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  • 1
    Electronic Resource
    Electronic Resource
    Fast radix-p discrete cosine transform (1992)
    Springer
    Applicable algebra in engineering, communication and computing 3 (1992), S. 39-46 
    Details
    ISSN: 1432-0622
    Keywords: Discrete cosine transform ; Discrete Vandermonde transform ; Radix-p-algorithm ; Chebyshev polynomials
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics , Technology
    Notes: Abstract A new fast radix-p-algorithm (p ≧ 2) for the discrete cosine transform (DCT) and its inverse is presented. It is based on the divide-and-conquer method and on the arithmetic with Chebyshev polynomials. The algorithm can be applied for the efficient calculation of DCT's of arbitrary transform lengths and for the implementation of other discrete Vandermonde transforms withO(N logN) arithmetical operations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01189022
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    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    Index transforms for multidimensional DFT's and convolutions (1989)
    Springer
    Numerische Mathematik 56 (1989), S. 513-528 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 94A11 ; 65T05 ; 20C05 ; 20K01 ; 10A10 ; CR: F.2.1
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Index transforms ofm-dimensional arrays inton-dimensional arrays play a significant role in many fast algorithms of multivariate discrete Fourier transforms (DFT's) and cyclic convolutions. The computation ofm-dimensional “long” DFT's or convolutions can be transfered to the parallel computation ofn-dimensional “short” DFT's or convolutions (n〉m). In this paper, the nature of index transforms is explored using group-theoretical ideas. We solve the open problems concerning index transforms posed recently by Hekrdla [5, 6].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396651
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    On symmetric radix representation of Gaussian integers (1989)
    Springer
    BIT 29 (1989), S. 563-571 
    Details
    ISSN: 1572-9125
    Keywords: 11A63 ; F.2.1 ; G.1.0 ; Gaussian integers ; symmetric residue ; symmetric radix representation ; symmetric mixed-radix representation ; residue arithmetic ; complex number system
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Symmetric radix representation and symmetric mixed-radix representation of Gaussian integers play a significant role in the residue arithmetic ofZ[i]. In the following, known results concerning corresponding representations of integers are generalized. It is shown that for any modulusmεZ[i] with $$m\bar m 〉 1$$ , except form=1±i, 2, there exists a unique symmetricm-radix representation of Gaussian integers.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02219240
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Preconditioners for Ill-Conditioned Toeplitz Matrices (1999)
    Springer
    BIT 39 (1999), S. 513-533 
    Details
    ISSN: 1572-9125
    Keywords: Ill-conditioned Toeplitz matrices ; CG-method ; clusters of eigenvalues ; preconditioners
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with the solution of systems of linear equations A N x = b, where $$\{ A_N \} _{N \in \mathbb{N}}$$ denotes a sequence of positive definite Hermitian ill-conditioned Toeplitz matrices arising from a (real-valued) nonnegative generating function f ∈ C 2π with zeros. We construct positive definite Hermitian preconditioners M N such that the eigenvalues of M N −1 A N are clustered at 1 and the corresponding PCG-method requires only O(N log N) arithmetical operations to achieve a prescribed precision. We sketch how our preconditioning technique can be extended to symmetric Toeplitz systems, doubly symmetric block Toeplitz systems with Toeplitz blocks and non-Hermitian Toeplitz systems. Numerical tests confirm the theoretical expectations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022322820082
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    A note on fast Fourier transforms for nonequispaced grids (1998)
    Springer
    Advances in computational mathematics 9 (1998), S. 337-352 
    Details
    ISSN: 1572-9044
    Keywords: fast Fourier transform ; nonequispaced grids ; B-splines ; Gaussian bells ; shift-invariant spaces ; 65T10 ; 41A15 ; 41A30 ; 42A16
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we are concerned with fast Fourier transforms for nonequispaced grids. We propose a general efficient method for the fast evaluation of trigonometric polynomials at nonequispaced nodes based on the approximation of the polynomials by special linear combinations of translates of suitable functions ϕ. We derive estimates for the approximation error. In particular, we improve the estimates given by Dutt and Rokhlin [7]. As a practical consequence, we obtain a criterion for the choice of the parameters involved in the fast transforms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018901926283
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    On multivariate attenuation factors (1995)
    Springer
    Numerical algorithms 9 (1995), S. 245-261 
    Details
    ISSN: 1572-9265
    Keywords: Multivariate discrete attenuation factors ; interpolation ; shift invariance ; Bernoulli functions ; box splines ; 65t10 ; 42B05 ; 65D05 ; 65D07 ; 41A05 ; 43A70
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper, we introduce the method of discrete attenuation factors for the approximate computation of multivariate discrete Fourier transforms. We consider attenuation factors related with multivariate discrete Bernoulli functions and deduce a best approximation property of the corresponding method of attenuation factors. Choosing a unique approach to the discrete and non-discrete settings, we emphasize the close relation between both cases and interpret results in the literature from a more general point of view.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02141590
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    Paper (German National Licenses)
    Fulltext
  • 7
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    Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphs (2024)
    Details
    Publication Date: 2024-03-22
    Description: We propose two graph neural network layers for graphs with features in a Riemannian manifold. First, based on a manifold-valued graph diffusion equation, we construct a diffusion layer that can be applied to an arbitrary number of nodes and graph connectivity patterns. Second, we model a tangent multilayer perceptron by transferring ideas from the vector neuron framework to our general setting. Both layers are equivariant with respect to node permutations and isometries of the feature manifold. These properties have been shown to lead to a beneficial inductive bias in many deep learning tasks. Numerical examples on synthetic data as well as on triangle meshes of the right hippocampus to classify Alzheimer's disease demonstrate the very good performance of our layers.
    Language: English
    Type: article , doc-type:article
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