PRH | Essay | 7.13 • Blur versus Fourier (and Friends)

Blur versus Fourier (and Friends) From Decomposition to Ignorance Buffers Aleksandar Perišić November 2025 Abstract Classical transforms—Fourier, Laplace, Mellin, spectral decompositions—break a known object into known building blocks. Blur does almost the opposite: it starts from a deliberately unknown or half–known presentor and lets it absorb what we do not (yet) wish to resolve. The useful information is what survives after this ignorance buffer is pushed through the object and then partially removed. This note compares these two ideologies. On one side stand orthogonal expansions, fixed dictionaries, and “sum over modes”. On the other side stand blur operators (Gaussian or Paley–Wiener, additive or multiplicative), Hilbert–Pólya realizations via blurred zero measures, Helson factorizations obtained by matching blurred Mellin samples, boundary guards that control deblur limits via Poisson kernels, and a discrete Lyapunov analysis of the accelerated Collatz map where blur appears as an averaged drift budget. The guiding message is: transforms decompose with respect to a fixed language; blur extracts what can be made irrelevant at a chosen resolution and measures what remains in the native coordinates of the problem. Wir können nicht wissen – wir werden dennoch wissen. 1 Two ideologies in one line At a caricature level one can place the two philosophies in a single row: Fourier / spectral: known object −→ known building blocks; Blur: known ignorance presentor −→ extract known bits, leave the rest. Transforms start from the function or distribution and project it onto a fixed dictionary (sines, exponentials, eigenfunctions, wavelets). Blur starts from a chosen “ignorance kernel” and asks: what part of the object can be absorbed into this kernel at a given scale, and what stubbornly refuses to disappear? The latter is the useful, structured remainder. In both cases there is a parameter: frequency or mode index for transforms, and a blur scale for blur. But the role of this parameter is different: transforms sort components in a predefined basis, while blur measures how costly it is to suppress detail. 2 Classical transforms as structured decomposition We briefly recall the standard picture. 2.1 Orthogonal expansions Let (H, ⟨·,·⟩)be a Hilbert space and {φn}n≥0an orthonormal basis. Any f∈Hadmits f=∞ X n=0 ⟨f, φn⟩φn, 1 with convergence in H. The transform is simply the coordinate map T:f7−→ ⟨f, φn⟩n≥0. Typical examples: •Fourier series on the circle: φn(x)=einx. •Fourier transform on R:φξ(x)=eiξx, parametrized continuously. •Eigenfunction expansions for self–adjoint operators, via the spectral theorem. The point is that both sides are completely specified: the dictionary {φn} and the notion of decomposition are fixed once for all. (Notice that the orthogonality so often imposed between components is, in a sense, blur in disguise: perpendicularity is the geometric face of an uncertainty principle, and the inner products between these vectors already encode the kind of uncertainty budget that blur keeps explicit.) 2.2 Integral transforms Integral transforms such as Fourier, Laplace, and Mellin fit the same pattern: (Ff)(ξ) = ZR f(x)e−2πixξ dx, (Lf)(s) = Z∞ 0 e−sxf(x)dx, (Mf)(s) = Z∞ 0 xs−1f(x)dx. Each transform has its own orthogonality or Plancherel theory and a family of inversion formulas. In all these cases one knows in advance what the building blocks look like: pure exponentials, power functions, or eigenfunctions of some operator. Even generalized transforms (wavelets, Gabor frames, coherent states) keep the same spirit: they still rely on a fixed generating function and a structured group action (translations, dilations, time–frequency shifts). 2.3 What is limited by design This design has two structural limitations: 1. The decomposition language is fixed. If the object naturally lives in a different grammar, one must first translate it (often painfully) into the transform’s language before any analysis is possible. 2. The unknown part is something to be approximated away. If the transform leaves a residual tail, the instinct is to make that tail as small as possible, not to interpret it as a resource. Blur turns both of these around. 3 Blur: ignorance buffers and resolution control Blur starts from the opposite end: instead of fixing a dictionary of building blocks, we fix a dictionary for ignorance. 