Lebesgue%20constants
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1: 1.8 Fourier Series
2: 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. …3: 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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4: 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 …5: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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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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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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6: 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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7: 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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8: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Zhang and Jin (1996, Table 3.8) tabulates for , to 8D or 8S.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
9: 20 Theta Functions
Chapter 20 Theta Functions
…10: 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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