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1: David M. Bressoud
 Wagon), published by Key College Press in 2000, and A Radical Approach to Lebesgue’s Theory of Integration, published by the Mathematical Association of America and Cambridge University Press in 2007. …
2: 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 , .
Riemann–Lebesgue Lemma
3: 1.1 Special Notation
x , y real variables.
L 2 ( X , d α ) the space of all Lebesgue–Stieltjes measurable functions on X which are square integrable with respect to d α .
4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
§1.18(ii) L 2 spaces on intervals in
For a Lebesgue–Stieltjes measure d α on X let L 2 ( X , d α ) be the space of all Lebesgue–Stieltjes measurable complex-valued functions on X which are square integrable with respect to d α , …The space L 2 ( X , d α ) becomes a separable Hilbert space with inner product … Eigenfunctions corresponding to the continuous spectrum are non- L 2 functions. … The well must be deep and broad enough to allow existence of such L 2 discrete states. …
5: 18.39 Applications in the Physical Sciences
Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being L 2 and forming a complete set. … The spectrum is mixed, as in §1.18(viii), the positive energy, non- L 2 , scattering states are the subject of Chapter 33. … with an infinite set of orthonormal L 2 eigenfunctions … The bound state L 2 eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the δ -function normalized (non- L 2 ) in Chapter 33, where the solutions appear as Whittaker functions. … The fact that non- L 2 continuum scattering eigenstates may be expressed in terms or (infinite) sums of L 2 functions allows a reformulation of scattering theory in atomic physics wherein no non- L 2 functions need appear. …
6: 1.4 Calculus of One Variable
Stieltjes, Lebesgue, and Lebesgue–Stieltjes integrals
A more general concept of integrability of a function on a bounded or unbounded interval is Lebesgue integrability, which allows discussion of functions which may not be well defined everywhere (especially on sets of measure zero) for x . …Similarly the Stieltjes integral can be generalized to a Lebesgue–Stieltjes integral with respect to the Lebesgue-Stieltjes measure d μ ( x ) and it is well defined for functions f which are integrable with respect to that more general measure. … … For α ( x ) nondecreasing on the closure I of an interval ( a , b ) , the measure d α is absolutely continuous if α ( x ) is continuous and there exists a weight function w ( x ) 0 , Riemann (or Lebesgue) integrable on finite subintervals of I , such that …
7: 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. …
8: 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.
  • 9: 1.16 Distributions
    , a function f on I which is absolutely Lebesgue integrable on every compact subset of I ) such that …More generally, for α : [ a , b ] [ , ] a nondecreasing function the corresponding Lebesgue–Stieltjes measure μ α (see §1.4(v)) can be considered as a distribution: … Since δ x 0 is the Lebesgue–Stieltjes measure μ α corresponding to α ( x ) = H ( x x 0 ) (see §1.4(v)), formula (1.16.16) is a special case of (1.16.3_5), (1.16.9_5) for that choice of α . …
    10: 18.2 General Orthogonal Polynomials
    More generally than (18.2.1)–(18.2.3), w ( x ) d x may be replaced in (18.2.1) by d μ ( x ) , where the measure μ is the Lebesgue–Stieltjes measure μ α corresponding to a bounded nondecreasing function α on the closure of ( a , b ) with an infinite number of points of increase, and such that a b | x | n d μ ( x ) < for all n . …