# Nonharmonic analysis

Harmonic analysis of pseudo-differential operators

Pseudo-differential operators on $\displaystyle \mathbb R^n$ [Kohn+Nirenberg, Hörmander 1965]:
$\displaystyle \widehat{f}(\xi) = \int_{\mathbb R^n} f(x)\ {e}^{-2\pi{i} x\cdot\xi} {d}x, \,\,\, Af(x) = \int_{\mathbb R^n} {e}^{2\pi{ i} x\cdot \xi} \sigma_A(x,\xi) \widehat{f}(\xi) {d}\xi,$
$\displaystyle \left| \partial_{\xi}^{\alpha} \partial_{x}^{\beta} \sigma_A(x,\xi) \right| \leq C_{\alpha \beta} \langle \xi \rangle^{m-|\alpha|}, \,\,\, \langle \xi \rangle =(1+|\xi|^2)^{1/2},\,\,\, { \xi \in \mathbb Z^n.}$

PDOs on the torus $\displaystyle \mathbb T^n= \mathbb R^n / \mathbb Z^n$: Fourier coefficients with $\xi \in \mathbb Z^n$
$\displaystyle \widehat{f}(\xi)= \int_{\mathbb T^n} f(x) {e}^{-{ i}2\pi x\cdot\xi} {d}x, \,\,\,\, Af(x) = \sum_{\xi\in\mathbb Z^n} e^{-2i\pi x\cdot\xi} \sigma_A(x, \xi) \widehat{f}(\xi),$
$\displaystyle \left| \triangle_{\xi}^{\alpha} \partial_x^{\beta} \sigma_A(x,\xi) \right| \leq C_{\alpha \beta} \langle \xi \rangle^{m-|\alpha|}, \,\,\, \xi \in\mathbb Z^n.$
[Agranovich 1990], [McLean 1991], [Turunen 2000], [R.+Turunen, JFAA, 2010].

PDOs on a compact Lie group $G$: [Ruzhansky and Turunen, Birkhaüser book, 2010]
$\displaystyle \widehat{f}(\xi) = \int_G f(x) \xi(x)^* dx, \,\,\, Af(x) = \sum_{[\xi]\in\widehat G} d_{\xi} {Tr}\left( \xi(x){ \sigma_A(x,\xi)} \widehat{f}(\xi) \right),$
$\displaystyle || \triangle_{\xi}^{\alpha} X^{\beta} \sigma_A(x,\xi) ||_{op} \leq C_{\alpha \beta} \langle \xi \rangle^{m-|\alpha|}, \,\,\, \xi\in \widehat G,\,\,\, \langle \xi\rangle =e.v., \,\,\, \Delta_\xi=diff.op., \cdots$

Harmonic vs nonharmonic analysis

Harmonic analysis: symmetries in the underlying space, e.g. working with $e^{2\pi i x\cdot\xi}$ on $\mathbb T^n$ with $\xi\in\mathbb Z^n$; more generally, working with representations of compact, nilpotent, or more general locally compact type I groups;

Nonharmonic analysis: no symmetries in the underlying space, e.g. working with $e^{2\pi i x\cdot\xi}$ on $\mathbb T^n$ with $\xi\not\in\mathbb Z^n$; This name was given by Paley and Wiener.

A survey of our recent works on nonharmonic analysis

Prof. Ruzhansky gave a talk about “Nonharmonic pseudo-differential analysis” at Seminar on Analysis, Differential Equations and Mathematical Physics, at the Southern Federal University, Russia.

Here we break a talk to timestamps:

• Overview of global (harmonic) quantization theories – 1:27
• Harmonic analysis of pseudo-differential operators – 3:39
• Harmonic vs nonharmonic analysis – 8:02
• Nonharmonic analysis of boundary value problems – 11:23
• General philosophy – 14:24
• Global Fourier analysis associated to L and L* – 22:58
• Convolution associated to L and L* – 27:52
• Fourier multipliers – 36:06
• Difference operators – 40:53
• Summary of results – 52:32

Nonharmonic analysis

The exponential systems $\{e^{2\pi i\lambda x}\}_{\lambda\in\Lambda}$ on $L^2(0,1)$ for a discrete set $\Lambda$ possibly containing $\lambda\not\in \mathbb Z$ have been considered by Paley and Wiener in their book R. E. A. C. Paley and N. Wiener. Fourier transforms in the complex domain. American Mathematical Society, New York, 1934.
They called such systems the nonharmonic Fourier series to emphasize the distinction with the usual (harmonic) Fourier series when $\Lambda=\mathbb Z.$

In our work we develop a global version of the Fourier analysis adapted to spectral decompositions. If such spectral decomposition comes from a non-self-adjoint operator, it may lead to biorthogonal systems, ‘nonharmonic’ in the sense of Paley and Wiener. For the analysis of such quantizations we need, among other things, a notion of convolution, and we also analyse these in general Hilbert spaces.

More explanations to be added soon. For now just some of our papers on this subject:

1. Our main first work on this subject:
Ruzhansky M., Tokmagambetov N., Nonharmonic analysis of boundary value problems, Int. Math. Res. Notices, (2016) 2016 (12), 3548-3615. offprint (open access), arxiv, link

Further related works:

• Ruzhansky M., Tokmagambetov N., Convolution, Fourier analysis, and distributions generated by Riesz bases, arxiv
• Delgado J., Ruzhansky M., Fourier multipliers, symbols and nuclearity on compact manifolds, J. Anal. Math., to appear, arxiv
• Ruzhansky M., Tokmagambetov N., Wave equation for operators with discrete spectrum and irregular propagation speed, Arch. Ration. Mech. Anal., 226 (2017), 1161-1207. offprint (open access), arxiv, link
• Kanguzhin B., Ruzhansky M., Tokmagambetov N., On convolutions in Hilbert spaces, Funct. Anal. Appl., 51 (2017), 221-224. link (eng) link (rus)
• Ruzhansky M., Tokmagambetov N., Nonharmonic analysis of boundary value problems without WZ condition, Math. Model. Nat. Phenom., 12 (2017), 115-140. arxiv, link
• Delgado J., Ruzhansky M., Tokmagambetov N., Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary, J. Math. Pures Appl.,107 (2017), 758-783. offprint (open access), arxiv, link
• Dasgupta A., Ruzhansky M., Eigenfunction expansions of ultradifferentiable functions and ultradistributions, Trans. Amer. Math. Soc., 368 (2016), 8481-8498. arxiv, link
• Dasgupta A., Ruzhansky M., Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations, Trans. Amer. Math. Soc., to appear, arxiv