Duffin and Schaeffer inequality revisited

We analyze the asymptotic rates of convergence of Chebyshev, Legendre and Jacobi polynomials. One complication is that there are many reasonable measures of optimality as enumerated here. Another is that there are at least three exceptions to the general principle that Chebyshev polynomials give the fastest rate of convergence from the larger family of Jacobi polynomials. When $$f(x)$$ f ( x ) is singular at one or both endpoints, all Gegenbauer polynomials (including Legendre and Chebyshev) converge equally fast at the endpoints, but Gegenbauer polynomials converge more rapidly on the interior with increasing order $$m$$ m . For functions on the surface of the sphere, associated Legendre functions, which are proportional to Gegenbauer polynomials, are best for the latitudinal dependence. Similarly, for functions on the unit disk, Zernike polynomials, which are Jacobi polynomials in radius, are superior in rate-of-convergence to a Chebyshev---Fourier series. It is true, as was conjectured by Lanczos 60 years ago, that excluding these exceptions, the Chebyshev coefficients $$a_{n}$$ a n usually decrease faster than the Legendre coefficients $$b_{n}$$ b n by a factor of $$\sqrt{n}$$ n . We calculate the proportionality constant for a few examples and restrictive classes of functions. The more precise claim that $$b_{n} \sim \sqrt{\pi /2} \sqrt{n} a_{n}$$ b n ~ ? / 2 n a n , made by Lanczos and later Fox and Parker, is true only for rather special functions. However, individual terms in the large $$n$$ n asymptotics of Chebyshev and Legendre coefficients usually do display this proportionality.

In earlier chapters we dealt with special sets of orthogonal polynomials, namely, Chebyshev and Hermite polynomials. In Chs. 9 and 10 we will study other orthogonal polynomials, namely, Laguerre and Legendre. All of these polynomial functions share many properties. This indicates that these polynomials are special cases of more general polynomials—Gegenbauer and Jacobi polynomials named after Leopold Gegenbauer (1849–1903) and Carl Gustav Jacob Jacobi (1804–1851). Gegenbauer polynomials are connected with axially symmetric potentials, while Jacobi polynomials are even more general, with Jacobi polynomials containing Gegenbauer polynomials as a special case. Collectively, these polynomials are called classical orthogonal polynomials. In this chapter we look at some of the elementary properties of these polynomials; the reader is referred to other texts for detailed descriptions.

It is well hlown that Chebyshev polynomials play a useful key role in giting close polynomial approximations to functions defined over finite intervalsBy employing Chebyshev polynomials, we can approximate a polynomial of very high degree (which'is practically the same as any continuous function) by another polynomial of rnuch lower degree, with great accuracy. This approximation is a consequence of the property that, for any finite interval [a,b], for fixed degree n, the Chebyshes polynomial (normalized to [a,b] ) has a maximum deviation from zero which is less than that of any other polynomial of degree n, and having the same leading coefficient of xn. But Chebyshev polynomials do not serve as well in approximations over infinite intervals, where an approximating polynomial will usually require a damping or weight factor. The simplest analogue of Chebyshev polynomials over infinite intervals would be polynomials harring th property that, for fixed degree n, when multiplied by e x or e x for [0,] or [_a),a)] respectively, they have a maximum deviation from zero which is less than

Markov’s inequality asserts that | | p n ′ | | ⩽ n 2 | | p n | | ||{p’_n}|| \leqslant {n^2}||{p_n}|| for any polynomial p n {p_n} of degree n n . (We denote the supremum norm on [ − 1 , 1 ] [ - 1,1] by | | . | | ||.|| .) In the case that p n {p_n} has all real roots, none of which lie in [ − 1 , 1 ] [ - 1,1] , Erdös has shown that | | p n ′ | | ⩽ e n | | p n | | / 2 ||{p’_n}|| \leqslant en||{p_n}||/2 . We show that if p n {p_n} has n − k n - k real roots, none of which lie in [ − 1 , 1 ] [ - 1,1] , then | | p n ′ ⩽ c n ( k + 1 ) | | p n | | ||{p’_n} \leqslant cn(k + 1)||{p_n}|| , where c c is independent of n n and k k . This extension of Markov’s and Erdös’ inequalities was conjectured by Szabados.

