A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Abstract. Preconditioned conjugate gradients (PCG) are widely and successfully used methods for solving a Toeplitz linear system $A_n\vec{x}=\vec{b}$ [59,9,20,5,34,62,6,10,28,45,44,46,49]. Frobenius-optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the Frobenius distance from $A_n$. The convergence features of these PCG have been naturally studied by means of the Weierstrass–Jackson Theorem [17,36,45], owing to the profound relationship between the spectral features of the matrices $A_n$, generated by the Fourier coefficients of a continuous function f, and the analytical properties of the symbol f itself. In this paper, we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f. On the other hand, by modifying the Korovkin Theorem to study the Frobenius-optimal preconditioning problem, we provide a new and unifying tool for analyzing all Frobenius-optimal preconditioners in any generic matrix algebra related to trigonometric transforms. Finally, the multilevel case is sketched and discussed by showing that a Korovkin-type Theory also holds in a multivariate sense.

[1]  Eugene E. Tyrtyshnikov,et al.  Circulant preconditioners with unbounded inverses , 1995 .

[2]  Stefano Serra.  How to Choose the Best Iterative Strategy for Symmetric Toeplitz Systems , 1999 .

[3]  U. Grenander,et al.  Statistical analysis of stationary time series , 1957 .

[4]  Eugene E. Tyrtyshnikov,et al.  Which circulant preconditioner is better? , 1996, Math. Comput..

[5]  Walter Gautschi.  The condition of Vandermonde-like matrices involving orthogonal polynomials☆ , 1983 .

[6]  Seymour V. Parter,et al.  On the distribution of the singular values of Toeplitz matrices , 1986 .

[7]  Fabio Di Benedetto,et al.  Analysis of Preconditioning Techniques for Ill-Conditioned Toeplitz Matrices , 1995, SIAM J. Sci. Comput..

[8]  Raymond H. Chan,et al.  Jackson's theorem and circulant preconditioned Toeplitz systems , 1992 .

[9]  I. P. Natanson.  Constructive function theory , 1964 .

[10]  Raymond H. Chan,et al.  Circulant preconditioners for Toeplitz matrices with piecewise continuous generating functions , 1992 .

[11]  P. Tilli.  Singular values and eigenvalues of non-hermitian block Toeplitz matrices , 1996 .

[12]  Stefano Serra Capizzano.  The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems , 1999, Numerische Mathematik.

[13]  Paola Favati,et al.  On a matrix algebra related to the discrete Hartley transform , 1993 .

[14]  A. M. Peterson.  Applications of digital signal processing , 1979 .

[15]  R. Chan.  Toeplitz Preconditioners for Toeplitz Systems with Nonnegative Generating Functions , 1991 .

[16]  E. E. Tyrtyshnikov.  A unifying approach to some old and new theorems on distribution and clustering , 1996 .

[17]  I. Gohberg,et al.  Convolution Equations and Projection Methods for Their Solution , 1974 .

[18]  Radakovič.  The theory of approximation , 1932 .

[19]  Fabio Di Benedetto,et al.  Preconditioning of Block Toeplitz Matrices by Sine Transforms , 1997, SIAM J. Sci. Comput..

[20]  F. Avram.  On bilinear forms in Gaussian random variables and Toeplitz matrices , 1988 .

[21]  Stefano Serra Capizzano,et al.  Optimal and Superoptimal Matrix Algebra Operators , 1997 .

[22]  R. Chan,et al.  Circulant preconditioners constructed from kernels , 1992 .

[23]  T. Chan.  An Optimal Circulant Preconditioner for Toeplitz Systems , 1988 .

[24]  Xiao-Qing Jin,et al.  Hartley preconditioners for Toeplitz systems generated by positive continuous functions , 1994 .

[25]  Thomas Huckle,et al.  Circulant and Skewcirculant Matrices for Solving Toeplitz Matrix Problems , 1992, SIAM J. Matrix Anal. Appl..

