measure with jumps

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1: 1.4 Calculus of One Variable

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Absolutely Continuous Stieltjes Measure

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Stieltjes Measure with $\alpha (x)$ Discontinuous

►The utility of the generalization implicit in the Stieltjes measure appears when $\alpha (x)$ is not everywhere continuous, but has discontinuous jumps at specific values of $x$, say $x_n\in (a,b)$. See Riesz and Sz.-Nagy (1990, Ch. 3). …

2: Funding

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Physical Measurement Laboratory (formerly Physics Laboratory)

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3: 18.39 Applications in the Physical Sciences

… ►where the orthogonality measure is now $dr$, $r\in [0,\mathrm{\infty }).$ … ►Orthogonality, with measure $dr$ for $r\in [0,\mathrm{\infty })$, for fixed $l$ … ►thus recapitulating, for $Z=1$, line 11 of Table 18.8.1, now shown with explicit normalization for the measure $dr$. …normalized with measure $r^{2}dr$, $r\in [0,\mathrm{\infty })$. … ►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 $x\to -1-$. …

4: Charles W. Clark

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5: 27.17 Other Applications

… ►Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.

6: 3.1 Arithmetics and Error Measures

§3.1 Arithmetics and Error Measures

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§3.1(v) Error Measures

… ►The relative precision is … ►The mollified error is … ►

7: Foreword

… ► The success of the original handbook, widely referred to as “Abramowitz and Stegun” (“A&S”), derived not only from the fact that it provided critically useful scientific data in a highly accessible format, but also because it served to standardize definitions and notations for special functions. …

8: 1.1 Special Notation

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$x,y$	real variables.
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$L^2(X,d\alpha )$	the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $d\alpha$.
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9: 18.36 Miscellaneous Polynomials

… ►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. … ►These are polynomials in one variable that are orthogonal with respect to a number of different measures. … ►These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. … ►This infinite set of polynomials of order $n\ge k$, the smallest power of $x$ being $x^k$ in each polynomial, is a complete orthogonal set with respect to this measure. … ►Exceptional type I $X_m$-EOP’s, form a complete orthonormal set with respect to a positive measure, but the lowest order polynomial in the set is of order $m$, or, said another way, the first $m$ polynomial orders, $0,1,\mathrm{\dots },m-1$ are missing. …

10: 18.2 General Orthogonal Polynomials

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§18.2(viii) Uniqueness of Orthogonality Measure and Completeness

… ►The measure is not necessarily absolutely continuous (i. …However, if OP’s have an orthogonality relation on a bounded interval, then their orthogonality measure is unique, up to a positive constant factor. … ►A system of OP’s with unique orthogonality measure is always complete, see Shohat and Tamarkin (1970, Theorem 2.14). … ►The moments for an orthogonality measure $d\mu (x)$ are the numbers …