Weighted mode evaluation

Why the basis values matter

Inverse simplex counts can control mode expansions when the basis values satisfy the evaluation criterion below. The reciprocal-sum identity supplies a useful bound.

For a specified scalar readout of the mode functions, write initially for finitely supported coefficients

$$f_c(x)=\sum_{n\ge1}c_n\phi_n(x),\qquad \|c\|_w^2=\sum_{n\ge1}w_n|c_n|^2,\qquad w_n>0.$$

This is a norm on coefficient sequences. Calling it a spatial $L^2$ norm would require an additional basis and measure identification. Its use for classical fields does not introduce any extra physical spatial dimensions.

Necessary and sufficient condition

At a fixed evaluation point, define

$$B(x)=\sum_{n\ge1}\frac{|\phi_n(x)|^2}{w_n}.$$

Cauchy–Schwarz gives

$$|f_c(x)|\le \|c\|_w\sqrt{B(x)}.$$

Conversely, for a finite cutoff $L$ with $B_L(x)>0$, choose

$$c_n=\frac{\overline{\phi_n(x)}}{w_n\sqrt{B_L(x)}} \quad(1\le n\le L).$$

These coefficients have norm one and evaluation $\sqrt{B_L(x)}$. Thus evaluation extends boundedly to the completed coefficient space if and only if $B(x)<\infty$, and its norm is exactly $\sqrt{B(x)}$. This proof also identifies the missing assumption in an argument based only on $\sum_n1/w_n$.

Applying the simplex weights

For one scalar mode per $n$, take $w_n=S(n,d)$ with $d>1$. If $|\phi_n(x)|\le C$ uniformly in both $n$ and $x$ on the region of interest, then

$$B(x)\le C^2\frac{d}{d-1}.$$

The exact reciprocal-count remainder gives a uniform tail bound:

$$\left|\sum_{n>L}c_n\phi_n(x)\right| \le C\left(\sum_{n>L}S(n,d)|c_n|^2\right)^{1/2} \left[\frac{d}{(d-1)S(L+1,d-1)}\right]^{1/2}.$$

This establishes uniform convergence for finite weighted norm under the stated uniform mode bound. It is a usable truncation estimate, not a general claim about every sector basis.

If band $n$ has several components, the actual criterion becomes

$$B(x)=\sum_{n,\alpha}\frac{|\phi_{n,\alpha}(x)|^2}{w_n}.$$

For example, give band $n$ multiplicity $g_d(n-1)=\binom{n+d-2}{d-1}$. There exist orthonormal families with all components having unit magnitude at a point: distinct Fourier modes on a circle with normalized measure provide a mathematical example. Then

$$B(0)=\sum_{n\ge1}\frac{g_d(n-1)}{S(n,d)} =\sum_{n\ge1}\frac{d}{n+d-1}=\infty.$$

The one-mode reciprocal sum converges while this evaluation is unbounded. This counterexample tests the inference from weights alone; the circle is not a replacement geometry for an IDWT particle.

Limits for contact calculations

V2's finite-band contact theorem uses a specified finite reproducing kernel. Passing to an infinite field requires controlling the corresponding evaluation sum and occupation weights. Derivative readouts need the sums of squared derivative values; interaction energies and determinants need their own estimates. Finite inverse-count weight alone is not a proof of universal ultraviolet control.

Evaluation on a homogeneous spatial space

Let a compact homogeneous space $X$ have invariant volume $\operatorname{Vol}X$. Take a rank-$N$ orthogonal projector onto smooth scalar functions or bundle sections, invariant under a transitive unitary symmetry action. For an orthonormal basis $\phi_j$ of its range the evaluation kernel is

$$K(x,y)=\sum_{j=1}^N\phi_j(x)\phi_j(y)^\dagger.$$

Equivariance conjugates $K(x,x)$ by the fibre symmetry. Its trace is therefore constant on the transitive orbit. Integrating that trace gives $N$, so

$$\operatorname{tr}K(x,x)=\frac{N}{\operatorname{Vol}X}.$$

For scalar functions this is the squared norm of point evaluation, and $|f(x)|^2\le N\|f\|^2/\operatorname{Vol}X$ in the projected space. For a spherical-harmonic shell on the unit $S^2$, it gives $K_\ell(x,x)=(2\ell+1)/(4\pi)$; summing shells $0\le\ell<n$ gives $n^2/(4\pi)$.

The identity fixes the summed evaluation weight. Individual basis values can vanish at a point. For bundle sections, a chosen source direction $v$ probes $v^\dagger K(x,x)v$, while summing an orthonormal set of fibre polarizations gives the trace. Scalar proportionality on the fibre requires an additional isotropy representation condition. The compact homogeneous hypotheses must hold for this bound to replace a direct evaluation sum.

The mode counting checks test saturation of the finite bound, generic complex coefficients, and the multiplicity counterexample.

Related

• Simplex reciprocal sum — supplies the exact tail used above.
• Counting mode tools — companion counting and spectral tools.