5.1 Introduction A set of functions {ϕ n (x)},n=0,1,2,…, is said to be orthogonal on the interval a<x<b , with respect to a weight function r(x)<0 , if ∫ b a r(x)ϕ n (x)ϕ k (x)dx=0k≠n Sets of orthogonal functions play an extremely important role in analysis, primarily because functions belonging to a very general class can be represented by series of orthogonal functions, called generalized Fourier series. A special class of orthogonal functions consists of the sets of orthogonal polynomials{p n (x)} , where n denotes the degree of the polynomial p n (x) . The Legendre polynomials discussed in Chap. 4 are probably the simplest set of polynomials belonging to this class. Other polynomial sets which commonly occur in applications are the Hermite, Laguerre, and Chebyshev polynomials. More general polynomial sets are defined by the Gegenbauer and Jacobi polynomials, which include the others as special cases. The study of general polynomial sets like the Jacobi polynomials facilitates the study of each polynomial set by focusing on those properties that are characteristic of all the individual sets.

Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.

The famous Russian mathematician Pafnutii Lvovich Tchebycheff (1821-1894) introduced T1(x) as the polynomial of least uniform norm on [-1, 1] amid the polynomials of degree n with fixed leading coefficient. The Tchebycheff polynomials appear prominently in various extremal problems posed in irn (the set of all polynomials of degree n). An illuminating example is the classical Markov inequality, Which shows that

In this work we introduce a new parameter,s≥1, in the well known Sobolev-Gagliardo-Nirenberg (abbreviated SGN) inequalities and show their validity (with an appropriates) for any compact subanalytic domain. The classical form of these SGN inequalities (s=1 in our formulation) fails for domains with outward pointing cusps. Our parameters measures the degree of cuspidality of the domain. For regular domainss=1. We also introduce an extension, depending on a parameter σ≥1, to several variables of a local form of the classical Markov inequality on the derivatives of a polynomial in terms of its own values, and show the equivalence of Markov and SGN inequalities with the same value of parameters, σ=s. Our extension of Markov's inequality admits, in the case of supremum norms, a geometric characterization. We also establish several other characterizations: the existence of a bounded (linear) extension ofC∞ functions with a homogeneous loss of differentiability, and the validity of a global Markov inequality. Our methods may broadly be classified as follows: 1. Desingularization and anLp-version of Glaeser-type estimates. In fact we obtain a bounds<-2d+1, whered is the maximal order of vanishing of the jacobian of the desingularization map of the domain. 2. Interpolation type inequalities for norms of functions and Bernstein-Markov type inequalities for multivariate polynomials (classical analysis). 3. Geometric criteria for the validity of local Markov inequalities (local analysis of the singularities of domains). 4. Multivariate Approximation Theory.

This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems. For this purpose, we establish new explicit formulas for the high-order derivatives of certain two classes of Jacobi polynomials in terms of their corresponding Jacobi polynomials. These two classes generalize the two celebrated non-symmetric classes of polynomials, namely, Chebyshev polynomials of third- and fourth-kinds. The idea of the derivation of such formulas is essentially based on making use of the power series representations and inversion formulas of these classes of polynomials. The derived formulas serve in converting the even-order linear differential equations with their boundary conditions into linear systems that can be efficiently solved. Furthermore, and based on the first-order derivatives formula of certain Jacobi polynomials, the operational matrix of derivatives is extracted and employed to present another algorithm to treat both linear and nonlinear two-point boundary value problems based on the application of the collocation method. Convergence analysis of the proposed expansions is investigated. Some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

