Explicit Sinc-Like Methods

Selected Topics in Approximation and Computation ◽

10.1093/oso/9780195080599.003.0007 ◽

1995 ◽

Author(s):

Marek A. Kowalski ◽

Krzystof A. Sikorski ◽

Frank Stenger

Keyword(s):

Rational Approximation ◽

Rational Functions ◽

Generalized Functions ◽

Elementary Transformation ◽

Delta Functions ◽

Simply Connected ◽

Methods Of Approximation

In this section we derive several methods of approximation using the function values {f(kh)}∞k=- ∞ . We present a family of simple rational functions, which make possible the explicit and arbitrarily accurate rational approximation of the filter, the step (Heaviside) and the impulse (delta) functions. The chief advantage of these methods is that they make it possible to write down a simple and explicit rational approximation corresponding to any desired accuracy. Also, the three families of approximations are very simply connected with one another—the filter being related to the Heaviside via an elementary transformation, and the impulse being the derivative of the Heaviside. Thus, these methods make it possible for us to approximate generalized functions. In this section we discuss various methods, some of which are new, for approximating a function f ( t ) using the values f(0), f(±h), f(±2h), . .., where h > 0.

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Possibility of Uniform Rational Approximation in the Spherical Metric

Canadian Journal of Mathematics ◽

10.4153/cjm-1976-013-0 ◽

1976 ◽

Vol 28 (1) ◽

pp. 112-115 ◽

Cited By ~ 8

Author(s):

P. M. Gauthier ◽

A. Roth ◽

J. L. Walsh

Keyword(s):

Compact Subset ◽

Rational Approximation ◽

Complex Plane ◽

Riemann Sphere ◽

Rational Functions ◽

Finite Complex ◽

Finite Plane ◽

Extended Plane ◽

Chordal Metric

Let ƒ b e a mapping defined on a compact subset K of the finite complex plane C and taking its values on the extended plane C ⋃ ﹛ ∞﹜. For x a metric on the extended plane, we consider the possibility of approximating ƒ x-uniformly on K by rational functions. Since all metrics on C ⋃ ﹛oo ﹜ are equivalent, we shall consider that x is the chordal metric on the Riemann sphere of diameter one resting on a finite plane at the origin.

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The Cauchy integral and analytic continuation

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100063076 ◽

1985 ◽

Vol 97 (3) ◽

pp. 491-498 ◽

Cited By ~ 1

Author(s):

James. E. Brennan

Keyword(s):

Analytic Continuation ◽

Polynomial Approximation ◽

Rational Functions ◽

Minimum Problem ◽

Weighted Polynomial Approximation ◽

Cauchy Integral ◽

Typical Problem ◽

Bernstein Problem ◽

Open Set ◽

Simply Connected

One of the most important concepts in the theory of approximation by analytic functions is that of analytic continuation. In a typical problem, for example, there is generally a region Ω, a Banach space B of functions analytic in Ω and a subfamily ℱ ⊂ B, each member of which is analytic in some larger open set, and one might be asked to decide whether or not ℱ is dense in B. It often happens, however, that either ℱ is dense or the only functions which can be so approximated have a natural analytic continuation across ∂Ω. A similar phenomenon is also known to occur even for approximation on sets without interior. In this article we shall describe a method for proving such theorems which can be applied in a variety of settings and, in particular, to: (1) the Bernštein problem for weighted polynomial approximation on the real line; (2) the completeness problem for weighted polynomial approximation on bounded simply connected regions; (3) the Shapiro overconvergence problem for sequences of rational functions with sparse poles; (4) the Akutowicz-Carleson minimum problem for interpolating functions. Although we shall present no new results, the method of proof, which is based on an argument of the author [6], seems sufficiently versatile to warrant exposition.

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INEQUALITIES OF BERNSTEIN TYPE FOR DERIVATIVES OF RATIONAL FUNCTIONS, AND INVERSE THEOREMS OF RATIONAL APPROXIMATION

Mathematics of the USSR-Sbornik ◽

10.1070/sm1985v052n02abeh002906 ◽

1985 ◽

Vol 52 (2) ◽

pp. 557-574 ◽

Cited By ~ 6

Author(s):

A A Pekarskiĭ

Keyword(s):

Rational Approximation ◽

Rational Functions ◽

Bernstein Type ◽

Inverse Theorems ◽

Derivatives Of

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The theory of generalized functions

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1955.0042 ◽

1955 ◽

Vol 228 (1173) ◽

pp. 175-190 ◽

Cited By ~ 42

Keyword(s):

Fourier Series ◽

Continuous Function ◽

Convergent Sequence ◽

Generalized Functions ◽

Generalized Function ◽

Partial Derivatives ◽

Full Complement ◽

Delta Functions ◽

Theory Of Generalized Functions

This paper gives an introductory account of the construction and properties of generalized functions f( x ) of real variables x 1 , x 2 , ..., x n . These are defined so as to ensure that (i) any generalized function f(x ) possesses its full complement of generalized partial derivatives D p f(x ) of all orders; (ii) any convergent sequence of generalized functions {f n (x)} has a generalized limit, f(x) , which is also a generalized function; (iii) the derived sequence {D p f n ( x )} converges to D p f ( x ). The construction of these generalized functions ensures that any continuous function possesses derivatives which are generalized functions, so that the delta functions of Dirac are included in the theory. The representation of generalized functions by Fourier series and integrals is considered as an example of the simplicity and generality of the theory.

