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Lebesgue constants

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1: 1.8 Fourier Series
Lebesgue Constants
1.8.8 L n = 1 π 0 π | sin ( n + 1 2 ) t | sin ( 1 2 t ) d t , n = 0 , 1 , .
2: 3.11 Approximation Techniques
to the maximum error of the minimax polynomial p n ( x ) is bounded by 1 + L n , where L n is the n th Lebesgue constant for Fourier series; see §1.8(i). … Moreover, the set of minimax approximations p 0 ( x ) , p 1 ( x ) , p 2 ( x ) , , p n ( x ) requires the calculation and storage of 1 2 ( n + 1 ) ( n + 2 ) coefficients, whereas the corresponding set of Chebyshev-series approximations requires only n + 1 coefficients. …
3: Bibliography F
  • C. L. Frenzen and R. Wong (1986) Asymptotic expansions of the Lebesgue constants for Jacobi series. Pacific J. Math. 122 (2), pp. 391–415.
  • 4: 2.4 Contour Integrals
    Then … … Then the Laplace transform …where σ ( c ) is a constant. … If this integral converges uniformly at each limit for all sufficiently large t , then by the Riemann–Lebesgue lemma (§1.8(i)) …
    5: 2.10 Sums and Sequences
    Formula (2.10.2) is useful for evaluating the constant term in expansions obtained from (2.10.1). …where γ is Euler’s constant5.2(ii)) and ζ is the derivative of the Riemann zeta function (§25.2(i)). e C is sometimes called Glaisher’s constant. … where α ( - 1 ) is a real constant, and … Hence by the Riemann–Lebesgue lemma (§1.8(i)) …
    6: 2.5 Mellin Transform Methods
    (The last order estimate follows from the Riemann–Lebesgue lemma, §1.8(i).) … where l ( 2 ) is an arbitrary integer and δ is an arbitrary small positive constant. … … where ψ ( z ) = Γ ( z ) / Γ ( z ) . …where γ is Euler’s constant5.2(ii)). …