Wavelets are proposed as appropriate analysis tool for the proposed NMEP, additionally to Fourier analysis technique. There are at least two approaches to wavelet analysis, both are addressing the somehow contradiction by itself, that a function over the one-dimensional space R can be unfolded into a function over the two-dimensional half-plane (HoM1):
the first approach is the interpretation of the wavelet transform as a time-frequency analysis tool, where a one-dimensional information (a one-parameter family of purely oscillations) is somehow 'unfolded' into a two-dimensional time-frequency plane, i.e. a function over the real line is mapped into a function over the time-frequency plane that tells 'when' and which 'frequency' occurs. The (human perception) hearing process of a concert is somehow reflecting this kind of compromise (!!) between the (mathematically correct) (either) time localization or frequency localization on the one hand side, and the human perceived melodies, and hence music, just based on a received one-dimensional signal on the other hand side. The interpretation of the wavelet transform of a one-dimensional signal in this context is about a time-frequency analysis with the physical parameters "time" and "frequency" with constant relative bandwidth.
The second approach uses the wavelet analysis as a mathematical microscope. The idea is to look at the details that are added if one goes from a scale "a" to a scale " a-da", where "da" is infinitesimally small. This second approach is closely linked to approximation theory, e.g. in the context of the building of Calderon-Zygmund operators, based on the truncation of kernels (MeY). This mathematical microscope tool 'unfolds' a function over the one-dimensional space R into a function over the two-dimensional half-plane of "positions" and "details" (where is which detail generated?). This two-dimensional parameter space may also be called the position-scale half-plane. The interpretation of the wavelet transform in this context is about a mathematical microscope with the physical parameters "position (parameter a)", "enlargement" and "optics (wavelet function g)".
In the context of this homepage (see also (FeP)) we propose the second approach based on the following rational:
1. (HoM1): the time-frequency approach is based on an (analytical) smoothing (cut-off) function v (compactly supported with a normed L(1)-norm). The related wavelet function g of the second approach is basically in the form g(w) = (d/dw)v(w) with corresponding reduced regularity requirements, whereby the normed L(1)-norm condition of the regular function v(w) is replaced by the admissibility condition of the wavelet function g(w).
The second approach overcomes current handicaps in the usage of cut-off frequency functions occurring for example
a. in relativistic scattering theory of spin-1/2 particles (key words: Coulomb potential, effective cross-section, cut-off frequency function)
handicap: infrared disaster
b. in microscopic dynamics of plasmas with its related Landau damping effect, i.e. the damping (exponential decrease as a function of time) of longitudinal space charge waves in plasma to prevent an instability from developing, and to create a region of stability in the parameter space. The effect is caused by energy exchange between an electromagnetic wave with given phase velocity and particles in the plasma with approximately (!) "equal" velocity: the mathematical tool is perturbation ("approximation") theory in the Vlasovian frame solving the Cauchy problem of the evolution (Vlasov-Poisson) equations
handicap: the mathematical assumption of "analytical" functions (w/o physical justifications/requirements) of the current proof of non-linear Landau damping (C. Villani).
2. (LoA) remark 1.1.10: The second mathematical microscope approach enables a purely (distributional) Hilbert scale framework where the "microscope observations" of two wavelet (optics) functions f, g can be compared with each other by the corresponding "reproducing" ("duality") formula (see also (*) below), whereby
- the "bra(c)"-wavelet transform W(f) is inverted by the adjoint operator of the "(c)ket"-wavelet transform W(g) (given corresponding admissibility conditions are valid)
- the identity (*) provides also some additional degree of freedom in the way that in order to analyze a signal s(t) the wavelet f can be chosen properly according to the special situation of the underlying mathematical model. The prize to be paid is only later, when the "re-building" wavelet g needs to be built accordingly to enable the corresponding "synthesis"
- the Hilbert transform operator (which is valid for every Hilbert scale) is a "natural" partner of the wavelet transform operator, as it is skew-symmetric, rotation invariant and each Hilbert transformed "function" has vanishing constant Fourier term. The example in the context above is the Hilbert transform of the Gaussian/Maxwellian distribution function, the (odd) Dawson function, with the "polynomial degree" point of zero at +/- infinite.
