A Corner Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Helmberg, Gilbert

doi:10.1006/jath.1998.3320

Cited by 7 publications

(7 citation statements)

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“…The Gibbs phenomenon also appears in higher dimensions, for example, in the case of multiple Fourier series and integrals [6,18] and in multidimensional wavelet expansions [30].…”

Section: The Approximation Of Discontinuous Functions Is a Difficult mentioning

confidence: 95%

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Constructive Approximation of Discontinuous Functions by Neural Networks

Llanas

Lantarón

Sáinz

2008

Neural Process Lett

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“…The Gibbs phenomenon also appears in higher dimensions, for example, in the case of multiple Fourier series and integrals [6,18] and in multidimensional wavelet expansions [30].…”

Section: The Approximation Of Discontinuous Functions Is a Difficult mentioning

confidence: 95%

“…Figure 9b shows the best result obtained by gradient descent. The test function f 3 is defined on [0, 20] by 18,20]. Figure 10a shows the graph of f 3.…”

Section: -(B) a Semi-iterative Procedures Based Onmentioning

confidence: 99%

Constructive Approximation of Discontinuous Functions by Neural Networks

Llanas

Lantarón

Sáinz

2008

Neural Process Lett

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“…This is but one of many details that have prompted much recent work on the multidimensional Gibbs phenomenon [99][100][101][102][103][104][105][106][107].…”

Section: Complexities Of Gibbs' Phenomenon and Coefficient Decays In mentioning

confidence: 98%

Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms

Boyd

2008

J Eng Math

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When a function is expanded asfor some set of basis functions φ j (x), its spectral coefficients a n generally have an asymptotic approximation, as n → ∞, in the form of an inverse power series plus terms that decrease exponentially with n. If f (x) is analytic on the expansion interval, then all the coefficients of the inverse power series are zero and the problem becomes one of "beyond-all-orders" or "exponential" asymptotics. The method of steepest descent for integrals and other complexpath integration techniques can successfully connect the exponentially small behavior of the spectral coefficients to the singularities of f (x) off the expansion interval. Many examples are given in both one and two dimensions.

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“…As an answer, in [1] the following result has been shown (the notation has been slightly adapted to fit the rest of this note).…”

Section: Problems and Answersmentioning

confidence: 99%

“…The first (Theorem 4 and Corollary 1) is a generalization of a localization result stated as Theorem 2 in [1]: under suitable conditions on the variation of a function f in two variables, periodic with period 2π in both variables, the Gibbs phenomenon at a corner point discontinuity only depends on the part of f in an arbitrarily small neighborhood thereof. The second result (Theorem 5) states that taking arithmetic means of the Dirichlet kernel D n over intervals [a, s] ⊂ [0, π] furnishes for fixed a a function p a,n of s whose variation on [−π, π] is bounded by a constant times n. As shown in a separate paper [2] this may be illustrated by the significant example in which integration starts at the origin; the function p 0,n in this special case turns out to be even monotonic on [0, π].…”

Section: Theoremmentioning

confidence: 99%

Localization of a Corner-Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Helmberg¹

2002

Journal of Fourier Analysis and Applications

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Let V be a convex neighborhood of the origin contained in the square Q = {(x, y) ∈ R 2 : |x| ≤ π, |y| ≤ π }. Let A be a 'sector' of V bounded by two convex or concave curves intersecting at the origin. Let f be an integrable function on Q, smooth on A and on V \ A but having a jump discontinuity at the origin, whose coordinate sections (i. e., the restrictions of f to x = const, resp. y = const) have uniformly bounded variation. Under essentially these conditions the partial sums S n,n of the Fourier series of f display for n → ∞ at the origin a corner point Gibbs phenomenon with an overshoot of up to 37,4% of half the jump size. This Gibbs phenomenon manifests itself in the pointwise convergence of S n,n ( x n , y n ; f ) as n → ∞ for all (x, y) ∈ R 2 to a non-constant limiting function only depending on the slopes of the boundary curves of A at the origin and on the jump of f (x, y) as (x, y) approaches (0, 0) within A resp. V \ A.

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A Corner Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Cited by 7 publications

References 7 publications

Constructive Approximation of Discontinuous Functions by Neural Networks

Constructive Approximation of Discontinuous Functions by Neural Networks

Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms

Localization of a Corner-Point Gibbs Phenomenon for Fourier Series in Two Dimensions

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