[26]  Eugene E. Tyrtyshnikov,et al.  Spectra of multilevel toeplitz matrices: Advanced theory via simple matrix relationships , 1998 .

[27]  H. Widom.  Szegö's limit theorem: The higher-dimensional matrix case☆ , 1980 .

[28]  Dario Bini,et al.  A new preconditioner for the parallel solution of positive definite Toeplitz systems , 1990, SPAA '90.

[29]  Francesco Aldo Costabile,et al.  Asymptotic expansion and extrapolation for Bernstein polynomials with applications , 1996 .

[30]  C. P. May.  Saturation and Inverse Theorems for Combinations of a Class of Exponential-Type Operators , 1976, Canadian Journal of Mathematics.

[31]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[32]  Stefano Serra,et al.  Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems , 1997 .

[33]  G. Strang.  A proposal for toeplitz matrix calculations , 1986 .

[34]  R. Chan,et al.  A Family of Block Preconditioners for Block Systems , 1992, SIAM J. Sci. Comput..

[35]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[36]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[37]  T. Kailath,et al.  Displacement structure approach to discrete-trigonometric-transform based preconditioners of G.Strang type and of T.Chan type , 1996 .

[38]  Stefano Serra.  The extension of the concept of the generating function to a class of preconditioned Toeplitz matrices , 1997 .

[39]  Raymond H. Chan,et al.  Cosine transform based preconditioners for total variation minimization problems in image processing , 1995 .

[40]  Stefano Serra,et al.  Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems , 1994 .

[41]  Stefano Serra.  Preconditioning Strategies for Hermitian Toeplitz Systems with Nondefinite Generating Functions , 1996 .

[42]  S. Capizzano.  Superlinear PCG methods for symmetric Toeplitz systems , 1999, Math. Comput..

[43]  O. Axelsson,et al.  On the rate of convergence of the preconditioned conjugate gradient method , 1986 .

[44]  Stefano Serra Capizzano,et al.  Korovkin theorems and linear positive Gram matrix algebra approximations of Toeplitz matrices , 1998 .

[45]  Dario Bini,et al.  Parallel Solution of Certain Toeplitz Linear Systems , 1984, SIAM J. Comput..

[46]  On some new Banach algebras and Wiener-Levy type theorems for Fourier series and integrals , 1970 .

[47]  Giuseppe Fiorentino,et al.  C. G. preconditioning for Toeplitz matrices , 1993 .

[48]  U. Grenander,et al.  Statistical analysis of stationary time series , 1958 .

[49]  Stefano Serra Capizzano,et al.  Korovkin tests, approximation, and ergodic theory , 2000, Math. Comput..

[50]  P. Butzer.  Linear Combinations of Bernstein Polynomials , 1953, Canadian Journal of Mathematics.

[51]  R. Chan,et al.  The circulant operator in the banach algebra of matrices , 1991 .

[52]  Gian-Carlo Rota,et al.  Linear Operators and Approximation Theory. , 1965 .

[53]  Stefano Serra Capizzano,et al.  A Korovkin-Based Approximation of Multilevel Toeplitz Matrices (With Rectangular Unstructured Blocks) via Multilevel Trigonometric Matrix Spaces , 1999 .

[54]  S. Serra Capizzano,et al.  Some theorems on linear positive operators and functionals and their applications , 2000 .

[55]  Harold Widom.  On the Singular Values of Toeplitz Matrices , 1989 .

[56]  G. Strang,et al.  Toeplitz equations by conjugate gradients with circulant preconditioner , 1989 .

[57]  Eugene E. Tyrtyshnikov,et al.  Optimal and Superoptimal Circulant Preconditioners , 1992, SIAM J. Matrix Anal. Appl..

[58]  Dario Bini,et al.  SPECTRAL AND COMPUTATIONAL PROPERTIES OF BAND SYMMETRIC TOEPLITZ MATRICES , 1983 .

[59]  Raymond H. Chan,et al.  Fast Band-Toeplitz Preconditioners for Hermitian Toeplitz Systems , 1994, SIAM J. Sci. Comput..