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<p>This paper develops a high-dimensional extension of the Tanh-Legendre basis, an orthonormal spectral framework specifically designed for unbounded domains. By utilizing a tensor-product construction of hyperbolic-mapped Legendre polynomials, we define a basis for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2(\mathbb {R}^d)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> that bypasses the limitations of domain truncation and classical Hermite functions. We present the mathematical derivation of the <i>d</i>-dimensional basis, prove its orthonormality, and apply it to Fredholm integral equations of the second kind on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>. For separable kernels, we demonstrate that the complexity of the method scales linearly with the dimension. Theoretical results confirm that the geometric convergence rate established in one dimension is preserved, providing spectral accuracy for multi-dimensional analytic problems.</p>

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The multi-dimensional Tanh-Legendre basis: an orthonormal spectral framework for integral equations on \(\mathbb {R}^d\)

  • 期刊论文

  • 发表日期：

    2026年6月2日

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  • Abdelaziz Mennouni

摘要

This paper develops a high-dimensional extension of the Tanh-Legendre basis, an orthonormal spectral framework specifically designed for unbounded domains. By utilizing a tensor-product construction of hyperbolic-mapped Legendre polynomials, we define a basis for \(L^2(\mathbb {R}^d)\) L 2 ( R d ) that bypasses the limitations of domain truncation and classical Hermite functions. We present the mathematical derivation of the d-dimensional basis, prove its orthonormality, and apply it to Fredholm integral equations of the second kind on \(\mathbb {R}^d\) R d . For separable kernels, we demonstrate that the complexity of the method scales linearly with the dimension. Theoretical results confirm that the geometric convergence rate established in one dimension is preserved, providing spectral accuracy for multi-dimensional analytic problems.