dimensional exte... more A general theory of best comonotone approximation in C[a, b] by elements of an «-dimensional extended Chebyshev subspace is described. In particular, theorems on the existence, (in general) nonuniqueness, and characterization of best comonotone approximations are established.
Transactions of the American Mathematical Society, 2003
The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for det... more The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. It achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the individual subspaces which make up the intersection. The main practical drawback of this algorithm, at least for some applications, is that the method is slowly convergent. In this paper, we consider a general class of iterative methods which includes the MAP as a special case. For such methods, we study an "accelerated" version of this algorithm that was considered earlier by and by . We show that the accelerated algorithm converges faster than the MAP in the case of two subspaces, but is, in general, not faster than the MAP for more than two subspaces! However, for a "symmetric" version of the MAP, the accelerated algorithm always converges faster for any number of subspaces. Our proof seems to require the use of the Spectral Theorem for selfadjoint mappings.
Best Approximation from the Intersection of a Closed Convex Set and a Polyhedron in Hilbert Space... more Best Approximation from the Intersection of a Closed Convex Set and a Polyhedron in Hilbert Space, Weak Slater Conditions, and the Strong Conical Hull Intersection Property. [SIAM Journal on Optimization 10, 252 (1999)]. Frank Deutsch, Wu Li, Joseph D. Ward. Abstract. ...
Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metri... more Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections. [SIAM Journal on Mathematical Analysis 14, 185 (1983)]. Frank Deutsch, Petar Kenderov. Abstract. Two new ...
... {0}, ifx = 0, F(x) = 0, ifx e X\{0}, (2.12) If we omit either one of the two hypotheses that ... more ... {0}, ifx = 0, F(x) = 0, ifx e X\{0}, (2.12) If we omit either one of the two hypotheses that ~D(F) = X or that F(xo) is a singleton for some x0 from Theorems 1.3 or 2.1 or from Corollary 2,2, then we Page 6. 102 FRANK DEUTSCH AND IVAN SINGER ...
Proceedings of the American Mathematical Society, 1978
A normed space E is an inner product space if and only if for every 2-dimensional subspace V and ... more A normed space E is an inner product space if and only if for every 2-dimensional subspace V and every segment / c V, the corresponding metric projections satisfy the commutative property P¡PV = PyP¡.
Numerical Functional Analysis and Optimization, 1998
Let T i (i = 1, 2, . . . , N ) be nonexpansive mappings on a Hilbert space H, and let Θ : H → R∪{... more Let T i (i = 1, 2, . . . , N ) be nonexpansive mappings on a Hilbert space H, and let Θ : H → R∪{∞} be a function which has a uniformly strongly positive and uniformly bounded second (Fréchet) derivative over the convex hull of T i (H) for some i. We first prove that Θ has a unique minimum over the intersection of the fixed point sets of all the T i 's at some point u * . Then a cyclic hybrid steepest descent algorithm is proposed and we prove that it converges to u * . This generalizes some recent results of minimization of Θ over the intersection ∩ N 1 C i of closed convex sets C i can be handled by taking T i to be the metric projection P C i onto C i . We also propose a modification of our algorithm to handle the inconsistent case (i.e., when ∩ N 1 C i is empty) as well.
ABSTRACT We construct polynomial approximations of Dzjadyk type (in terms of the k-th modulus of ... more ABSTRACT We construct polynomial approximations of Dzjadyk type (in terms of the k-th modulus of continuity, k⩾1) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E. Furthermore, the error in this approximation is decaying as e−cnα strictly inside E, where c and α are positive constants independent of the degree n of the approximating polynomial.
This is a survey (without proofs except for verifying a few new facts) of the slowest possible ra... more This is a survey (without proofs except for verifying a few new facts) of the slowest possible rate of convergence of a sequence of linear operators that converges pointwise to a linear operator. A sequence of linear operators (L n ) is said to converge to a linear operator Larbitrarily slowly (resp., almost arbitrarily slowly) provided that (L n ) converges to L pointwise, and for each sequence of real numbers (ϕ(n)) converging to 0, there exists a point x = x ϕ such that ||</font >Ln(x) -</font > L(x)||</font > ³</font > f</font >(n)\|{L}_{n}(x) - L(x)\| \geq \phi (n) for all n (resp., for infinitely many n). Two main “lethargy” theorems are prominent in this study, and they have numerous applications. The first lethargy theorem (Theorem 11.16) characterizes almost arbitrarily slow convergence. Applications of this lethargy theorem include the fact that a large class of polynomial operators (e.g., Bernstein, Hermite–Fejer, Landau, Fejer, and Jackson operators) all converge almost arbitrarily slowly to the identity operator. Also all the classical quadrature rules (e.g., the composite Trapezoidal Rule, composite Simpson’s Rule, and Gaussian quadrature) converge almost arbitrarily slowly to the integration functional. The second lethargy theorem (Theorem 11.21) gives useful sufficient conditions that guarantee arbitrarily slow convergence. In the particular case when the sequence of linear operators is generated by the powers of a single linear operator, there is a “dichotomy” theorem (Theorem 11.27) which states that either there is linear (fast) convergence or arbitrarily slow convergence; no other type of convergence is possible. Some applications of the dichotomy theorem include generalizations and sharpening of (1) the von Neumann-Halperin cyclic projections theorem, (2) the rate of convergence for intermittently (i.e., “almost” randomly) ordered projections, and (3) a theorem of Xu and Zikatanov.
