Persistent Fourier Analysis

• Persistent Fourier Analysis is a framework that stabilizes Fourier representations across added structures like topology, resolution, adaptive basis, and memory.
• It enables robust tracking of spectral features over varying resolutions and supports adaptive multiscale analysis for complex, nonstationary signals.
• Applications include ECG classification via persistent homology, adaptive AM–FM decomposition, generalized Fourier series, and recurrence criteria in persistent random walks.

Searching arXiv for recent and foundational papers relevant to “Persistent Fourier Analysis”. Persistent Fourier Analysis denotes a family of constructions in which a Fourier representation is made stable, comparable, or analytically tractable across an additional organizing structure. In the cited literature, that additional structure appears in several distinct forms: topological filtration on a Fourier-space point cloud, adaptive resolution changes within a redundant Fourier representation, decomposition into mono-component Fourier intrinsic band functions, change of basis to nonsinusoidal quasi-harmonics, and Fourier-analytic criteria for random walks with persistence or memory (Ni et al., 2021, 0802.1348, Singh et al., 2015, Török et al., 2020, Cénac et al., 2017). The literature summarized here does not present a single unified formalism under that exact expression. This suggests that the term is best understood as an umbrella designation for methods that preserve Fourier structure while extending it across topology, scale, basis, or stochastic dependence.

1. Conceptual scope and recurrent meanings of persistence

A first meaning of persistence is topological. In the ECG classification framework of Yao and collaborators, sliding-window fast Fourier transformation produces a point cloud in Fourier coefficient space, and persistent homology then measures how loops in that cloud are born and die across a Vietoris–Rips filtration. The resulting descriptors—normalized persistent entropy, maximum life of time, and maximum life of Betty number—summarize periodicity, stability, and disorder in the underlying signal geometry (Ni et al., 2021).

A second meaning is persistence across resolution. The adaptive-resolution Fourier framework replaces a single static time–frequency choice by a family of exact algebraic transforms linking short-window, high-time-resolution spectra to longer-window, high-frequency-resolution spectra. In that setting, spectral structure is not recomputed from scratch at each scale; rather, it is transformed through a redundant representation and a resolution transform, so features can be tracked as resolution changes (0802.1348).

A third meaning is persistence of Fourier analysis beyond its conventional stationary sinusoidal form. The Fourier Decomposition Method recasts the signal as a finite sum of band-limited analytic components with variable amplitudes and instantaneous frequencies, while remaining entirely within a Fourier construction. The nonsinusoidal periodic-series framework generalizes the usual sine–cosine basis to complete systems generated by even and odd quasi-harmonics, preserving the Hilbert-space machinery of expansion, orthogonality after Gram–Schmidt, and convergence theory (Singh et al., 2015, Török et al., 2020).

A fourth meaning is persistence in the probabilistic sense of directional memory. For persistent random walks generated by variable-length Markov chains, Fourier analysis no longer acts on i.i.d. increments. Instead it is lifted to Fourier-perturbed Markov operators whose dominant eigenvalue replaces the scalar characteristic function in a Chung–Fuchs-type criterion for recurrence or transience (Cénac et al., 2017).

These usages are mathematically different, but they share a common structural theme. Fourier coefficients, spectra, or harmonic expansions remain the central representation, while persistence enters through a second axis—filtration, scale hierarchy, adaptive decomposition, generalized basis, or memory state.

2. Persistent homology of Fourier embeddings

In "A Novel Heart Disease Classification Algorithm based on Fourier Transform and Persistent Homology" (Ni et al., 2021), persistent Fourier analysis takes a concrete geometric form. The raw ECG is first denoised with a Butterworth filter that “cut off noisy portions with spectral power over 50 Hz,” after which a local search algorithm detects R-peaks and “successfully segment[s] continuous ECG into single heartbeat.” All subsequent processing is applied to single-beat segments.

