Lebesgue
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1: David M. Bressoud
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► 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.
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2: 1.8 Fourier Series
3: 1.1 Special Notation
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real variables. | |
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the space of all Lebesgue–Stieltjes measurable functions on which are square integrable with respect to . | |
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4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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§1.18(ii) spaces on intervals in
… ►For a Lebesgue–Stieltjes measure on let be the space of all Lebesgue–Stieltjes measurable complex-valued functions on which are square integrable with respect to , …The space becomes a separable Hilbert space with inner product … ►Eigenfunctions corresponding to the continuous spectrum are non- functions. … ►The well must be deep and broad enough to allow existence of such discrete states. …5: 18.39 Applications in the Physical Sciences
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►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being and forming a complete set.
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►The spectrum is mixed, as in §1.18(viii), the positive energy, non-, scattering states are the subject of Chapter 33.
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►with an infinite set of orthonormal eigenfunctions
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►The bound state eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the -function normalized (non-) in Chapter 33, where the solutions appear as Whittaker functions.
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►The fact that non- continuum scattering eigenstates may be expressed in terms or (infinite) sums of functions allows a reformulation of scattering theory in atomic physics wherein no non- functions need appear.
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6: 1.4 Calculus of One Variable
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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 . …Similarly the Stieltjes integral can be generalized to a Lebesgue–Stieltjes integral with respect to the Lebesgue-Stieltjes measure and it is well defined for functions which are integrable with respect to that more general measure. … … ►For nondecreasing on the closure of an interval , the measure is absolutely continuous if is continuous and there exists a weight function , Riemann (or Lebesgue) integrable on finite subintervals of , such that …7: 3.11 Approximation Techniques
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►to the maximum error of the minimax polynomial is bounded by , where is the th Lebesgue constant for Fourier series; see §1.8(i).
… Moreover, the set of minimax approximations requires the calculation and storage of coefficients, whereas the corresponding set of Chebyshev-series approximations requires only coefficients.
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8: Bibliography F
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Asymptotic expansions of the Lebesgue constants for Jacobi series.
Pacific J. Math. 122 (2), pp. 391–415.
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9: 1.16 Distributions
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►, a function on which is absolutely Lebesgue integrable on every compact subset of ) such that
…More generally, for a nondecreasing function the corresponding Lebesgue–Stieltjes measure (see §1.4(v)) can be considered as a distribution:
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►Since is the Lebesgue–Stieltjes measure corresponding to (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 .
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