Note on degree of trigonometric and polynomial approximation to an analytic function

Note on Degree of Trigonometric and Polynomial Approximation to an Analytic Function § J. L. Walsh and W. E. Sewell

/ d e i \ (is) y — r*?W' = -/"Y'. \dx* c h / The factor — r on the right-hand side of this equation shows that the Dirac equation for an electron is not invariant under inversions. However, if we set 0? = O and JU = 0, then equation (15) is numerically invariant under inversions. This is done in the neutrino theory of light. Hence that theory has the same invariance properties as the Maxwell th...

NOTE ON DEGREE OF TRIGONOMETRIC AND POLYNOMIAL APPROXIMATION TO AN ANALYTIC FUNCTION, IN THE SENSE OF LEAST pTK POWERS

Note on Invariance of Degree of Polynomial and Trigonometric Approximation under Change of Independent Variable.

We postpone to another occasion the proof of THEOREM 7. Let the function f(x) be of class CO on a closed bounded interval E, and let pn(x) (0 f (x) on E) be a polynomial of degree n which for some p, 0 < p < 1, minimizes (3). Let f(x) pn(X) have zeros of respective inultiplicities Kj, 1 < j < r, and let Kj* be the smallest integer > f(1p) Kj] and which for zeros interior to E is also of the sam...

Nonperiodic Trigonometric Polynomial Approximation

The most common approach for approximating non-periodic function defined on a finite interval is based on considering polynomials as basis functions. In this paper we will address the non-optimallity of polynomial approximation and suggest to switch from powers of x to powers of sin(px) where p is a parameter which depends on the dimension of the approximating subspace. The new set does not suf...

A Note on the Degree of Polynomial Approximation*

Let C be a rectifiable Jordan curve of the finite z plane. We shall say that a function f(z) belongs to the class Lip (C, j , a) if ƒ (z) is regular in the limited region bounded by C (which we shall call the interior of C), if f{z) is continuous in the corresponding closed region, and if the jth. derivative of f(z) is also continuous in this closed region and satisfies a Lipschitz condition wi...