2 3.1 Abstract blur Definition 3.1 (Blur operator).Let X be a measure space and let {kϑ}ϑ>0 be a family of kernels on X (depending possibly on a geometry on X ). A blur at scale ϑ is an operator B ϑ acting on functions or measures Fon Xof the form Bϑ[F](x) = ZX kϑ(x, y)F(y)dy, with the following informal properties: 1. Normalization: RXkϑ(x, y)dx = 1 for each y(total mass 1). 2. Symmetry or controlled asymmetry, depending on the application. 3. Approximate identity: as ϑ↓0the family {kϑ}tends to the identity in a suitable sense. When Fis a measure we define Bϑ[µ]:=kϑ∗µ. One should think of ϑ as a resolution scale: small ϑ means fine resolution, large ϑ means coarse resolution. Crucially, kϑ is not designed to be orthogonal to anything; it is designed to absorb what we do not insist on tracking. Remark 3.2 (Ignorance first, structure later).Given an object F and a blur family { B ϑ} , the primary question is not “what is the coefficient of the nth mode?” but rather: How much of Fcan be quietly parked inside Bϑat scale ϑwithout affecting the property of interest? Whatever refuses to be parked is the structured residue. 3.2 Blur versus transform: different asymmetries To make the contrast precise, consider: • A transform T is typically linear, invertible or almost invertible, and symmetric in the sense that both fand T(f)are on equal footing; we move between them freely. • A blur B ϑ is intentionally asymmetric: only the forward map is canonically defined. Any attempt to “deblur” requires auxiliary information, guards, or rigidity principles. The failure of naive invertibility is not a bug; it is where blur stores uncertainty. In practice, blur is rarely used in isolation; it interacts with transforms. We blur in a coordinate that is natural for the problem (additive on the line, multiplicative on primes, phase in a spectral variable), then use standard analytic tools to read off the surviving structure. 4 Four blur prototypes from number theory and dynamics We now recall four situations where blur plays a central role while remaining deliberately fluid: it is applied in a way that does not reduce to a single explicit inequality, but rather feeds a whole analytic or dynamical pipeline. 3 4.1 Hilbert–Pólya via blurred zero measures Let ξ ( s )be the completed Riemann zeta function and s = 1 2 + it . Write Θ( t ) = arg ξ ( 1 2 + it ) (calibrated argument) and consider the calibrated zero measure µ:= 1 πdΘ−dµ∞, where dµ∞ is a smooth archimedean reference measure. Under RH this is a positive, pure point measure supported on the ordinates {±γn}. Rather than attacking µ directly, one introduces an admissible blur family {fϑ}ϑ>0 (Gaussian or Paley–Wiener) and forms µϑ:= fϑ∗µ. On the blurred side one defines a Cauchy–type integral mϑ(z) := ZR1 λ−z−λ 1+λ2dµϑ(λ) + c0, z ∈C+, which is a Herglotz function whenever µϑ is positive. This mϑ is the Weyl function of a canonical self–adjoint operator Hϑwhose spectral measure is µϑ. The punchline is the deblur limit: as ϑ↓ 0the family mϑ approaches a Herglotz limit m , and the operators Hϑ converge (in strong resolvent sense) to a self–adjoint operator H whose spectrum is supported exactly on the zero ordinates. Here blur is the regularization mechanism that makes the Weyl function well–behaved, while the limit recovers the desired Hilbert–Pólya realization. The proofs use: •positivity of blurred measures at each scale, •convergence theorems for Herglotz functions, •spectral convergence (Jacobi matrices, Kre˘ın strings, de Branges spaces). The blur is not encoded as “one inequality with explicit constants”; it is a soft device that keeps singularities under control while we move between measures, Weyl functions, and operators. 4.2 Helson factorization via multiplicative blur In the Helson–zeta setting one wants to represent an analytic function fon {ℜs > 1}as f(s) = −ζ′ χ(s) ζχ(s)+g(s), where ζχ is a Helson zeta (Dirichlet series with unimodular multiplicative coefficients) and g is entire. Conceptually this is a multiplicative analogue of a Mittag–Leffler expansion. Blur enters in the following way: 1. One passes from f to a Mellin–side distribution q on (1 ,∞ )so that f ( s ) = Rq ( x ) x−sdx + g0 ( s ) on {ℜs>1}. 