The Euler transformation is a linear sequence transformation to accelerate the convergence of an alternating series. The sequence of weights of the transformation is extended to a continuous weight function which can accelerate Fourier type integrals including Hankel transforms with a slowly convergent integrand. We show that the continuous weight function can also be used to compute the Fourie...

In our last work, we formulate a Fourier transformation on the infinitedimensional space of functionals. Here we first calculate the Fourier transformation of infinite-dimensional Gaussian distribution exp ( −πξ ∞ −∞ α 2(t)dt ) for ξ ∈ C with Re(ξ) > 0, α ∈ L2(R), using our formulated Feynman path integral. Secondly we develop the Poisson summation formula for the space of functionals, and defi...

In this work, the method of Fourier descriptors has been extended to produce a set of normalized coefficients which are invariant under any affine transformation (translation, rotation, scaling, and shearing). The method is based on a parameterized boundary description which is transformed to the Fourier domain and normalized there to eliminate dependencies on the affine transformation and on t...

It is shown that a lossless first-order optical system whose real symplectic ray transformation matrix can be diagonalized and has only unimodular eigenvalues is similar to a separable fractional Fourier transformer in the sense that the ray transformation matrices of the unimodular system and the separable fractional Fourier transformer are related by means of a similarity transformation. More...

Rotational transformation based on coordinate rotation in Fourier space is a useful technique for simulating wave field propagation between nonparallel planes. This technique is characterized by fast computation because the transformation only requires executing a fast Fourier transform twice and a single interpolation. It is proved that the formula of the rotational transformation mathematical...

The purpose is to formulate a Fourier transformation for the space of functionals, as an infinitesimal meaning. We extend R to ( ∗R) under the base of nonstandard methods for the construction. The domain of a functional is the set of all internal functions from a ∗-finite lattice to a ∗-finite lattice with a double meaning. Considering a ∗-finite lattice with a double meaning, we find how to tr...

We derive the I transformation, an iterative sequence transformation that is useful for the convergence acceleration of certain Fourier series. The derivation is based on the concept of hierarchical consistency in the asymptotic regime. We show that this sequence transformation is a special case of the J transformation. Thus, many properties of the I transformation can be deduced from the known...

The idea of two-dimensional Fourier transformation was originated by Jeener (1) in 1971, but the very wide significance of his proposal does not seem to have been generally realized until several years later, when several elegant applications were described (2-6). The basic idea is very simple. In modern nmr spectrometers, spectra are normally obtained by Fourier transformation of a transient f...

The publication of the Cooley-Tukey fast Fourier transform (FIT) algorithm in 1965 has opened a new area in digital signal processing by reducing the order of complexity of some crucial computational tasks like Fourier transform and convolution from N 2 to N log2 N, where N is the problem size. The development of the major algorithms (Cooley-Tukey and split-radix FFT, prime factor algorithm and...