The maximum iiumber of ones arid zeros and their distribution among the coefficients of some polynomials is given. It is shown that the coefficients of the poly- nomials gq(x) = (Xn + l)/p(x) and g2(x) = (Xn + 1)/((X + l)p(x)) and other related polynomials have special distribution when p(x) is a primitive polynomial, and these coefficients contain the maximum possible number of ones and zeros. Introduction. Let F be the field of two elements, 0 and 1, and let g(x) be a polynomial of degree r over F. Let n be the least integer for which g(x) divides xn + 1. We suppose g(x) to have no multiple zeros, so that n is odd. Denote by Rn the polynomial ring F(x)/(xn + 1). Let k = n - r. The set of polynomials I = (f(x)q(x) mod (x' + 1)) = Rng(x) is an ideal (cyclic code) of Rn. The following facts about I are used repeatedly (1), (2). (a) I is a subspace of the vector space of Rn ; vector addition is ordinary polynomial addition (coefficients in F). (b) a(x) E I implies xta(x) (mod (xn + 1)) E I for i = 1, 2, * * *, n -1. In words, I contains with a(x) every cyclic shift of a(x). (c) g(x) is the unique polynomial of least degree (r) in I. (d) It follows from (b) and (c) that iio polynomial of I can contain a run of k ( = n - r) consecutive zero coefficieiits; otherwise a suitable cyclic shift would make it into a polynomial of degree < r. The first part of this paper is concerned with the distribution of ones and zeros among the coefficients of g(x). We then take up a special case. Let p (x) be an irreducible factor of degree k of xn + 1, and a a zero of p(x). F(x)/p(x) = F(a) is isomorphic to the finite field GF(2k). If a is a generator of the multiplicative group of GF(2k), we say that p(x) is primi- tive. In this case n = 2k _ 1. We consider in particular the cases in which p(x) is primitive, and (i) gl(X) = (xn + l)/p(x), (ii) g2(x) = (xn + 1)/(p(x)(x + 1)) = gi(x)/(x + 1). In general,

The complex Bernoulli spiral is connected to Grandi curves and Chebyshev polynomials. In this framework, pseudo-Chebyshev polynomials are introduced, and some of their properties are borrowed to form classical trigonometric identities; in particular, a set of orthogonal pseudo-Chebyshev polynomials of half-integer degree is derived.

A classic result of Ritt describes polynomials invertible in radicals: they are compositions of power polynomials, Chebyshev polynomials and polynomials of degree at most 4. In this paper we prove that a polynomial invertible in radicals and solutions of equations of degree at most k is a composition of power polynomials, Chebyshev polynomials, polynomials of degree at most k and, if $$k\le 14$$ , certain polynomials with exceptional monodromy groups. A description of these exceptional polynomials is given. The proofs rely on classification of monodromy groups of primitive polynomials obtained by Müller based on group-theoretical results of Feit and on previous work on primitive polynomials with exceptional monodromy groups by many authors.

