Efficient multivariate approximation on the cube

Abstract

We combine a periodization strategy for weighted $$L_{2}$$ L 2 -integrands with efficient approximation methods in order to approximate multivariate non-periodic functions on the high-dimensional cube $$\left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}$$ - 1 2 , 1 2 d . Our concept allows to determine conditions on the d-variate torus-to-cube transformations $${\psi :\left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}\rightarrow \left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}}$$ ψ : - 1 2 , 1 2 d → - 1 2 , 1 2 d such that a non-periodic function is transformed into a smooth function in the Sobolev space $${\mathcal {H}}^{m}(\mathbb {T}^{d})$$ H m ( T d ) when applying $$\psi $$ ψ . We adapt $$L_{\infty }(\mathbb {T}^{d})$$ L ∞ ( T d ) - and $$L_{2}(\mathbb {T}^{d})$$ L 2 ( T d ) -approximation error estimates for single rank-1 lattice approximation methods and adjust algorithms for the fast evaluation and fast reconstruction of multivariate trigonometric polynomials on the torus in order to apply these methods to the non-periodic setting. We illustrate the theoretical findings by means of numerical tests in up to $$d=5$$ d = 5 dimensions.

A note on transformed Fourier systems for the approximation of non-periodic signals

“…We consider Chebyshev-and tent-transformed rank-1 lattices in the context of Chebyshev and cosine approximation methods [15,9]. Finally, we outline the transformed Fourier system for the approximation of nonperiodic signals, as introduced in [11], and provide two examples of parameterized transformations.…”

“…We reflect the ideas of a particular family of parameterized torus-to-cube transformations as suggested in [10,11], that generalize the construction idea of the Chebyshev system in composing a mapping with a multiple of its inverse. We call a continuously differentiable, increasing and odd mapping ˜ : (0, 1) → R with ˜ ( ) → ±∞ for → {0, 1} a transformation to R. We obtain a parameterized torus-to-cube transformation (•, )…”

“…Applying the univariate tent-transformation (11) to a sampling set of equispaced nodes, can be interpreted as mirroring the given function at its right boundary point and approximating the resulting function by means of a half-periodic cosine system. However, the produced continuous periodic function generally won't be smooth.…”

“…The also invertible parametrized torus-to-cube transformation (15) adapts the idea of the Chebyshev-transformation by mirroring the original function into an even, continuous and periodic function. Additionally, the involved parameter controls the smoothening effect on the periodized function, see [11].…”

“…For the approximation of non-periodic functions defined on the cube [0, 1] , fast algorithms based on Chebyshev-and tent-transformed rank-1 lattice methods have been introduced in [15,9]. Recently, we suggested a general framework for transformed rank-1 lattice approximation, in which functions defined on the cube [− 1 2 , 1 2 ] are periodized onto R or T , [10,11]. In these approaches we define parameterized families (•, ) : [0, 1] → [0, 1] , ∈ R + of transformations that, depending on the parameter choice, yield a certain smoothening effect when composed with a given non-periodic function.…”