Riesz Basis Expansions in Hilbert Spaces

• Riesz basis expansions are representations in Hilbert spaces using non-orthogonal yet stable sequences that guarantee unique, invertible decompositions.
• They are employed in trigonometric collocation and Lippmann-Schwinger equation discretizations to achieve spectral diagonalization and computational acceleration.
• Error estimates and well-posedness analyses confirm that these expansions maintain stability and convergence under practical discretization parameters.

A Riesz basis expansion refers to the representation of functions or solutions in Hilbert spaces via sequences that possess Riesz basis properties—bounded invertibility and unconditional convergence but not necessarily orthogonality. In applied analysis and numerical mathematics, Riesz bases are crucial for discretizing space-time integral and evolution equations, enabling stable and accurate solution algorithms. In particular, trigonometric systems and eigenfunction expansions are regularly employed as Riesz bases to diagonalize convolution-type and integral operators, as exemplified in the context of Lippmann-Schwinger equation discretizations via trigonometric collocation (Lechleiter et al., 2014).

1. Mathematical Foundation of Riesz Bases

Let $\mathcal{H}$ be a separable Hilbert space. A sequence $\{\varphi_j\}_{j\in\mathbb{Z}^d}$ is a Riesz basis for $\mathcal{H}$ if it is the image under a bounded invertible operator of an orthonormal basis. Explicitly, every $f\in\mathcal{H}$ admits a unique expansion $f = \sum_j c_j \varphi_j$ with convergence in $\mathcal{H}$, and there exist constants $A,B > 0$ such that

$$A \sum_j |c_j|^2 \leq \left\|\sum_j c_j \varphi_j\right\|^2 \leq B \sum_j |c_j|^2.$$

The lack of orthogonality is compensated by these bounds ensuring numerical and analytic stability. In contrast to orthogonal bases, Riesz bases admit invertibility of the corresponding synthesis-operator, guaranteeing well-posed projections in discretizations.

2. Riesz Basis in Trigonometric Collocation and Integral Equations

Trigonometric systems $$\varphi_j(x)=\frac1{(4\rho)^{3/2}}\exp\left(\frac{i\pi}{2\rho}j\cdot x\right)$$ for $j\in\mathbb{Z}^3_N$ function as periodized Riesz bases in domains $G_{2\rho}\subset\mathbb{R}^3$, due to their completeness and bounded invertibility properties on uniformly discretized grids. In the discretization of time-domain Lippmann-Schwinger equations, these Riesz bases diagonalize the periodized volume integral operator $\hat V_p$, facilitating efficient Fourier-space solution and preserving bounded operator norm equivalence between discrete and continuous settings (Lechleiter et al., 2014).

3. Expansion Schemes and Spectral Diagonalization

Within the framework of CQ (Convolution Quadrature) and trigonometric collocation, the problem is reformulated so that the solution $\hat u_{p,N}$ is sought in the span $T_N$ of the Riesz basis $\{\varphi_j\}$. The action of $\hat V_p$ on $\varphi_j$ yields

$$\hat V_p\,\varphi_j = (4\rho)^{3/2}\,\hat\kappa_{p,s}(j)\;\varphi_j,$$

with $\hat\kappa_{p,s}(j)$ given by local periodized Fourier transform. Thus, the Lippmann-Schwinger equation reduces, in the collocation scheme, to solving a system where expansion coefficients correspond to Riesz basis elements, the system matrix being diagonal in the coefficient space. This enables algorithmic acceleration via fast Fourier transforms and stability via invertibility of the synthesis-operator.

4. Well-Posedness and Error Estimates for Riesz-Based Expansions

Analysis in (Lechleiter et al., 2014) establishes that the discrete operator, constructed via Riesz basis expansions, is boundedly invertible in the weighted $L^2_{q_c}(D)$ norm: $$I+\frac{s^2}{c_0^2}\,\hat V_{q_c}: L^2_{q_c}(D)\;\to\;L^2_{q_c}(D),$$ with norm bounds governed by spectral parameters of the basis and contrast function. Full discretization—CQ in time and trigonometric Riesz basis collocation in space—admits error estimates of the form

$$\left(\Delta t\sum_{n=0}^M\|u^s(t_n)-u^{s,\Delta t,N}_n\|^2_{L^2_{q_c}(D)}\right)^{1/2}\leq C_2\left((\Delta t)^p+N^{-1}\right),$$

where the stability and convergence hold provided the expansion degree $N$ and time step $\Delta t$ meet conservative stability constraints. The implication is that, in practice, Riesz basis expansion preserves spectral accuracy, avoids spurious modes, and retains well-posedness under moderate discretization parameters (Lechleiter et al., 2014).

5. Algorithmic Implications and Implementation Strategies

Algorithmic use of Riesz bases proceeds by collocating the solution on the Riesz grid, forming expansion coefficients via interpolation operators $Q_N$, and exploiting the diagonal structure in Fourier-space by inverse FFTs. The procedure

• selects time steps $\Delta t$ and Riesz degree $N$,
• evaluates discrete Laplace-transformed incident fields in the span of the basis,
• solves the diagonally-structured matrix system per frequency,
• reconstructs the solution via inverse discrete Fourier transforms, underscores the computational advantage conferred by Riesz basis expansions in large-scale simulation settings (Lechleiter et al., 2014). Notably, observed error behavior aligns with theoretical estimates, and the stability predicted by Riesz basis structure is confirmed, even for discontinuous media.

6. Extensions and Context in Operator Theory

While the explicit treatment in (Lechleiter et al., 2014) centers on trigonometric Riesz bases in scattering problems, analogous expansion strategies extend to eigenfunction families of differential operators and to relativistic quantum scattering (via generalized resolvents) (Sakhnovich, 2019). The spectral diagonalization approach, foundational to Riesz basis expansions, underpins both stationary and dynamical formulations of scattering, ensuring a rigorous link between time-domain dynamics and energy-domain amplitude via expansion in complete, invertible bases. This suggests a broad utility of Riesz basis methodology across integral equations, operator theory, and computational wave propagation.