On the Lebesgue constant of subperiodic trigonometric interpolation

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1 On the Lebesgue constant of subperiodic trigonometric interpolation Gaspare Da Fies and Marco Vianello November 4, 202 Abstract We solve a recent conjecture, proving that the Lebesgue constant of Chebyshev-like angular nodes for trigonometric interpolation on a subinterval [ ω,ω] of the full period [ π,π] is attained at ±ω, its value is independent of ω and coincides with the Lebesgue constant of algebraic interpolation at the classical Chebyshev nodes in (, ) AMS subject classification: 42A5, 65T40. Keywords: subperiodic trigonometric interpolation, Lebesgue constant. Introduction In several recent papers, subperiodic trigonometric interpolation and quadrature have been studied, i.e., interpolation and quadrature formulas exact on T n ([ ω,ω]) = span{,cos(kθ),sin(kθ), k n, θ [ ω,ω]}, () where 0 < ω π; cf. [2, 5, 6]. These are related by a simple nonlinear transformation to interpolation and quadrature on [, ], and have been called subperiodic since they concern subintervals of the period of trigonometric polynomials. For any fixed trigonometric degree n, consider the 2n+ angles θ j := θ j (n,ω) = 2arcsin(sin(ω/2)τ j ) ( ω,ω), j = 0,,...,2n, (2) where 0 < ω π, and τ j := τ j,2n+ = cos ( ) (2j +)π (,), j = 0,,...,2n (3) 2(2n+) Supported by the ex-60% funds of the University of Padova, and by the GNCS- INdAM. Dept. of Mathematics, University of Padova, Italy marcov@math.unipd.it

2 are the zeros of the 2n+-th Chebyshev polynomial T 2n+ (x). Denoting by l j (x) = T 2n+ (x)/(t 2n+ (τ j)(x τ j )) (4) the j-th algebraic Lagrange polynomial (of degree 2n) for the nodes {τ j }, l j (τ k ) = δ jk, in [2] it has been proved that the cardinal functions for trigonometric interpolation at the angular nodes (2) can be written explicitly as and for j n where with inverse and a j (θ) = 2 L n (θ) = L ω n (θ) = l n(x) (5) L j (θ) = L ω j (θ) = 2 (l j(x)+l 2n j (x)) ( + τ2 j sin(θ j ) ) sin(θ) x 2 = a j (θ)l j (x)+b j (θ)l 2n j (x), (6) x = x(θ) = sin(θ/2) [,] (7) sin(ω/2) θ = θ(x) = 2arcsin(sin(ω/2)x) [ ω,ω], (8) ( + cos(θ/2) ) cos(θ j /2), b j (θ) = 2 ( cos(θ/2) ) = a j (θ). (9) cos(θ j /2) It is worth recalling that the key role played by the transformation (7) on subintervals of the period was also recognized in [, E.3, p. 235], and more recently in [8], in the context of trigonometric polynomial inequalities. Moreover, in [2] stability of such Chebyshev-like subperiodic trigonometric interpolation has been studied, proving that its Lebesgue constant increases logarithmically in the degree L j (θ) α 2 l j (x) α 2 (+ 2π log(2n+) ) where α = sin(ω/2), ω < π. This estimate is useless for ω π (α ), but in view of numerical evidences (see Figure ), it has been there conjectured essentially that: the Lebesgue constant of the angular nodes (2) is attained at θ = ±ω, its value is independent of ω and coincides with the Lebesgue constant of algebraic interpolation at the classical Chebyshev nodes (3). In this note we prove that the conjecture holds, so that the Lebesgue constant has a logarithmic bound independent of ω., 2

3 Figure : Lebesgue functions for degree n = 5 corresponding to the angular nodes (2) for ω = π/3 (left) and ω = π/2 (right). 2 Bounding the Lebesgue constant We begin with the following Lemma Let us consider the angles {θ j } in (2) and the corresponding cardinal functions {L j (θ)} in (6). Moreover, let us consider {φ j } and {L π j (φ)}, i.e., the (equally spaced) angles in ( π, π) φ j = 2(j n)π 2n+ = 2arcsin(τ j ), j = 0,,...,2n, (0) and the corresponding cardinal functions for ω = π, where with inverses φ = 2arcsin(x) = 2arcsin(sin(θ/2)/sin(ω/2)) () x = sin(φ/2), θ = 2arcsin(sin(ω/2)x) = 2arcsin(sin(ω/2)sin(φ/2)), (2) cf. (7)-(8). Then, for every ω (0,π) the following inequality holds L j (θ) + L 2n j (θ) L π j (φ) + Lπ 2n j (φ), j = 0,,...,2n, and in particular for θ θ j, (i.e., x τ j and φ φ j ) L j (θ) + L 2n j (θ) = l j (x) + l 2n j (x) = L π j(φ) + L π 2n j(φ). Proof. First, notice that in view of (4), if x τ j the sign of l j (x) is the same of l 2n j (x), whereas if x < τ j the sign of l j (x) is opposite to that of l 2n j (x). Indeed, T 2n+ (x) is odd, T 2n+ (x) is even, and τ 2n j = τ j. 3