The sine and cosine functions have unbounded support and they do not vanish at infinity. Their spectra are very local consisting of a finite sum of Dirac measures. Conversely, if one use approximations based on finite sum of Dirac measures the spectrum of the corresponding basis "functions" (which is basically the (cosine(x*s) + i * sine(x*s) function) does not vanishes at infinity in the frequency domain.
The wavelet concept is trying to overcome this issue, while basically looking for an orthogonal basis of a Hilbert space (e.g. L(2)=H(0) or H(-1/2)), constructed from a unique generation function g (the scaling function), via translation, dilation and linear combinations, whereby g can be localized in x (space variable) and s (Fourier variable). The admissibility condition for a wavelet governs the behavior of the wavelets in the neighborhood of the frequency zero. The (wavelet) admissibility condition is obviously related to the H(-1/2) Hilbert space norm in case of space dimension m=1.
We note that the hypothesis that a function g has compact support is essential to become a wavelet. Otherwise, it can be shown that there are infinitely supported solutions of the corresponding scaling equation. For instance, the Hilbert transform of the function g satisfies the scaling recursion whenever g does.
We further note the two fundamental examples of universal scaling functions (scaling functions for every rank), the sinc and the Haar scaling functions, which are Fourier transforms of each other.
The wavelet transform W(g)(v) of a function v with respect to a wavelet function g is an isometric mapping, whereby the corresponding adjoint operator is given by the inverse wavelet transform on its range. Let u,v denote two elements of a Hilbert space with inner product (u,v), let ((*,*)) denote the inner product of the Hilbert space H(-1/2). Let further f,g denote two wavelets with bounded inner product ((f,g)) and let (((*,*))) denote the inner product of the corresponding wavelet transforms W(f)(u), W(g)(v) with respect to the underlying Haar measure. Then (up to a constant) it holds
(*) (((W(f)(u),W(g)(v)))) = ((f,g)) * (u,v) .
This identity (in combination with the below) enables a combined wave-wavelet ((H(0),H(-1)) concept for analysis of the H(-1/2) = H(0) * H(0)(ortho) framework, whereby in this specific case it holds (u,v):=((u,v)).
In (PaR) the wavelet transform for a class of distributions is provided, whereby the corresponding inversion formula is established by interpreting convergence in a weak distributional sense. In the context of above we note that log2(sin(x/2)) (with its corresponding 1st and 2nd derivatives, the cot(x) and the 1/(sin(x)*sin(x)) functions) is a L(2) function fulfilling the admissibility condition.
The Gaussian function stands out since it minimizes the Heisenberg uncertainty principle (DaS). The corresponding windowed Fourier (integral) transform is e.g. applied in quantum physics, where it is used for defining and investigating coherent states. It is related to the Weyl-Heisenberg group, while the corresponding wavelet (integral) transform is related to the affine group. In other words, from a group theory perspective windowed Fourier transforms and wavelet transforms are identical.
The wavelet mother function, which is directly connected to the Gaussian function (which is not a wavelet) is the Mexican hat function. It is basically the second derivative of the Gaussian function. In (DaS) a new interpretation of the Mexican hat function is provided: it can be interpreted as a minimizing function of an uncertainty principle, in case its rotation invariant form "A" has a certain form/representation.
The affine-linear group (where each element of that group has two components, while the Weyl-Heisenberg group has three components) of unitary operators equipped with the Haar measure is locally compact, i.e. the group multiplication and the inverse operation of the group are continuous mappings. For local compact groups there is an orthogonality relationship valid, which provides the common group theoretical denominator of windowed Fourier and wavelet transforms (GrA).