For a single-beat ECG represented as a discrete time series

$S=\{s_i\}_{i=1}^n,$

the method forms overlapping child-signals by a sliding-window map

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$

with window length $d$ and sliding speed $t$. Each window

$P_k=[s_{kt},s_{kt+1},\ldots,s_{kt+(d-1)}]$

is transformed by FFT. Writing the complex Fourier coefficients as $a_i+b_i j$, with amplitude

$A_i=|a_i+b_i j|$

and phase

$D_i=\operatorname{atan2}(b_i,a_i),$

the paper defines the Fourier embedding

$$\mathrm{Pc}(P_k)=\big[A_1\cos(D_1),\ldots,A_d\cos(D_d),A_1\sin(D_1),\ldots,A_d\sin(D_d)\big].$$

Although the text states that each child signal is embedded as a point in $\mathbb{R}^d$, the explicit formula has $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$0 coordinates, so the effective feature dimension is $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$1. The authors emphasize that “SWFFT embedding … retains almost all ECG information due to the orthogonality of trigonometric functions.” The point cloud is then equipped with the Euclidean metric

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$2

Persistent homology is computed on the resulting cloud $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$3 via Vietoris–Rips complexes $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$4, in which a simplex is included when the maximum pairwise distance between its vertices is at most $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$5. As $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$6 increases, one obtains a filtration

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$7

with homology maps

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$8

Although the framework is stated generally, the paper explicitly “focus[es] on obtaining 1-dimensional persistent homology diagram and barcode.” The stated reason is that, in this setting, the Vietoris–Rips complex “shows the complete information of graph, while the information of other high-dimensional skeletons is incidental.”

Three 1-dimensional topological features are then extracted from the persistence diagram $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$9. The normalized persistent entropy is defined from lifetimes $d$0 by

$d$1

followed by

$d$2

The motivation given is that “Entropy is then introduced to [reveal] chaoticity and disorder,” but ordinary entropy is said to deviate under “few significant lifetime,” motivating persistent entropy instead. The second feature, maximum life, is

$d$3

used to express “the most lasting cycle.” The third feature, maximum life of Betti number, is described as the longest filtration interval over which the 1-dimensional Betti number remains constant; operationally, it captures the most stable global loop-count regime.

The interpretation is physiological as well as geometric. Stable, nearly periodic ECGs yield coherent low-dimensional clouds with dominant loop structure, whereas arrhythmic or irregular beats yield scattered clouds with broken periodicity. In the reported results, ventricular flutter has “the minimum max Betty life,” interpreted as greater instability of the generated complex; PVC “owns the longest max life,” associated with a distinctive stable loop induced by the tall and deep QRS complex; healthy ECGs show “a more concentrated distribution of persistence entropy whose center is 2” (Ni et al., 2021).

The classification stage maps each beat to

$d$4

and applies SVMs. The dataset, drawn from the MIT–BIH Arrhythmia Database, contains 100 healthy beats, 50 left bundle branch block beats, 50 premature ventricular contraction beats, and 50 ventricular flutter beats after preprocessing. Binary classification accuracies are reported as $d$5 and $d$6 for ventricular flutter with linear and Gaussian SVM respectively; $d$7 and $d$8 for left bundle branch block; and $d$9 and $t$0 for PVC. In 4-class classification with a linear SVM, the reported accuracies are $t$1 for healthy, $t$2 for ventricular flutter, $t$3 for left bundle branch block, and $t$4 for P.V.C. The exact train/test split is not specified, and the Gaussian kernel is described as visually overfitted (Ni et al., 2021).

This construction is the most direct instance in the cited corpus of persistent Fourier analysis in the topological sense: local Fourier spectra define a Euclidean trajectory, and persistence quantifies its loop structure across scale.

3. Resolution persistence and multiresolution Fourier transforms

"Fourier-Based Spectral Analysis with Adaptive Resolution" (0802.1348) addresses a different problem: the fixed time–frequency trade-off of standard Fourier methods. For a sampled signal with sampling rate $t$5, an FFT over a block of length $t$6 yields time resolution $t$7 and frequency resolution $t$8. In classical DFT, FFT, or STFT usage, that choice is fixed once $t$9 is fixed, and obtaining a different resolution ordinarily requires recomputation.

