cardinal function

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1: 24.17 Mathematical Applications

… ►The members of ${𝒮}_n$ are called cardinal spline functions. The functions … ►

Bernoulli Monosplines

►A function of the form $x^n-S(x)$, with $S(x)\in {𝒮}_{n-1}$ is called a cardinal monospline of degree $n$. …For each $n=1,2,\mathrm{\dots }$ the function $M_n(x)$ is also the unique cardinal monospline of degree $n$ satisfying (24.17.6), provided that …

2: 3.3 Interpolation

… ►For interpolation of a bounded function $f$ on $ℝ$ the cardinal function of $f$ is defined by ►

3.3.43 $$C(f,h)(x)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}f(kh)S(k,h)(x),$$

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Defines:

$C(f,h)$: cardinal function of $f$ (locally)

Symbols:

$f(z)$: function and $S(k,h)(x)$: Sinc function

Permalink:

http://dlmf.nist.gov/3.3.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(vi), §3.3 and Ch.3

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3: 26.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$x$	real variable.
…
$\left|A\right|$	number of elements of a finite set $A$.
…

►The main functions treated in this chapter are: ► ►►►►

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
…
$p\left(n\right)$	number of partitions of $n$.
$p_k\left(n\right)$	number of partitions of $n$ into at most $k$ parts.
…

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4: Bibliography S

… ►

Cardinal Spline Interpolation. Society for Industrial and Applied Mathematics, Philadelphia, PA.

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Notes:

Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12

External Links:

MathReview (H. L. Loeb), ZentralBlatt

Referenced by:

§24.17(ii).

Encodings:

BibTeX

… ►

Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.

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Notes:

Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)

External Links:

ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§14.30(i), 3rd item.

Encodings:

BibTeX

… ►

On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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External Links:

ZentralBlatt

Referenced by:

§12.13(i).

Encodings:

BibTeX

… ►

Topics in Complex Function Theory. Vol. III: Abelian Functions and Modular Functions of Several Variables. Interscience Tracts in Pure and Applied Mathematics, No. 25, Wiley-Interscience, [John Wiley & Sons, Inc], New York-London-Sydney.

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Notes:

Translated from the original German by E. Gottschling and M. Tretkoff. Reprinted by John Wiley & Sons, New York, 1989.

External Links:

ISBN 0-471-79090-7, MathReview, ZentralBlatt

Referenced by:

Ch.21.

Encodings:

BibTeX

… ►

Automatic computing methods for special functions. IV. Complex error function, Fresnel integrals, and other related functions. J. Res. Nat. Bur. Standards 86 (6), pp. 661–686.

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Notes:

Includes Fortran code, maximum accuracy 18S.

External Links:

ISSN 0022-4340, MathReview, ZentralBlatt

Referenced by:

§7.25(iii).

Encodings:

BibTeX

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5: 26.18 Counting Techniques

… ►

26.18.1

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Symbols:

$\left|A\right|$: number of elements of a finite set $A$ , $\cap$: intersection, $\setminus$: set subtraction, $\cup$: union, $j$: nonnegative integer, $n$: nonnegative integer, $A_k$: subsets and $S$: set

Permalink:

http://dlmf.nist.gov/26.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18 and Ch.26

… ►For further examples in the use of generating functions, see Stanley (1997, 1999) and Wilf (1994). …