Regularity and approximation of Gaussian random fields evolving temporally over compact two-point homogeneous spaces

Abstract

We consider Gaussian random fields on the product of a compact two-point homogeneous space cross the time, which are space isotropic and time stationary. We study regularity properties of these random fields in terms of function spaces whose elements have different smoothness in the space and time domain. Namely, we express the norm of the corresponding covariance kernel functions in terms of the summability of the associated spectral coefficients. Furthermore, we define an approximation method based on the truncation of the expansion related to the spectral representation of a given random field. The accuracy of this approximation is measured in the \(L^p\) sense. Finally, we model a space–time dataset of ozone concentrations in Mexico City using a seasonal temporal covariance function constructed through an expansion of Jacobi polynomials. We find that we need relatively few Jacobi polynomials to get the best fit to the data in terms of the deviance information criterion. We discuss the characteristics of this model, including seasonality, decay and approximate conditional independencies.

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Notes

  1. 1.

    Ozone levels are given hourly but are, in fact, an average of the 60 minute-by-minute measurements. Missing ozone levels were imputed before receiving the data using the measurements at the nearest station within the same region, when available. When no simultaneous measurements were available within the same region, we use the nearest station in a different region.

References

  1. Adams RA, Fournier JJ (2003) Sobolev spaces. Elsevier, Amsterdam

    Google Scholar 

  2. Barbosa VS, Menegatto VA (2016) Differentiable positive definite functions on two-point homogeneous spaces. J Math Anal Appl 434(1):698–712. https://doi.org/10.1016/j.jmaa.2015.09.040

    MathSciNet  Article  MATH  Google Scholar 

  3. Barbosa VS, Menegatto VA (2017) Strict positive definiteness on products of compact two-point homogeneous spaces. Integral Transforms Spec Funct 28(1):56–73

    MathSciNet  Article  Google Scholar 

  4. Besse AL (2012) Manifolds all of whose geodesics are closed, vol 93. Springer Science & Business Media, Cham

    Google Scholar 

  5. Broadbridge P, Kolesnik AD, Leonenko N, Olenko A, Omari D (2020) Spherically restricted random hyperbolic diffusion. Entropy 22(2):Paper No. 217, 31. https://doi.org/10.3390/e22020217

    MathSciNet  Article  Google Scholar 

  6. Clarke J, Alegria A, Porcu E (2018) Regularity properties and simulations of Gaussian random fields on the sphere cross time. Electron J Stat 12:399–426

    MathSciNet  Article  Google Scholar 

  7. Cleanthous G, Georgeadis A, Lang A, Porcu E (2019) Regularity, continuity and approximation of isotropic Gaussian random fields on on compact two-point homogeneous spaces. arXiv

  8. Cleanthous G, Georgiadis AG, Lang A, Porcu E (2020) Regularity, continuity and approximation of isotropic Gaussian random fields on compact two-point homogeneous spaces. Stochastic Processes and their Applications

  9. Da Prato G, Zabczyk J (1992) Stochastic equations in infinite dimensions. Encyclopedia of mathematics and its applications, vol 44. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511666223

  10. Direccion de Monitoreo Atmosferico SEDEMA, Ciudad de México (2017) Indice de la Calidad del Aire (horarios) http://www.aire.cdmx.gob.mx

  11. Gangolli R (1967) Positive definite kernels on homogeneous spaces and certain stochastic processes related to Levy’s Brownian motion of several parameters. Ann Inst H Poincare 3:121–226

    MATH  Google Scholar 

  12. Haario H, Saksman E, Tamminen J (2001) An adaptive metropolis algorithm. Bernoulli 7(2):223–242

    MathSciNet  Article  Google Scholar 

  13. Helgason Su (1965) The radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds. Acta Math 113:153–180. https://doi.org/10.1007/BF02391776

    MathSciNet  Article  MATH  Google Scholar 

  14. Katzfuss M, Guinness J (2017) A general framework for Vecchia approximations of Gaussian processes. arXiv preprint arXiv:170806302

  15. Kerkyacharian G, Ogawa S, Petrushev P, Picard D (2018) Regularity of Gaussian processes on Dirichlet spaces. Constr Approx 47(2):277–320. https://doi.org/10.1007/s00365-018-9416-8

    MathSciNet  Article  MATH  Google Scholar 

  16. Klenke A (2013) Probability theory: a comprehensive course. Springer Science & Business Media, Cham

    Google Scholar 

  17. Lang A, Schwab C (2013) Isotropic random fields on the sphere: regularity, fast simulation and stochastic partial differential equations. Ann Appl Probab 25:3047–3094

    MathSciNet  Article  Google Scholar 

  18. Lang A, Schwab C (2015) Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations. Ann Appl Probab 25(6):3047–3094

    MathSciNet  Article  Google Scholar 

  19. Lu T, Ma C (2020) Isotropic covariance matrix functions on compact two-point homogeneous spaces. J Theor Probab 33(3):1630–1656. https://doi.org/10.1007/s10959-019-00920-1