The paper’s central claim is that Fourier analysis can be generalized to adaptive resolution while remaining backward compatible with classical spectral techniques. The construction begins with the Redundant Spectral Transform,

$P_k=[s_{kt},s_{kt+1},\ldots,s_{kt+(d-1)}]$0

where

$P_k=[s_{kt},s_{kt+1},\ldots,s_{kt+(d-1)}]$1

Here $P_k=[s_{kt},s_{kt+1},\ldots,s_{kt+(d-1)}]$2 is the base window length and $P_k=[s_{kt},s_{kt+1},\ldots,s_{kt+(d-1)}]$3 is the redundancy factor. Each of the $P_k=[s_{kt},s_{kt+1},\ldots,s_{kt+(d-1)}]$4 consecutive segments of length $P_k=[s_{kt},s_{kt+1},\ldots,s_{kt+(d-1)}]$5 is represented by $P_k=[s_{kt},s_{kt+1},\ldots,s_{kt+(d-1)}]$6 frequency samples rather than the usual $P_k=[s_{kt},s_{kt+1},\ldots,s_{kt+(d-1)}]$7. When $P_k=[s_{kt},s_{kt+1},\ldots,s_{kt+(d-1)}]$8, the construction reduces to the classical DFT.

Resolution changes are produced by the Resolution Transform,

$P_k=[s_{kt},s_{kt+1},\ldots,s_{kt+(d-1)}]$9

with $a_i+b_i j$0 a natural number dividing $a_i+b_i j$1. This operator merges $a_i+b_i j$2 consecutive redundant spectra into a longer-window spectrum. The effective window increases from $a_i+b_i j$3 to $a_i+b_i j$4, frequency resolution improves by a factor $a_i+b_i j$5, redundancy decreases from $a_i+b_i j$6 to $a_i+b_i j$7, and time resolution worsens accordingly. Repeating the process until $a_i+b_i j$8 yields the standard DFT on the full $a_i+b_i j$9-sample block.

The significance of the construction lies in the fact that resolution is changed algebraically within one coherent Fourier representation. The paper explicitly states that the resolution-transforming iterations can be applied to a partial set of frequencies, so different frequency ranges may be transformed to different resolutions, subject to the restriction that all frequencies involved in a given inverse transform must have the same resolution. This gives a direct formal mechanism for selective refinement in frequency bands or time segments.

Several technical properties make this a genuine persistent multiresolution framework. First, the full transform is an exact factorization of the DFT, so standard Fourier meaning is preserved. Second, the resolution transform is associative and commutative in the sense that

$A_i=|a_i+b_i j|$0

which implies consistency of different resolution paths. Third, the method is real-time capable: one may begin with short-window spectra and refine them incrementally as more samples arrive. The paper illustrates this with a three-close-frequency example, proceeding from $A_i=|a_i+b_i j|$1 to $A_i=|a_i+b_i j|$2 and then $A_i=|a_i+b_i j|$3, where initially unresolved peaks become progressively separated (0802.1348).

The computational analysis distinguishes direct redundant processing from a postponed-redundancy optimization. Direct implementation has complexity

$A_i=|a_i+b_i j|$4

whereas postponed redundancy may reduce complexity to $A_i=|a_i+b_i j|$5 in favorable adaptive-use regimes. The paper also emphasizes that the framework stays within linearity, the Fourier kernel, and exact invertibility.

In this sense, persistence does not refer to topological birth and death, but to continuity of spectral description across resolution levels. A spectral feature can be followed through a chain

$A_i=|a_i+b_i j|$6

and the same data supports multiple mutually consistent views rather than mutually disconnected transforms. That is the paper’s specific form of persistent Fourier analysis (0802.1348).

4. Adaptive AM–FM decomposition within the Fourier framework

"The Fourier Decomposition Method for nonlinear and nonstationary time series analysis" (Singh et al., 2015) develops persistent Fourier analysis in yet another direction: it preserves Fourier orthogonality and completeness while reorganizing the spectrum into adaptive mono-component AM–FM modes. The authors explicitly challenge the widespread view that Fourier methods are unsuitable for nonlinear and nonstationary data.

