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Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
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TL;DR: In this article, it was shown that a growth function X(r) is a function defined for 0, r < <*> that is positive, nondecreasing, and continuous.
Abstract: log | / ( re\") | = Y,ck(r)eTM. The growth of/, measured by the Nevanlinna characteristic T(r,f)t and the growth of the coefficients Ck(r) are closely related, even in the case of functions of infinite order. The ck(r) can be expressed in terms of the local behaviour of ƒ near the origin, as well as the distribution of the zeros and poles of ƒ. Thus, results on the zero and pole distribution of meromorphic functions of a given rate of growth are obtained that involve the angular distribution as well as the radial distribution. In particular, entire functions with prescribed zeros are constructed without the use of canonical products. Also, it is proved that each meromorphic function of a reasonably general rate of growth is the quotient of two entire functions of the same rate of growth. Some of the results in this note were obtained for a special case in [2]. DEFINITION. A growth function X(r) is a function defined for 0 ^r < <*> that is positive, nondecreasing, and continuous. DEFINITION. A meromorphic function ƒ is said to be of finite \\-type if there exist constants A and B such that T(r,f) ^A\\(Br). REMARK. An entire function ƒ is of finite \\-type if and only if there exist constants A and B such that \\f(z)\\ ^exp(A\\(B\\z\\ )). In case X(r) =r , p > 0 , then the functions of finite X-type are precisely the functions of growth at most order p, finite type. Our considerations include f unctions/(z) that grow like exp(exp(exp(|2;|))) for example. We consider sequences Z = {zn} of complex numbers, zn distinct from 0, such that |2n|—»°° as n—»oo. DEFINITION. We say that Z is of finite \\-density if there exist constants A and B such that N(r, Z) i£A\\(Br), where
56 citations
TL;DR: In this article, the Fourier expansion, Hecke eigenvalues, Atkin-Lehner pseudo-eigenvalues and other data associated to these Eisenstein series, with arbitrary integer weight, level, and nebentypus, were derived in explicit form.
56 citations
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TL;DR: In this article, the covariance function is modeled as a random variable and a variational distribution is used to transform it into a Gaussian distribution to fit the data, which can capture complex functions better than standard approaches and avoid overfitting.
Abstract: Standard sparse pseudo-input approximations to the Gaussian process (GP) cannot handle complex functions well. Sparse spectrum alternatives attempt to answer this but are known to over-fit. We suggest the use of variational inference for the sparse spectrum approximation to avoid both issues. We model the covariance function with a finite Fourier series approximation and treat it as a random variable. The random covariance function has a posterior, on which a variational distribution is placed. The variational distribution transforms the random covariance function to fit the data. We study the properties of our approximate inference, compare it to alternative ones, and extend it to the distributed and stochastic domains. Our approximation captures complex functions better than standard approaches and avoids over-fitting.
56 citations
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TL;DR: In this paper, the basic properties of Fourier integral operators on the torus are studied by using the global representations by Fourier series instead of local representations, which can be applied to weakly hyperbolic partial differential equations.
Abstract: Basic properties of Fourier integral operators on the torus are studied by using the global representations by Fourier series instead of local representations. The results can be applied to weakly hyperbolic partial differential equations.
56 citations
TL;DR: In this article, an analytical solution using homotopy analysis method is developed to describe the nonlinear progressive waves in water of finite depth, where the velocity potential of the wave is expressed by Fourier series and nonlinear free surface boundary conditions are satisfied by continuous mapping.
56 citations