# Dirichlet series from automorphic forms by Garrett P.

By Garrett P.

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Extra resources for Dirichlet series from automorphic forms

Example text

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.