Starting from the classical Fourier series

$A_i=|a_i+b_i j|$7

with analytic representation

$A_i=|a_i+b_i j|$8

the method partitions the positive-frequency harmonics into contiguous bands and rewrites

$A_i=|a_i+b_i j|$9

Each analytic component

$D_i=\operatorname{atan2}(b_i,a_i),$0

is an analytic Fourier intrinsic band function, and the real signal is reconstructed as

$D_i=\operatorname{atan2}(b_i,a_i),$1

FIBFs are defined by three properties. They have zero mean, they are orthogonal, and each has an analytic counterpart

$D_i=\operatorname{atan2}(b_i,a_i),$2

with

$D_i=\operatorname{atan2}(b_i,a_i),$3

The monotone-phase constraint is the central admissibility condition; it ensures that each AFIBF is a mono-component signal with a well-defined nonnegative instantaneous frequency. In discrete time the paper uses

$D_i=\operatorname{atan2}(b_i,a_i),$4

or, alternatively, $D_i=\operatorname{atan2}(b_i,a_i),$5. The algorithm chooses the maximum possible $D_i=\operatorname{atan2}(b_i,a_i),$6 for each band such that the monotonic-phase condition holds, thereby obtaining the smallest possible number of AFIBFs (Singh et al., 2015).

The method also has an explicit filter-bank interpretation. Each AFIBF corresponds to passing the analytic signal through an ideal zero-phase bandpass filter selecting $D_i=\operatorname{atan2}(b_i,a_i),$7. For multivariate data, the multivariate FDM applies the same zero-phase filter bank to every channel, producing multivariate FIBFs that preserve scale alignment, trend, and instantaneous frequency. The paper gives a cutoff-design rule using the narrowbandness factor

$D_i=\operatorname{atan2}(b_i,a_i),$8

and shows that consecutive center frequencies, cutoffs, and bandwidths can be related by a constant ratio $D_i=\operatorname{atan2}(b_i,a_i),$9 (Singh et al., 2015).

The time–frequency–energy representation follows from the analytic signal of each FIBF. The Fourier–Hilbert spectrum $$\mathrm{Pc}(P_k)=\big[A_1\cos(D_1),\ldots,A_d\cos(D_d),A_1\sin(D_1),\ldots,A_d\sin(D_d)\big].$$0, marginal spectrum

$$\mathrm{Pc}(P_k)=\big[A_1\cos(D_1),\ldots,A_d\cos(D_d),A_1\sin(D_1),\ldots,A_d\sin(D_d)\big].$$1

and instantaneous energy density

$$\mathrm{Pc}(P_k)=\big[A_1\cos(D_1),\ldots,A_d\cos(D_d),A_1\sin(D_1),\ldots,A_d\sin(D_d)\big].$$2

are all defined from instantaneous amplitude and instantaneous frequency. Because the FIBFs are orthogonal and band-limited, energy is preserved across the decomposition.

The paper repeatedly contrasts this framework with EMD and its variants. FIBFs are presented as Fourier-based analogues of intrinsic mode functions, but without sifting, envelope interpolation, or empirical stopping criteria. The authors argue that the method reduces mode mixing, avoids end effects associated with spline envelopes, and is computationally efficient. Concrete examples include a quadri-variate tone–noise mixture decomposed in $$\mathrm{Pc}(P_k)=\big[A_1\cos(D_1),\ldots,A_d\cos(D_d),A_1\sin(D_1),\ldots,A_d\sin(D_d)\big].$$3 s versus $$\mathrm{Pc}(P_k)=\big[A_1\cos(D_1),\ldots,A_d\cos(D_d),A_1\sin(D_1),\ldots,A_d\sin(D_d)\big].$$4 s for MEMD, an intermittency example with FDM computed in about $$\mathrm{Pc}(P_k)=\big[A_1\cos(D_1),\ldots,A_d\cos(D_d),A_1\sin(D_1),\ldots,A_d\sin(D_d)\big].$$5 s versus about $$\mathrm{Pc}(P_k)=\big[A_1\cos(D_1),\ldots,A_d\cos(D_d),A_1\sin(D_1),\ldots,A_d\sin(D_d)\big].$$6 s for EEMD with ensemble size $$\mathrm{Pc}(P_k)=\big[A_1\cos(D_1),\ldots,A_d\cos(D_d),A_1\sin(D_1),\ldots,A_d\sin(D_d)\big].$$7, and a unit-sample sequence whose energy concentrates at $$\mathrm{Pc}(P_k)=\big[A_1\cos(D_1),\ldots,A_d\cos(D_d),A_1\sin(D_1),\ldots,A_d\sin(D_d)\big].$$8 in the derived TFE representation (Singh et al., 2015).