The concept of a visible point of a convex set relative to a given point is introduced. A number ... more The concept of a visible point of a convex set relative to a given point is introduced. A number of basic properties of such visible point sets is developed. In particular, it is shown that this concept is useful in the study of best approximation, and it also seems to have potential value in the study of robotics.
Transactions of the American Mathematical Society, 1994
A general theory of best comonotone approximation in C[a, b] by elements of an «-dimensional exte... more A general theory of best comonotone approximation in C[a, b] by elements of an «-dimensional extended Chebyshev subspace is described. In particular, theorems on the existence, (in general) nonuniqueness, and characterization of best comonotone approximations are established.
Transactions of the American Mathematical Society, 2003
The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for det... more The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. It achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the individual subspaces which make up the intersection. The main practical drawback of this algorithm, at least for some applications, is that the method is slowly convergent. In this paper, we consider a general class of iterative methods which includes the MAP as a special case. For such methods, we study an "accelerated" version of this algorithm that was considered earlier by and by . We show that the accelerated algorithm converges faster than the MAP in the case of two subspaces, but is, in general, not faster than the MAP for more than two subspaces! However, for a "symmetric" version of the MAP, the accelerated algorithm always converges faster for any number of subspaces. Our proof seems to require the use of the Spectral Theorem for selfadjoint mappings.
Best Approximation from the Intersection of a Closed Convex Set and a Polyhedron in Hilbert Space... more Best Approximation from the Intersection of a Closed Convex Set and a Polyhedron in Hilbert Space, Weak Slater Conditions, and the Strong Conical Hull Intersection Property. [SIAM Journal on Optimization 10, 252 (1999)]. Frank Deutsch, Wu Li, Joseph D. Ward. Abstract. ...
Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metri... more Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections. [SIAM Journal on Mathematical Analysis 14, 185 (1983)]. Frank Deutsch, Petar Kenderov. Abstract. Two new ...
... {0}, ifx = 0, F(x) = 0, ifx e X\{0}, (2.12) If we omit either one of the two hypotheses that ... more ... {0}, ifx = 0, F(x) = 0, ifx e X\{0}, (2.12) If we omit either one of the two hypotheses that ~D(F) = X or that F(xo) is a singleton for some x0 from Theorems 1.3 or 2.1 or from Corollary 2,2, then we Page 6. 102 FRANK DEUTSCH AND IVAN SINGER ...
Proceedings of the American Mathematical Society, 1978
A normed space E is an inner product space if and only if for every 2-dimensional subspace V and ... more A normed space E is an inner product space if and only if for every 2-dimensional subspace V and every segment / c V, the corresponding metric projections satisfy the commutative property P¡PV = PyP¡.
Numerical Functional Analysis and Optimization, 1998
Let T i (i = 1, 2, . . . , N ) be nonexpansive mappings on a Hilbert space H, and let Θ : H → R∪{... more Let T i (i = 1, 2, . . . , N ) be nonexpansive mappings on a Hilbert space H, and let Θ : H → R∪{∞} be a function which has a uniformly strongly positive and uniformly bounded second (Fréchet) derivative over the convex hull of T i (H) for some i. We first prove that Θ has a unique minimum over the intersection of the fixed point sets of all the T i 's at some point u * . Then a cyclic hybrid steepest descent algorithm is proposed and we prove that it converges to u * . This generalizes some recent results of minimization of Θ over the intersection ∩ N 1 C i of closed convex sets C i can be handled by taking T i to be the metric projection P C i onto C i . We also propose a modification of our algorithm to handle the inconsistent case (i.e., when ∩ N 1 C i is empty) as well.
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