Real analytic generalized functions

The stress-deformed state of the underground main pipeline under the action of seismic waves of an earthquake is considered. The generalized functions of seismic impulses are constructed. The pipeline motion equations are solved with used Laplace transformation by the time. Tensions and deformations of the pipeline have been determined. A numerical example is reviewed. Diagrams of change of the tension depending on earthquake force are provided in earthquake-points.

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Random generalized functions of locally finite order

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1968-0233409-1 ◽

1968 ◽

Vol 19 (6) ◽

pp. 1457-1457 ◽

Cited By ~ 2

Author(s):

D. M. Eaves

Keyword(s):

Finite Order ◽

Generalized Functions ◽

Locally Finite

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On finite sums of periodic functions

Georgian Mathematical Journal ◽

10.1515/gmj-2019-2076 ◽

2020 ◽

Vol 27 (2) ◽

pp. 265-269

Author(s):

Alexander Kharazishvili

Keyword(s):

Analytic Function ◽

Real Line ◽

Real Analytic Function ◽

Periodic Functions ◽

Uniform Limit ◽

The Real ◽

Pointwise Limit ◽

Finite Sums ◽

Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

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Certain fundamental properties of generalized natural transform in generalized spaces

Advances in Difference Equations ◽

10.1186/s13662-021-03328-6 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Shrideh Khalaf Al-Omari ◽

Serkan Araci

Keyword(s):

Inversion Formula ◽

Acta Math ◽

Generalized Functions ◽

General Nature ◽

Qualitative Behavior ◽

Fundamental Properties ◽

Integrable Functions ◽

Extended Operator ◽

Definition Of ◽

Space Of Integrable Functions

AbstractThis paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386–1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed.

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Modular invariance in finite temperature Casimir effect

Journal of High Energy Physics ◽

10.1007/jhep10(2020)134 ◽

2020 ◽

Vol 2020 (10) ◽

Author(s):

Francesco Alessio ◽

Glenn Barnich

Keyword(s):

Partition Function ◽

Chemical Potential ◽

Casimir Effect ◽

Temperature Inversion ◽

Massless Scalar ◽

Partition Functions ◽

Parallel Plates ◽

Modular Transformations ◽

Real Analytic ◽

Set Up

Abstract The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series. These results become transparent after explicitly showing equivalence of the partition functions for Maxwell’s theory between perfectly conducting parallel plates and for a massless scalar with periodic boundary conditions.

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Eisenstein series and an asymptotic for the K-Bessel function

The Ramanujan Journal ◽

10.1007/s11139-020-00358-8 ◽

2021 ◽

Author(s):

Jimmy Tseng

Keyword(s):

Asymptotic Expansion ◽

Bessel Function ◽

Eisenstein Series ◽

Uniform Estimate ◽

Positive Real ◽

Dominant Term ◽

Complex Order ◽

Fixed Parameter ◽

Real Argument ◽

Real Analytic

AbstractWe produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$ K r + i t ( y ) with positive, real argument y and of large complex order $$r+it$$ r + i t where r is bounded and $$t = y \sin \theta $$ t = y sin θ for a fixed parameter $$0\le \theta \le \pi /2$$ 0 ≤ θ ≤ π / 2 or $$t= y \cosh \mu $$ t = y cosh μ for a fixed parameter $$\mu >0$$ μ > 0 . In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$ K r + i t ( y ) as $$y \rightarrow \infty $$ y → ∞ . When t and y are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $$E_0^{(j)}(z, r+it)$$ E 0 ( j ) ( z , r + i t ) for each inequivalent cusp $$\kappa _j$$ κ j when $$1/2 \le r \le 3/2$$ 1 / 2 ≤ r ≤ 3 / 2 .

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