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The hardest halfspace

Computational Complexity ◽

10.1007/s00037-021-00211-4 ◽

2021 ◽

Vol 30 (2) ◽

Author(s):

Alexander A. Sherstov

Keyword(s):

Lower Bounds ◽

Rational Approximation ◽

Communication Complexity ◽

Rational Functions ◽

Explicit Construction ◽

Upper Bounds ◽

Infinity Norm ◽

Communication Problem

AbstractWe study the approximation of halfspaces $$h:\{0,1\}^n\to\{0,1\}$$ h : { 0 , 1 } n → { 0 , 1 } in the infinity norm by polynomials and rational functions of any given degree. Our main result is an explicit construction of the “hardest” halfspace, for which we prove polynomial and rational approximation lower bounds that match the trivial upper bounds achievable for all halfspaces. This completes a lengthy line of work started by Myhill and Kautz (1961). As an application, we construct a communication problem that achieves essentially the largest possible separation, of O(n) versus $$2^{-\Omega(n)}$$ 2 - Ω ( n ) , between the sign-rank and discrepancy. Equivalently, our problem exhibits a gap of log n versus $$\Omega(n)$$ Ω ( n ) between the communication complexity with unbounded versus weakly unbounded error, improving quadratically on previous constructions and completing a line of work started by Babai, Frankl, and Simon (FOCS 1986). Our results further generalize to the k-party number-on-the-forehead model, where we obtain an explicit separation of log n versus $$\Omega(n/4^{n})$$ Ω ( n / 4 n ) for communication with unbounded versus weakly unbounded error.

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BERNSTEIN FRACTAL RATIONAL APPROXIMANTS WITH NO CONDITION ON SCALING VECTORS

Fractals ◽

10.1142/s0218348x18500457 ◽

2018 ◽

Vol 26 (04) ◽

pp. 1850045 ◽

Cited By ~ 3

Author(s):

N. VIJENDER

Keyword(s):

Rational Approximation ◽

Iterated Function System ◽

Rational Functions ◽

Compact Interval ◽

Approximation Properties ◽

Weierstrass Theorem ◽

Scaling Factors ◽

New Class ◽

Fractal Functions ◽

The One

Fractal functions defined through iterated function system have been successfully used to approximate any continuous real-valued function defined on a compact interval. The fractal dimension is a quantifier (or index) of irregularity (non-differentiability) of fractal approximant and it depends on the scaling factors of the fractal approximant. Viswanathan and Chand [Approx. Theory 185 (2014) 31–50] studied fractal rational approximation under the hypothesis “magnitude of the scaling factors goes to zero”. In this paper, first, we introduce a new class of fractal approximants, namely, Bernstein [Formula: see text]-fractal functions which converge to the original function for every scaling vector. Using the proposed class of fractal approximants and imposing no condition on the corresponding scaling factors, we establish self-referential Bernstein [Formula: see text]-fractal rational functions and their approximation properties. In particular, (i) we study the fractal analogue of the Weierstrass theorem and Müntz theorem of rational functions, (ii) we study the one-sided approximation by Bernstein [Formula: see text]-fractal rational functions, (iii) we develop copositive Bernstein fractal rational approximation, (iv) we investigate the existence of a minimizing sequence of fractal rational approximation to a continuous function defined as a real compact interval. Finally, we introduce the non-self-referential Bernstein [Formula: see text]-fractal approximants.

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Generalized functions for applications

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000004562 ◽

1985 ◽

Vol 26 (3) ◽

pp. 362-374 ◽

Cited By ~ 3

Author(s):

B. D. Craven

Keyword(s):

Real Line ◽

Generalized Functions ◽

Generalized Function ◽

The Real ◽

Square Roots ◽

Rigorous Approach ◽

Schwartz Distributions ◽

Delta Functions

AbstractA simple rigorous approach is given to generalized functions, suitable for applications. Here, a generalized function is defined as a genuine function on a superset of the real line, so that multiplication is unrestricted and associative, and various manipulations retain their classical meanings. The superset is simply constructed, and does not require Robinson's nonstandard real line. The generalized functions go beyond the Schwartz distributions, enabling products and square roots of delta functions to be discussed.

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ABOUT THE RATE OF RATIONAL APPROXIMATION OF SOME ANALYTIC FUNCTIONS

Mathematical Modelling and Analysis ◽

10.3846/13926292.1998.9637100 ◽

1998 ◽

Vol 3 (1) ◽

pp. 168-176

Author(s):

A. YA. Radyno

Keyword(s):

Analytic Function ◽

Unit Disk ◽

Rational Approximation ◽

Analytic Functions ◽

Hankel Operator ◽

Rational Functions ◽

Best Approximations ◽

Singular Numbers

The article is devoted to results relating to the theory of rational approximation of an analytic function. Let ƒ be an analytic function on the disk {z : |z| < ñ), ñ > 1. The rate of decrease of the best approximations ñn of a function ƒ by the rational functions of order at most n in the uniform metric on the unit disk E with the center z = 0 is investigated. The theorem connecting the rate of decrease of ñn with the order ó > 0 of ƒ in the disk {z : |z| < ñ} is proved. The proof of this results is based on an analysis of behavior of the singular numbers of the Hankel operator constructed from the function ƒ.

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