Linking back to the primary topic of this homepage we note that the continuous, periodic Riemann function (Fourier) series representation belongs to the space C(1/2), (HoM). It is best analyzed by a continuous wavelet transform using the specific complex wavelet g(x):=1/(x+i). The corresponding wavelet transform is given by the Jacobi Theta function. S. Jaffard (JaS) drew his attention to the irrational points of the Riemann function, building on the fundamental result of J. Gerver, that the Riemann function belongs to the space C(1) on rational p/q with p,q odd numbers (i.e. the derivative of the Riemann function R still does exist and equal -1 at each rational point of the type t=p/q where both numbers p and q are odd), while the Riemann function is non-differentiable elsewhere.
The Davenport and Chowla identity is about the identity of two infinite series, the Riemann function on the one hand side, and an infinite series built from the Liouville function (a prime number-theoretical entity) and the saw-tooth Fourier series on the other side (ChK). "The corresponding integrated identity can be derived from the functional equation only, but to differentiate it, one needs the estimate for the error term for the Liouville function. This is as deep as the prime number theorem and is known to be very difficult."
From the RH related papers we recall that the periodic saw-tooth function, as well as its Hilbert transform, belongs to the periodical L(2)-Hilbert space. The Hilbert transform corresponds to the log(2sin(x)) function, i.e. its first derivative is given by the cot(x)-function.
A natural extension of this result is given in (OsK), where the (generalized) solution of the Cauchy initial value problem for the Schrödinger equation is analyzed. The real part of the restriction of this solution on the line x=0 is given by the Riemann function. A basic role in (OsK) is played by a representation of the differences of the function via Poisson's summation formula and the oscillatory Fresnel integral.
Looking to open questions about the irrationality/transcendence of certain numbers like the Euler constant (in the context of the overall "solution concept of a fractional Hilbert space H(a) framework of this homepage to solve/answer the RH-, NSE-, YME-Millenium problems") we note the following: the Riemann function (continuous, periodic, Fourier representation) belongs to the H(0) Hilbert space, while its dual representation with respect to the inner product of the H(-1/2) Hilbert space is given by the Fourier series representation of the cot-function. The latter one belongs to the Hilbert space H(-1), same as the entire Zeta function on the critical line, i.e. their corresponding weak forms belong to the Hilbert space H(-1/2).
The Sobolev embedding theorem provides the relationship to the continuous (and the corresponding n-time differentiable) space(s) C(n). In this sense Gerver's theorem (resp. its generalization for the solution of the Cauchy initial value problem of the Schrödinger equation, (OsK)) might provide opportunities for new related irrationality proof techniques.
(ChK) Chakraborty K., Kanemitsu S., Long L. H., Quadratic reciprocity and Riemann's "non-differentiable" function, Research in Number Theory, Springer Open Journal, (2015) 1-14
(DaS) Dahlke S., Maass P., The Affine Uncertainty Principle in One and Two Dimensions, Computers Math. Applic. Vol. 30, No. 3-6, (1995), pp. 293-305
(FeP) Federbush P., Navier and Stokes Meet the Wavelet, Commun. Math. Phys. 155, 219-248 (1993)
(GrA) Grossmann A., Morlet J., Paul T., Transforms associated to square integrable group representations I: General results, J. Math. Phys. 26, (1985) pp. 2473-2479
(HoM) Holschneider M., Tchamitchian P., Pointwise analysis of Riemann's "non-differentiable" function, Invest. Math. 105, (1991) pp. 157-175
(HoM1) Holschneider M., Wavelets, An Analysis Tool, Oxford Science Publications, Clarendon Press, Oxford, 1995
(LoA) Louis A. K., Maaß P., Rieder A., Wavelets, Theorie und Anwendungen, B. G. Teubner Verlag, Stuttgart, 1998
(MeY) Meyer Y., Coifman R., Wavelets, Calderon-Zygmund and multilinear operators, Cambridge studies in advanced mathematics 48, Cambridge University Press, 1996
(OsK) Oskolkov K. I., Chakhkiev M. A., On Riemann "Nondifferentiable" Equation and Schrödinger Equation, Proceedings of the Steklov Institute of Mathematics, 2010, Vol. 269, pp. 186-196
(PaR) Pathak R. S., Singh A., Distributional Wavelet Transform, Proc. Natl. Acad. Sci., India, Sect. A. Phys. Sci. 86(2), (2016) 273-277