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1: 1.15 Summability Methods
Poisson Kernel
Fejér Kernel
1.15.17 K n ( θ ) 0 ,
Poisson Kernel
Fejér Kernel
2: Bibliography W
  • J. Waldvogel (2006) Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.
  • C. S. Whitehead (1911) On a generalization of the functions ber x, bei x, ker x, kei x. Quart. J. Pure Appl. Math. 42, pp. 316–342.
  • 3: 10.62 Graphs
    See accompanying text
    Figure 10.62.2: ker x , kei x , ker x , kei x , 0 x 8 . Magnify
    See accompanying text
    Figure 10.62.4: e x / 2 ker x , e x / 2 kei x , e x / 2 N ( x ) , 0 x 8 . Magnify
    4: 31.10 Integral Equations and Representations
    Kernel Functions
    The kernel 𝒦 must satisfy …
    Kernel Functions
    The kernel 𝒦 must satisfy … leads to the kernel equation …
    5: 10.67 Asymptotic Expansions for Large Argument
    §10.67(i) ber ν x , bei ν x , ker ν x , kei ν x , and Derivatives
    10.67.1 ker ν x e - x / 2 ( π 2 x ) 1 2 k = 0 a k ( ν ) x k cos ( x 2 + ( ν 2 + k 4 + 1 8 ) π ) ,
    The contributions of the terms in ker ν x , kei ν x , ker ν x , and kei ν x on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). …
    10.67.14 ker x kei x - ker x kei x - π 2 x e - x 2 ( 1 2 - 1 8 1 x + 9 64 2 1 x 2 - 39 512 1 x 3 + 75 8192 2 1 x 4 + ) ,
    10.67.15 ker x ker x + kei x kei x - π 2 x e - x 2 ( 1 2 + 3 8 1 x - 15 64 2 1 x 2 + 45 512 1 x 3 + 315 8192 2 1 x 4 + ) ,
    6: 10.61 Definitions and Basic Properties
    When ν = 0 suffices on ber , bei , ker , and kei are usually suppressed. Most properties of ber ν x , bei ν x , ker ν x , and kei ν x follow straightforwardly from the above definitions and results given in preceding sections of this chapter. …
    ker - ν x = cos ( ν π ) ker ν x - sin ( ν π ) kei ν x ,
    kei - ν x = sin ( ν π ) ker ν x + cos ( ν π ) kei ν x .
    ker - n x = ( - 1 ) n ker n x , kei - n x = ( - 1 ) n kei n x .
    7: 10.63 Recurrence Relations and Derivatives
    §10.63(i) ber ν x , bei ν x , ker ν x , kei ν x
    ker ν x , kei ν x ;
    kei ν x , - ker ν x .
    2 ker x = ker 1 x + kei 1 x ,
    Equations (10.63.6) and (10.63.7) also hold when the symbols ber and bei in (10.63.5) are replaced throughout by ker and kei , respectively. …
    8: 12.16 Mathematical Applications
    PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …
    9: 10.70 Zeros
    zeros of  ker ν x 2 ( t + f ( - t ) ) , t = ( m - 1 2 ν - 5 8 ) π ,
    10: 13.27 Mathematical Applications
    The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. …