Dirichlet series from automorphic forms by Garrett P.

By Garrett P.

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The associated inverse differential relation is: φ=φ+θ ∂φ ∂t . 33) The corresponding convolution filter is: φ= 1 θ t −∞ φ(x, t ) exp − t−t θ dt . 34) It is easily seen that this filter commutes with time and space derivatives. This filter is causal, because it incorporates no future information, and therefore is applicable to real-time or post-processing of the data. 1 Definition and Properties of the Filter in the Homogeneous Case 21 Helmholtz Elliptic Filter. An elliptic filter is obtained by taking: 2∂ φ=φ−∆ 2 φ ∂x2l .

Introduction problems related to the effects of the numerical method used in the simulation. The representation of the numerical error in the form of an additional filter is introduced, along with the problem of the relative weight of the various filters used in the numerical simulation. 10). This approach is briefly presented in this book in Sect. 4. The interested reader can refer to [276] for an exhaustive description. The fact that the ideal solution uΠ is associated to a non-vanishing projection error eΠ (u, uΠ ) raises the problem of the reliability of data obtained via Large-Eddy Simulation for practical purposes.

71) For filters with compact support, the following criterion holds: lim l−→∞ (kmax ∆)|Ml+1 (x)| <1 . 72) For symmetric filters, the analogous criterion is (kmax ∆)2 |M2l+2 (x)| <1 . 73) Pruett et al. 73). This proof is now presented. If the filter is non-negative, following an integral mean value theorem, there exists a value c, −2π ≤ −π − x ≤ c ≤ π − x ≤ 2π, such that |M2l+2 | = = 4 |c| ∆ 2 |c| ∆ 2 (−π−x)/∆ ξ 2l G(ξ)dξ (π−x)/∆ |M2l | , It is recalled that the filter is said to be non-negative if G(x) ≥ 0, ∀x.

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