Discrete orthogonal polynomials: anomalies of time series and boundary effects of polynomial filters
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A. A. Kytmanov
Abstract
We describe a new result in the classical theory of univariate discrete orthogonal polynomials (Hahn polynomials on an equidistant grid with unit weights): extremely fast decay of their values on the grid near the interval boundary for polynomials of sufficiently high degree. This effect dramatically differs from the behavior of continuous orthogonal polynomials, which are much more popular in mathematical curricula. The practical importance of this new result for the theory of discrete polynomial filters (widely applied for detection of anomalies of time series of measurements) is demonstrated on the practical example of detection of outliers and small discontinuities in the publicly available GPS and GLONASS trajectories. Discrete polynomial filters, on one hand, can detect very small anomalies in sparse time series (with amplitude of order 10−11 relative to the typical values of the time series). On the other hand, our general result limits sensitivity of polynomial filters near the boundary of the time series. The main problem in practical applications of the discussed method is numerical instability of construction of the discrete orthogonal polynomials of high degree. In this paper, we give a review of our previous research and prove the following additional properties of Hahn polynomials on an equidistant grid with unit weights: - such polynomials have extremely large values between the grid points (near the endpoints); - such polynomials have their roots very close to the grid points (near the endpoints); - these results imply important general boundary effects for stable linear polynomial filters (we call this property “rapid boundary attenuation”); this also applies to other types of stable filters, e. g., based on discrete Chebyshev polynomials. These rigorous results appear to be new; their explanation in the framework of the wellknown asymptotic theory of discrete orthogonal polynomials could not be found in the literature cited.
Abstract
We describe a new result in the classical theory of univariate discrete orthogonal polynomials (Hahn polynomials on an equidistant grid with unit weights): extremely fast decay of their values on the grid near the interval boundary for polynomials of sufficiently high degree. This effect dramatically differs from the behavior of continuous orthogonal polynomials, which are much more popular in mathematical curricula. The practical importance of this new result for the theory of discrete polynomial filters (widely applied for detection of anomalies of time series of measurements) is demonstrated on the practical example of detection of outliers and small discontinuities in the publicly available GPS and GLONASS trajectories. Discrete polynomial filters, on one hand, can detect very small anomalies in sparse time series (with amplitude of order 10−11 relative to the typical values of the time series). On the other hand, our general result limits sensitivity of polynomial filters near the boundary of the time series. The main problem in practical applications of the discussed method is numerical instability of construction of the discrete orthogonal polynomials of high degree. In this paper, we give a review of our previous research and prove the following additional properties of Hahn polynomials on an equidistant grid with unit weights: - such polynomials have extremely large values between the grid points (near the endpoints); - such polynomials have their roots very close to the grid points (near the endpoints); - these results imply important general boundary effects for stable linear polynomial filters (we call this property “rapid boundary attenuation”); this also applies to other types of stable filters, e. g., based on discrete Chebyshev polynomials. These rigorous results appear to be new; their explanation in the framework of the wellknown asymptotic theory of discrete orthogonal polynomials could not be found in the literature cited.
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- About the editors IX
- List of contributors XI
- Intermediate systems and invariants 1
- Folding in fluids 19
- Vortex models of plane turbulent flows 37
- Equations of dyon electromagnetic field 53
- Minimal action principle for gravity and electrodynamics, Einstein lambda, and Lagrange points 65
- Methods for constructing invariant conservative finite-difference schemes for hydrodynamic-type equations 83
- On exact analytical solutions of equations of Maxwell incompressible viscoelastic medium 111
- Approximate solution of a boundary-value problem for a model of the far momentumless turbulent wake 125
- Analysis of overdetermined system that describes the special class of two-dimensional motion of an ideal fluid 135
- Discrete orthogonal polynomials: anomalies of time series and boundary effects of polynomial filters 143
- Index 165
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- About the editors IX
- List of contributors XI
- Intermediate systems and invariants 1
- Folding in fluids 19
- Vortex models of plane turbulent flows 37
- Equations of dyon electromagnetic field 53
- Minimal action principle for gravity and electrodynamics, Einstein lambda, and Lagrange points 65
- Methods for constructing invariant conservative finite-difference schemes for hydrodynamic-type equations 83
- On exact analytical solutions of equations of Maxwell incompressible viscoelastic medium 111
- Approximate solution of a boundary-value problem for a model of the far momentumless turbulent wake 125
- Analysis of overdetermined system that describes the special class of two-dimensional motion of an ideal fluid 135
- Discrete orthogonal polynomials: anomalies of time series and boundary effects of polynomial filters 143
- Index 165
