Stieltjes–Perron inversion
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1: 4.37 Inverse Hyperbolic Functions
§4.37 Inverse Hyperbolic Functions
… ►Inverse Hyperbolic Sine
… ►Inverse Hyperbolic Cosine
… ►Inverse Hyperbolic Tangent
… ►Other Inverse Functions
…2: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
… ►Inverse Sine
… ►Inverse Cosine
… ►Inverse Tangent
… ►Other Inverse Functions
…3: 22.15 Inverse Functions
§22.15 Inverse Functions
►§22.15(i) Definitions
… ►Each of these inverse functions is multivalued. The principal values satisfy … ►4: 18.40 Methods of Computation
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§18.40(ii) The Classical Moment Problem
… ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain . ►Stieltjes Inversion via (approximate) Analytic Continuation
… ►Histogram Approach
… ►Derivative Rule Approach
…5: 1.14 Integral Transforms
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§1.14(vi) Stieltjes Transform
►The Stieltjes transform of a real-valued function is defined by … … ►Inversion
… ►Laplace Transform
…6: 18.39 Applications in the Physical Sciences
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18.39.50
, .
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►The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as .
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►The equivalent quadrature weight, , also forms the foundation of a novel inversion of the Stieltjes–Perron moment inversion discussed in §18.40(ii).
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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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inverse of the square matrix | |
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8: 3.10 Continued Fractions
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►For example, by converting the Maclaurin expansion of (4.24.3), we obtain a continued fraction with the same region of convergence (, ), whereas the continued fraction (4.25.4) converges for all except on the branch cuts from to and to .
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Stieltjes Fractions
… ►is called a Stieltjes fraction (-fraction). … ►For the same function , the convergent of the Jacobi fraction (3.10.11) equals the convergent of the Stieltjes fraction (3.10.6). …9: 31.15 Stieltjes Polynomials
§31.15 Stieltjes Polynomials
… ►§31.15(ii) Zeros
… ►This is the Stieltjes electrostatic interpretation. … ►§31.15(iii) Products of Stieltjes Polynomials
…10: Bibliography R
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Fourier analysis and signal processing by use of the Möbius inversion formula.
IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.
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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.
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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering 23 (4), pp. 91.
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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering 23 (3), pp. 56–64.
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