    MathSciNet  Article  MATH  Google Scholar 

  20. Ma C (2016) Time varying isotropic vector random fields on spheres. J Theor Probab. https://doi.org/10.1007/s10959-016-0689-1

    Article  Google Scholar 

  21. Ma C, Malyarenko A (2020) Time-varying isotropic vector random fields on compact two-point homogeneous spaces. J Theor Probab 33(1):319–339. https://doi.org/10.1007/s10959-018-0872-7

    MathSciNet  Article  MATH  Google Scholar 

  22. Malyarenko AA (2003) Abelian and Tauberian theorems for random fields on two-point homogeneous spaces. Teor Ĭmovīr Mat Stat 69:106–118. https://doi.org/10.1090/S0094-9000-05-00619-8

    Article  MATH  Google Scholar 

  23. Malyarenko A (2013) Invariant random fields on spaces with a group action. Springer, New York

    Google Scholar 

  24. Marinucci D, Peccati G (2013) Mean-square continuity on homogeneous spaces of compact groups. Electron Commun Probab 18(37):10. https://doi.org/10.1214/ECP.v18-2400

    MathSciNet  Article  MATH  Google Scholar 

  25. Mastrantonio G, Jona Lasinio G, Pollice A, Capotorti G, Teodonio L, Genova G, Blasi C (2019) A hierarchical multivariate spatio-temporal model for clustered climate data with annual cycles. Ann Appl Stat 13(2):797–823

    MathSciNet  Article  Google Scholar 

  26. Porcu E, Zastavnyi V, Bevilacqua M, Emery X (2019) Stein hypothesis and screening effect for covariances with compact support. Submitted

  27. Porcu E, Furrer R, Nychka D (2020) \(30\) years of space-time covariance functions. Technical report, Trinity College Dublin Submitted for publication to WIRES

  28. Sahu SK, Gelfand AE, Holland DM (2007) High-resolution space–time ozone modeling for assessing trends. J Am Stat Assoc 102(480):1221–1234

    MathSciNet  Article  Google Scholar 

  29. Schmeisser HJ (2007) Recent developments in the theory of function spaces with dominating mixed smoothness. NAFSA 8–nonlinear analysis, function spaces and applications, vol 8. Czech. Acad. Sci, Prague, pp 144–204

  30. Shirota S, Gelfand A (2017) Space and circular time log Gaussian cox processes with application to crime event data. Ann Appl Stat 11(2):481–503

    MathSciNet  Article  Google Scholar 

  31. Spiegelhalter DJ, Best NG, Carlin BP, Van Der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc Ser B (Stat Methodol) 64(4):583–639

    MathSciNet  Article  Google Scholar 

  32. Stein ML (1999) Statistical interpolation of spatial data: some theory for Kriging. Springer, New York

    Google Scholar 

  33. Stein ML (2005) Space–time covariance functions. J Am Stat Ass 100(469):310–321

    MathSciNet  Article  Google Scholar 

  34. Szegő G (1939) Orthogonal polynomials. Colloquium Publications., vol XXIII. American Mathematical Society, Providence

    Google Scholar 

  35. Wang HC (1952) Two-point homogeneous spaces. Ann Math 55:177–191. https://doi.org/10.2307/1969427

    Article  Google Scholar 

  36. White P, Porcu E (2019) Nonseparable covariance models on circles cross time: a study of Mexico City ozone. Environmetrics 30:e2558

    MathSciNet  Google Scholar 

  37. White PA, Gelfand AE, Rodrigues ER, Tzintzun G (2019) Pollution state modelling for Mexico City. J R Stat Soc Ser A (Stat Soc) 182(3):1039–1060

    MathSciNet  Article  Google Scholar 

  38. Xu Y (2018) Approximation by polynomials in sobolev spaces with Jacobi weight. J Fourier Anal Appl 24(6):1438–1459

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors would like to express their gratitude to the two anonymous referees for their careful reading of their manuscript and the several suggestions they proposed, which improved significantly the paper. The research of GC has been supported by the individual grant “New function spaces in harmonic analysis and their applications in statistics” by the University of Cyprus. Emilio Porcu acknowledges the support of Fondecyt Project Number 1170290 from the Chilean Ministry of Education, and Millennium Science Initiative of the Ministry of Economy, Development, and Tourism, grant “Millenium Nucleus Center for the Discovery of Structures in Complex Data.”