Within the present theme, the significance of FDM is that Fourier analysis is made persistent with respect to nonstationarity and nonlinearity: the basis remains Fourier, but the operational objects are adaptive, band-limited, mono-component modes rather than global fixed-frequency sinusoids.

5. Nonsinusoidal quasi-harmonic expansions

"Nonsinusoidal periodic Fourier series" (Török et al., 2020) broadens persistent Fourier analysis at the level of basis selection. The paper begins with the classical fact that a $$\mathrm{Pc}(P_k)=\big[A_1\cos(D_1),\ldots,A_d\cos(D_d),A_1\sin(D_1),\ldots,A_d\sin(D_d)\big].$$9-periodic function $\mathbb{R}^d$0 satisfying Dirichlet conditions or belonging to $\mathbb{R}^d$1 admits the real Fourier expansion

$\mathbb{R}^d$2

It then replaces the trigonometric basis by a pair of periodic generating functions.

The even generator $\mathbb{R}^d$3 and odd generator $\mathbb{R}^d$4, called fundamental quasi-harmonics, are assumed $\mathbb{R}^d$5-periodic, square-integrable or Dirichlet-regular, and of zero mean. They admit classical Fourier expansions

$\mathbb{R}^d$6

From these, the secondary quasi-harmonics are defined by scaling,

$\mathbb{R}^d$7

with period $\mathbb{R}^d$8. The key structural result is that $\mathbb{R}^d$9 spans the even zero-mean subspace and $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$00 spans the odd subspace, provided the relevant nondegeneracy condition such as $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$01 holds.

For an even zero-mean function $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$02, Theorem 1 gives

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$03

and for an odd function $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$04, Theorem 2 gives

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$05

Combining these with the mean $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$06, Theorem 3 states that any $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$07 can be expanded as

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$08

yielding a complete, generally non-orthogonal basis

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$09

The generalized coefficients are obtained by an explicit change of basis from classical Fourier coefficients. For instance, if

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$10

then the equality

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$11

induces an infinite triangular linear system relating $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$12 and $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$13. The first formulas are

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$14

with analogous formulas in the odd case. This shows that generalized Fourier analysis is, in this formulation, an invertible linear repackaging of the classical trigonometric coefficient space (Török et al., 2020).

Because the basis $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$15 or $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$16 is usually non-orthogonal, the paper also constructs orthogonalized families $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$17 and $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$18 by Gram–Schmidt: $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$19 and similarly for $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$20. Together with the constant function $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$21, these form a biorthogonal basis, and coefficients can then be computed by Euler-type formulas

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$22

The convergence theory is stated to inherit the standard Fourier results: Riemann–Lebesgue decay, Parseval identity in the orthogonalized basis, pointwise convergence to the half-sum at discontinuities, uniform convergence for continuous piecewise $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$23 functions, $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$24 convergence, and almost-everywhere convergence. The paper also presents several examples in rectangular, triangular, sawtooth, polynomial, and exponential quasi-harmonic bases, and suggests applications to the approximation of functions and to the numerical and analytical solution of differential equations (Török et al., 2020).

This line of work extends persistence from scale or topology to basis flexibility. The underlying Fourier framework persists, but the privileged sine–cosine basis is replaced by tailor-made periodic generators that may better match the target function class.