4 Then, the sign of L j (θ) and L π j (φ) is the same of l j(x). Consider the representation (6)-(9), and observe that a j (θ) 0 since θ/2,θ j /2 ( π/2,π/2). Moreover, for x τ j we have θ θ j and b j (θ) 0, whereas for x < τ j we have θ < θ j and b j (θ) < 0. It follows that l j (x) has the same sign of a j (θ)l j (x) and of b j (θ)l 2n j (x), and thus also of L j (θ). The case of the sign of L π j (φ) is completely analogous. Consider now L j (θ) + L 2n j (θ). If x τ j (i.e., θ θ j and φ φ j ) then L j (θ) and L 2n j (θ) have the same sign, along with L π j (φ) and L π 2n j (φ), hence but since L j (θ) + L 2n j (θ) = L j (θ)+l 2n j (θ) = l j (x)+l 2n j (x) = L π j(φ)+l π 2n j(φ) = L π j(φ) + L π 2n j(φ), l j (x)+l 2n j (x) = l j (x) + l 2n j (x) the equality case follows immediately. On the contrary, if x < τ j then L j (θ) and L 2n j (θ) have opposite sign and the same holds for L π j (φ) and Lπ 2n j (φ). Thus L j (θ) + L 2n j (θ) = L j (θ) L 2n j (θ) = cos(θ/2) cos(θ j /2) (l j(x) l 2n j (x)) = cos(θ/2) cos(θ j /2) l j(x) l 2n j (x) < cos(φ/2) cos(φ j /2) l j(x) l 2n j (x) = cos(φ/2) cos(φ j /2) (l j(x) l 2n j (x)) = Lπ j (φ) Lπ 2n j (φ) = Lπ j (φ) + Lπ 2n j (φ), as soon as we prove that cos(θ/2) cos(θ j /2) < cos(φ/2) cos(φ j /2) for x < τ j. In fact, the latter becomes that is or Now, this is equivalent to cos(arcsin(sin(ω/2)x)) cos(arcsin(sin(ω/2)τ j )) < cos(arcsin(x)) cos(arcsin(τ j )), sin 2 (ω/2)x 2 sin 2 (ω/2)τ 2 j sin 2 (ω/2)x 2 sin 2 (ω/2)τ 2 j < x 2 τ 2 j < x2 τ 2 j sin 2 (ω/2)x 2 τ 2 j +sin2 (ω/2)x 2 τ 2 j < x2 sin 2 (ω/2)τ 2 j +sin2 (ω/2)x 2 τ 2 j,. that is x 2 ( sin 2 (ω/2)) < τ 2 j ( sin2 (ω/2)) 4

5 which holds true since we have assumed x < τ j. We can now state and prove the main result of this note. Theorem The maximum of the Lebesgue function of the subperiodic interpolation angles {θ j } in (2), i.e., their Lebesgue constant, is attained at θ = ±ω, its value is independent of ω, and satisfies Λ n = max θ [ ω,ω] = L j (±ω) = L j (θ) = max φ [ π,π] l j (±) = max L π j (φ) = L j (±π) x [,] l j (x). (3) Proof. From Lemma it follows immediately that and in particular for θ = ±ω L j (±ω) = L j (θ) L π j (φ) L π j (±π) = l j (±). Now, by a classical result by Ehlich and Zeller [7], the maximum of the Lebesgue function of trigonometric interpolation of degree at most n at the 2n+ equally spaced angular nodes {φ j } in (0), is attained (also) at φ = ±π. In fact, their Lebesgue function is periodic with period 2π/(2n+), and takes its maximum value at the midpoint of each interval [φ j,φ j+ ], j =,0,...,2n. On the other hand, also the maximum of the Lebesgue function of algebraic interpolation at the classical Chebyshev nodes (3) is attained at x = ±, cf. [3]. Then L j (±ω) max = θ [ ω,ω] L j (±ω) = L j (θ) max φ [ π,π] l j (±) = max x [,] L π j (φ) = L π j (±π) l j (x). 5

6 Remark Observe that in view of well-known estimates (see [3, 4]) and Theorem above Λ n = max θ [ ω,ω] = max φ [ π,π] L j (θ) = max x [,] l j (x) L π j (φ) 2 π log(n)+δ n, where the sequence δ n decreases monotonically from 5/3 to its infimum (2/π)(log(6/π)+γ) = , γ being the Euler-Mascheroni constant. References [] P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, New York, 995. [2] L. Bos and M. Vianello, Subperiodic trigonometric interpolation and quadrature, Appl. Math. Comput. 28 (202), [3] L. Brutman, Lebesgue functions for polynomial interpolation a survey, Ann. Numer. Math. 4 (997), 27. [4] E.W. Cheney and T.J. Rivlin, A note on some Lebesgue constants, Rocky Mountain J. Math. 6 (976), [5] G. Da Fies and M. Vianello, Algebraic cubature on planar lenses and bubbles, Dolomites Res. Notes Approx. 5 (202), 7 2. [6] G. Da Fies and M. Vianello, Trigonometric Gaussian quadrature on subintervals of the period, Electron. Trans. Numer. Anal. 39 (202), [7] H. Ehlich and K. Zeller, Auswertung der Normen von Interpolationsoperatoren, Math. Ann. 64 (966), [8] M. Vianello, Norming meshes by Bernstein-like inequalities, preprint (online at: marcov/publications.html). 6