2. One fixes a multiplicative Gaussian Kρand considers blurred samples of qalong rays: Bσ,t;ρ,Y [q] := ⟨q(x), x−(σ+it)Kρ(log x−Y)⟩. These are Mellin–Fourier coefficients localized around x≈eY. 4 3. One constructs χ(p)stagewise so that the corresponding blurred prime sum Bσ,t;ρ,Y [Hχ] = X n≥1 χ(n)Λ(n) nσ+it Kρ(log n−Y) matches Bσ,t;ρ,Y [q]on a dense sampling set in (Y, t). Technically the construction uses: •smoothed prime number theorems at relative scale in Y, •a time–frequency sampling theorem for Gaussians (Gabor frame), •a diagonal argument that builds χon disjoint prime windows. The blur is crucial: instead of controlling the raw Dirichlet series on a vertical line (which would require extremely strong prime distribution input) one matches blurred Mellin samples and then deblurs using the sampling theorem. Again the blur is fluid: it comes with a scale parameter ρ , but the argument does not hinge on an isolated inequality; it is all about how much multiplicative Gaussian ignorance we can afford at each step and still reconstruct f up to an entire correction. 4.3 Boundary guards, Poisson blur, and BP2 In the Hilbert–Pólya realization one wants not only exact positivity (under RH) but also a more flexible notion of approximate positivity of blurred measures on compacts, encoded in a condition often denoted (BP2). Roughly, inf z∈Kℑmϑ(z)≥ −εϑ(K)with εϑ(K)→0, for each compact K⊂C+ . Here ℑmϑ is a Poisson blur of µϑ , and the defect εϑ ( K )is the amount of negativity we still tolerate at scale ϑ. The way blur appears is: • On the boundary side, one controls logarithmic derivatives of ζ via a logarithmic Rouché guard in a collar around a rectangle. This guard says that the phase increment is close to that of a model Dirichlet series (Helson proxy). • The difference between the model and the genuine ζ is encoded in a phase difference measure, which is then convolved with blur kernels and Poisson kernels. • One tracks a small family of budgets (coming from model mismatch, quadratic terms, truncated moments, and prime tails). The sum of these budgets bounds the negativity of ℑmϑon any fixed compact. Here blur plays two roles at once: 1. It regularizes boundary data into something that can be propagated harmonically into the interior with Poisson kernels. 2. It packages all residual errors into a single parameter εϑ ( K )which can be made small by choosing the blur schedule carefully. This is a much softer use of blur than in a fully explicit discrete certificate. One does not display “the inequality that proves everything”; instead one proves that along a chosen program of blur scales the combined budgets are summable, forcing any Herglotz limit to exist and to inherit positivity. 5 4.4 Collatz drift under residue blur The accelerated Collatz map T(n) = 3n+ 1 2ν2(3n+1) acts on positive odd integers and encodes in ν2 (3 n + 1) the length of the “vertical drop” in each step. A successful Lyapunov analysis needs control of the typical size of this valuation along orbits, but the raw pointwise dynamics is extremely irregular. The blur used here is discrete and combinatorial: • Fix a modulus 2 k and work with residue classes r∈S = { 1 , 3 ,..., 2 k− 1 } . For any odd n≡r ( mod 2 k )the valuation ν2 (3 n + 1) is determined by r , except for a single exceptional residue r∗ . Thus the map T induces a finite Markov chain on S with transition r7→ Fk ( r ) and weight a(r) := log 3 −ν2(3r+ 1) log 2 for non-exceptional r. • The residue blur is the decision to track only this coarse Markov structure: we average over all n≡r ( mod 2 k )and treat them as indistinguishable at that scale. The unknown variation inside each residue class is pushed into a single scalar “drift budget”. •One then seeks a potential ϕ:S→Rand parameters ρ≥0,δ > 0such that for all r∈S, a(r)+ρ+ϕ(Fk(r)) ≤ϕ(r)−δ. This is a one-step Lyapunov inequality on the blurred state space S. The constant log 4 3= 2 log 2 −log 3 plays the role of an average vertical drop: under the residue blur one can show that the expected change in log n per accelerated step is at most −log (4 / 3), up to the finite corrections captured by ϕ and ρ . This constant is not a trivial artefact; it combines multiplication by 3and division by powers of 2into a net contraction rate that is only visible after averaging over residue classes. In this example blur appears in yet another guise: • It is not a Gaussian or Poisson integral, but a coarse–graining of the state space into diadic residue classes. • The “ignorance buffer” is the decision to ignore which representative of a residue class we are on, and to work purely with the coarse Markov chain and its drift. • The deblur step corresponds to showing that the averaged Lyapunov drift still bounds the true dynamics from above when we return to individual integers. What survives the blur is exactly the contraction encoded by log (4 / 3) and the residue certificate; the intricate local