6. Fourier criteria for persistent stochastic walks and general synthesis

"Recurrence of Multidimensional Persistent Random Walks. Fourier and Series Criteria" (Cénac et al., 2017) develops persistent Fourier analysis for stochastic processes with memory. The increments are generated not by an i.i.d. sequence but by a Variable Length Markov Chain on a context tree. Breaking times are defined when the direction changes, and the process observed at those times yields a Markov random walk skeleton $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$25, where $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$26 is an internal Markov chain describing direction changes and $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$27 is the accumulated displacement at breakpoints.

The central Fourier object is no longer a scalar characteristic function. For $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$28, the paper defines the Fourier-perturbed operator

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$29

Under irreducibility, aperiodicity, positive recurrence, and a spectral-gap framework on a Banach space, there exists a neighborhood of $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$30 and a continuous function $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$31 such that $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$32 is the unique maximal-modulus eigenvalue of $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$33, has algebraic multiplicity $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$34, and satisfies $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$35 with equality only at $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$36. The decomposition

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$37

separates the dominant spectral contribution from a remainder of spectral radius less than $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$38 (Cénac et al., 2017).

This leads to a Chung–Fuchs-type recurrence criterion for Markov random walks. Under an additional sector condition,

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$39

the walk is recurrent or transient according as

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$40

is infinite or finite. The paper also proves equivalence with the Green-series criterion

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$41

for an initial distribution $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$42 dominated by the stationary measure. In the classical i.i.d. setting, the scalar characteristic function $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$43 controls recurrence. Here it is replaced by the dominant eigenvalue of a Fourier-perturbed Markov operator, which is the precise mechanism by which Fourier analysis accommodates persistence or memory (Cénac et al., 2017).

The paper further studies two-dimensional persistent random walks, including directionally reinforced random walks and non-backtracking directionally reinforced random walks. Under assumptions of independent horizontal and vertical persistence times and specified transition probabilities $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$44 and $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$45, recurrence of the full persistent walk is characterized by divergence of one of two series of crossing probabilities. In the symmetric transient case this criterion is equivalent to a more explicit Fourier integral involving

$$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$46

A key technical ingredient is a new upper bound for the Lévy concentration function associated with symmetric distributions, expressed in terms of Fourier-analytic estimates and Esseen-type inequalities (Cénac et al., 2017).

The paper also constructs a counterexample to the conjecture of Mauldin, showing that certain generalized directionally reinforced random walks in $$\operatorname{Pr}_{t,d}(S)=\{P_k\}_{k=1}^m,\qquad m=\left\lfloor \frac{n-d}{t}\right\rfloor+1,$$47 can be recurrent while their Markov-random-walk skeletons are transient. This establishes that persistence at the path level and recurrence of the skeleton need not coincide, even though the difference is described as “extremely thin” (Cénac et al., 2017).

Taken together, the cited works support a broad synthesis. Persistent Fourier Analysis is not a single theorem or algorithm, but a coherent research direction in which Fourier methods are extended without abandoning their core algebraic and spectral structure. In one line of work, local spectra become point clouds whose persistent homology yields robust descriptors of periodicity and disorder. In another, redundant spectra are transformed across resolutions rather than discarded. In another, Fourier harmonics are grouped into adaptive AM–FM modes suited to nonlinear and nonstationary data. In another, trigonometric bases are replaced by complete quasi-harmonic systems while preserving Fourier-series machinery. In yet another, persistence as memory in stochastic motion is analyzed through operator-valued Fourier transforms and dominant spectral data (Ni et al., 2021, 0802.1348, Singh et al., 2015, Török et al., 2020, Cénac et al., 2017).

A common misconception is that persistent Fourier analysis must mean persistent homology applied to Fourier coefficients. The literature here indicates a broader picture. Topological persistence is one important realization, but resolution persistence, adaptive AM–FM persistence, basis persistence, and persistence of directional memory each produce mathematically distinct Fourier frameworks. A plausible implication is that the most durable unifying definition is structural rather than procedural: Fourier analysis is made persistent when a single spectral formalism remains valid across an additional hierarchy—of scales, topologies, bases, or state-dependent dynamics.