Suppose \(\{P_{n}^{(\alpha , \beta )}(x)\} _{n=0}^\infty \) is a sequence of Jacobi polynomials with \( \alpha , \beta >-1.\) We discuss special cases of a question raised by Alan Sokal at OPSFA in 2019, namely, whether the zeros of \( P_{n}^{(\alpha ,\beta )}(x)\) and \( P_{n+k}^{(\alpha + t, \beta + s )}(x)\) are interlacing if \(s,t >0\) and \( k \in {\mathbb {N}}.\) We consider two cases of this question for Jacobi polynomials of consecutive degree and prove that the zeros of \( P_{n}^{(\alpha ,\beta )}(x)\) and \( P_{n+1}^{(\alpha , \beta + 1 )}(x),\) \( \alpha> -1, \beta > 0, \) \( n \in {\mathbb {N}},\) are partially, but in general not fully, interlacing depending on the values of \(\alpha , \beta \) and n. A similar result holds for the extent to which interlacing holds between the zeros of \( P_{n}^{(\alpha ,\beta )}(x)\) and \( P_{n+1}^{(\alpha + 1, \beta + 1 )}(x),\) \( \alpha>-1, \beta > -1.\) It is known that the zeros of the equal degree Jacobi polynomials \( P_{n}^{(\alpha ,\beta )}(x)\) and \( P_{n}^{(\alpha - t, \beta + s )}(x)\) are interlacing for \( \alpha -t> -1, \beta > -1, \) \(0 \le t,s \le 2.\) We prove that partial, but in general not full, interlacing of zeros holds between the zeros of \( P_{n}^{(\alpha ,\beta )}(x)\) and \( P_{n}^{(\alpha + 1, \beta + 1 )}(x),\) when \( \alpha> -1, \beta > -1.\) We provide numerical examples that confirm that the results we prove cannot be strengthened in general. The symmetric case \(\alpha = \beta = \lambda -1/2\) of the Jacobi polynomials is also considered. We prove that the zeros of the ultraspherical polynomials \( C_{n}^{(\lambda )}(x)\) and \( C_{n + 1}^{(\lambda +1)}(x),\) \( \lambda > -1/2,\) are partially, but in general not fully, interlacing. The interlacing of the zeros of the equal degree ultraspherical polynomials \( C_{n}^{(\lambda )}(x)\) and \( C_{n}^{(\lambda +3)}(x),\) \( \lambda > -1/2,\) is also discussed.

In this paper, we consider the second-order differential expression $$\begin{aligned} \ell [y](x)=(1-x^{2})(-(y^{\prime }(x))^{\prime }+k(1-x^{2})^{-1} y(x))\quad (x\in (-1,1)). \end{aligned}$$ This is the Jacobi differential expression with nonclassical parameters $$\alpha =\beta =-1$$ in contrast to the classical case when $$\alpha ,\beta >-1$$ . For fixed $$k\ge 0$$ and appropriate values of the spectral parameter $$\lambda ,$$ the equation $$\ell [y]=\lambda y$$ has, as in the classical case, a sequence of (Jacobi) polynomial solutions $$\{P_{n}^{(-1,-1)} \}_{n=0}^{\infty }.$$ These Jacobi polynomial solutions of degree $$\ge 2$$ form a complete orthogonal set in the Hilbert space $$L^{2}((-1,1);(1-x^{2})^{-1})$$ . Unlike the classical situation, every polynomial of degree one is a solution of this eigenvalue equation. Kwon and Littlejohn showed that, by careful selection of this first-degree solution, the set of polynomial solutions of degree $$\ge 0$$ are orthogonal with respect to a Sobolev inner product. Our main result in this paper is to construct a self-adjoint operator $$T$$ , generated by $$\ell [\cdot ],$$ in this Sobolev space that has these Jacobi polynomials as a complete orthogonal set of eigenfunctions. The classical Glazman–Krein–Naimark theory is essential in helping to construct $$T$$ in this Sobolev space as is the left-definite theory developed by Littlejohn and Wellman.

It has long been recognized that Fibonacci-type recurrence relations can be used to define a set of versatile polynomials $$\{ p_{n} (z)\}$$ that have Fibonacci numbers and Chebyshev polynomials as special cases. We show that a tridiagonal matrix, which can be factored into the product $$AB$$ of two special matrices $$A$$ and $$B$$ , is associated with these polynomials. We apply tools that have been developed to study the supersymmetry of Hamiltonians that have a tridiagonal matrix representation in a basis to derive a set of partner polynomials $$\{ p_{n}^{( + )} (z)\}$$ associated with the matrix product $$BA$$ . We find that special cases of these polynomials share similar properties with the Fibonacci numbers and Chebyshev polynomials. As a result, we find two new sum rules that involve the Fibonacci numbers and their product with Chebyshev polynomials.