variations in ν2(3n+ 1) are parked inside the blur. 4.5 Observer–based Ricci flow: geometric blur by heat kernel Blur also has a genuinely geometric avatar in curvature flows. In an observer–based Ricci flow (OBRF) one starts from the classical Ricci flow on a Riemannian manifold (M, gt), ∂tgt=−2 Ric(gt), but insists that the observer never sees the raw curvature. Instead, the observer sees a blurred curvature obtained by applying the heat semigroup to the Ricci tensor: Hσ(gt) Ric(gt) := exp σ2 2∆gtRic(gt), 6 where ∆ gt is the Laplace–Beltrami operator on tensors and σ > 0is an observer scale. The family Hσ ( gt )plays the role of a Gaussian blur on curvature: small σ means fine geometric resolution, large σmeans a coarse, smoothed view. One then measures curvature only through a blurred energy, for example Eσ(t) =  H1/2 σ(gt) Ric(gt) 2 L2(M,gt), and derives evolution inequalities for Eσ ( t )rather than for the raw ∥Ric ( gt ) ∥L2 . The heat kernel absorbs all small–scale irregularities of the metric; the analysis tracks how much curvature can be hidden inside Hσ at each fixed σ , and how fast the visible part is forced to decay along the flow. Conceptually this is exactly the blur pattern: •the ignorance buffer is the Gaussian smoothing Hσ(gt)on curvature; • the guards are monotonicity and differential inequalities for Eσ ( t )(and related quantities) that hold uniformly in time; • the deblur limit appears in blow–up analysis: near a putative singularity one rescales the flow, and on the rescaled scale the blur parameter σ becomes negligible so that Hσ tends to the identity. In such a scheme, any nontrivial blow–up limit must solve the unblurred Ricci flow and still inherit the vanishing of the blurred energy. This can be used to show that blow–ups are flat and therefore cannot occur in dimension three. One then obtains a global, nonsingular flow that converges to a round metric, recovering the Poincaré conclusion on 3–manifolds in a language where blur is present from the outset and does all the regularizing work. The role of Fourier–style decomposition is minimal; the key actors are the heat–kernel blur, its associated energy budgets, and the rigidity of limit flows under deblurring. 5 Blur as a meta–transform Blur does not replace Fourier and other transforms; it sits “above” them as a meta–transform. 5.1 Transforms as bases, blur as flow A classical transform is a static change of basis. It answers the question: “How does flook in the eigenbasis of operator A?” Blur, by contrast, is a controlled flow of information: “How much of fcan be absorbed into our chosen ignorance buffer at scale ϑ, and what is the shape of what remains?” In the zeta and Collatz contexts this manifests as: • For each ϑ (or each modulus 2 k ) we build a perfectly rigid object (self–adjoint operator, Helson ζχ, finite Markov chain with Lyapunov function) from blurred data. • We then send ϑ↓ 0(or k→ ∞ ) under guards that ensure the limit is stable and dominates the original dynamics. The transforms (Fourier, Mellin, Poisson) appear as internal tools: they express blur in coordinates where convolution is manageable or sampling theorems are available. But the conceptual direction is opposite: we choose the blur first and then see which transforms help us analyze it. 7 5.2 New information from blur A key difference is that blur can create new information when combined with a suitable rigidity principle. In the Helson factorization story, the Gaussian blur plus sampling theorem allows one to manufacture a completely multiplicative χ with prescribed logarithmic derivative pattern on {ℜs > 1 } . In the Hilbert–Pólya story, approximate positivity of blurred measures, together with Herglotz theory, forces the existence of a self–adjoint realization whose spectrum is the deblur limit of the zeros. In the Collatz story, residue blur plus a finite certificate yields a global Lyapunov contraction for the discrete dynamical system. The transforms involved (Mellin, Fourier, Poisson) by themselves do not say “there exists a self–adjoint operator”, “there exists a Helson ζχ with this exact pole pattern”, or “there exists a global Lyapunov function with this drift”. It is the combination of blur, guards, and rigidity that pushes us beyond mere reexpression of known data. 6 Comparison table and outlook For quick reference: Classical transforms Blur framework Primary object function for measure µignorance kernel +object Dictionary fixed (exponentials, eigenfunctions) chosen blur kernels, often problem–specific Goal decompose into known modes absorb unknown part, read persistent structure Symmetry near–invertible, basis ↔ function asymmetric: forward blur canonical, deblur conditional Use of scale frequency / mode index resolution parameter, budget for ignorance Typical tools Plancherel, inversion formulae approximate identities, Poisson kernels, guards, rigidity Number/dynamics avatars explicit formula, Mellin transforms Hilbert–Pólya blur, Helson blur, BP2, Collatz drift Blur is still a young language compared to Fourier. What seems to be emerging is a pattern: • In problems where classical transforms reach their limits (Riemann zeros, Twin primes, highly structured dynamics such as Collatz), blur offers a way to systematically separate what must be paid for (information budgets) from what can be safely ignored. • In the prototypes above, the different blurs—additive on zero ordinates, multiplicative on primes, Poisson in the upper half–plane, discrete on residue classes—interact smoothly with standard transforms but are not reducible to them. • As the technology matures, one can imagine “blur calculus” becoming as standard as Fourier analysis: pick a blur, track its budgets, deblur under guards, and see which structural features of the object survive. From this viewpoint, many classical transforms can be reinterpreted as extremely rigid, orthogonalized versions of blur: they correspond to the special case where the ignorance buffer has already been pushed to zero and we insist on resolving every mode. Blur keeps a tunable entry for the unknown and asks systematically what can be proved without ever forcing that parameter to vanish. 8 7 Predictive motives: why transforms exist, and what blur actually does Classical integral transforms did not enter analysis as abstract gadgets; they were introduced to solve equations and make predictions. The archetype is the heat or wave equation on a simple domain. One writes the PDE, chooses boundary conditions, and then looks for a basis in which the evolution is as simple as possible. Orthogonal eigenfunctions of the Laplacian diagonalize the dynamics, so that each mode evolves independently (typically by a scalar factor e−λt or e±i√λt). The whole game is: •choose a basis {φn}in which the operator of interest is simple (often diagonal); •expand fas f=Pn⟨f, φn⟩φn; •evolve each coefficient ⟨f, φn⟩by an explicit formula; •reconstruct by inverting the transform. Orthogonality is crucial here. It gives: 1. a clean bookkeeping of energy or variance (Parseval): ∥f∥2=Pn|⟨f, φn⟩|2; 2. a guarantee that solving for the basis functions “once and for all” pays off indefinitely: new initial data are just new coordinates in the same stable frame; 3. a strong form of stability: small perturbations of f lead to small perturbations of each coefficient, so long as the norm is controlled. This makes the classical transforms extremely well suited for prediction: they turn time evolution into coefficient-wise algebra. At first glance, blur appears to do the opposite. Instead of sharpening information into a crisp basis, we apply a smoothing operator Bτthat literally loses resolution: f7−→ Bτf, and we explicitly keep track of what the blur could not resolve. It is natural to worry that this makes prediction harder: if some remnant (the blurred-out part) still encodes complicated long-term effects, we might be missing exactly the structure that matters most. The actual situation is more benign, and this is where no free information becomes a useful organizing principle: • Blur is not about ignoring information arbitrarily; it is about minimizing the unknown portion under resource constraints. One chooses Bτ and the scale τ so that everything that can be reliably extracted at the given budget is pulled into the explicit part, and what remains is explicitly labelled as unknown. • In this view, blur neither improves nor worsens the true predictability of the system. It simply gives the best attainable description at the current resolution; any future method, including a Fourier transform, must respect the same global constraints on information and energy. • The unknown remainder is not a mysterious reservoir of future catastrophes. Its contribution is bounded by an explicit blur budget. If we know, for instance, that the unknown component has norm at most ε or carries at most δ units of energy, then no evolution compatible with our model can make that part suddenly dominate the known f -part without paying for it in the same budget. 9