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A Appendix: Proofs

A Appendix: Proofs

A.1 Mathematical background

We recall some properties of Jacobi polynomials (see Lemma 4.3 Cleanthous et al. 2020; Szegő 1939):

1. Orthogonality. For every \(k,\ell \in {\mathbb N}_0\),

$$\begin{aligned} \Big \langle P_{k}^{(\alpha ,\beta )},P_{\ell }^{(\alpha ,\beta )}\Big \rangle _{\alpha ,\beta }=\delta _{k,\ell }h_{\ell }^{(\alpha ,\beta )}, \end{aligned}$$
(A.1)

where we set

$$\begin{aligned} h_{\ell }^{(\alpha ,\beta )}:=\frac{2^{\alpha +\beta +1}}{2\ell +\alpha +\beta +1}\frac{\Gamma (\ell +\alpha +1)\Gamma (\ell +\beta +1)}{\ell !\Gamma (\ell +\alpha +\beta +1)}\sim (\ell +1)^{-1}, \end{aligned}$$
(A.2)

where \(\Gamma \) denotes the special function Gamma. Note that the constants are depending only on \(\alpha \) and \(\beta \).

From the above, it turns out that the sequence

$$\begin{aligned} \Big \{\tilde{P}_{\ell }^{(\alpha ,\beta )}:=\frac{P_{\ell }^{(\alpha ,\beta )}}{\sqrt{h_{\ell }^{(\alpha ,\beta )}}}:\ell \ge 0\Big \} \end{aligned}$$
(A.3)

forms an orthonormal basis for \(L^2([-1,1],\omega _{\alpha ,\beta })\).

2. Maximum. Jacobi polynomials are maximized for \(u=1\). Moreover, for every \(\ell \in {\mathbb N}_0\),

$$\begin{aligned} \Vert P_{\ell }^{(\alpha ,\beta )}\Vert _{\infty }=P_{\ell }^{(\alpha ,\beta )}(1)\sim (\ell +1)^{\alpha }, \end{aligned}$$
(A.4)

provided that \(\alpha \ge \beta \). Here, the constants depend only on \(\alpha \).

3. Derivatives. The Jacobi polynomials \(P_{\ell }^{(\alpha ,\beta )}\) have algebraic degree equal with \(\ell \). Let \(0\le n\le \ell \). We have the following formula for the derivatives of these polynomials:

$$\begin{aligned} \frac{\mathrm{d}^n}{ \mathrm{d} u^n}P^{(\alpha ,\beta )}_{\ell }(u)\sim (\ell +1)^{n}P^{(\alpha +n,\beta +n)}_{\ell -n}(u), \end{aligned}$$
(A.5)

where the similarity constants depend on n, \(\alpha \) and \(\beta \) and are independent of \(\ell \) and u.

Let us list the corresponding properties here. For a comprehensive study of Hermite polynomials, the reader is referred to Szegő (1939).

Let \(\ell \in {\mathbb N}_0\), the normalized \(\ell \)-th degree Hermite polynomial is defined as

$$\begin{aligned} H_\ell (t)=\big (\sqrt{\pi }2^{\ell } \ell !\big )^{-1/2}(-1)^{\ell } e^{x^2}\frac{\mathrm{d} ^{\ell }}{\mathrm{d}x^{\ell }}e^{-x^2}. \end{aligned}$$
(A.6)

Properties:

(i) Orthogonality. For every \(n,m\in {\mathbb N}_0\)

$$\begin{aligned} \langle H_n, H_m\rangle _{\gamma }=\delta _{n,m}. \end{aligned}$$
(A.7)

(ii) Derivatives. Note that the \(H_n\) is a polynomial of degree n. Let \(n,m\in {\mathbb N}_0\) with \(n\ge m\). Then,

$$\begin{aligned} \frac{\mathrm{d}^m}{\mathrm{d}t^m} H_n (t)=\left( \frac{2^m n!}{(n-m)!}\right) ^{1/2}H_{n-m}(t)\sim (n+1)^{m/2} H_{n-m}(t) \end{aligned}$$
(A.8)

where the constants above depend on m, but are independent of n and t.

A.2 A convergence argument for identity (3.9)

Let us now consider the convergence of \(K_{IS}(\cdot ,t)\) on \(L^2_{(\alpha ,\beta )}\). Since \(P_{k_1}^{(\alpha ,\beta )}=\tilde{P}_{k_1}^{(\alpha ,\beta )}\sqrt{h_{k_1}^{(\alpha ,\beta )}}\) and \(\tilde{P}_{k_1}^{(\alpha ,\beta )}\) is an orthonormal basis for \(L^2_{(\alpha ,\beta )}\), the convergence of the series

$$\begin{aligned} \sum _{k_1=0}^{\infty }\Big |B_{k_1}(t)\sqrt{h_{k_1}^{(\alpha ,\beta )}}\Big |^2 \end{aligned}$$

guarantees the \(L^2_{(\alpha ,\beta )}\)-convergence of (2.2), for every \(t\in {\mathbb R}\).

But from (2.3) and (A.4), there exists a constant \(c>0\) such that

$$\begin{aligned} B_{k_1}(0)\le c(k_1+1)^{-\alpha },\;\;\text {for every}\;k_1\in {\mathbb N}_0. \end{aligned}$$
(A.9)

Since now \(\alpha =(d-2)/2\ge -1/2\) for any two-point homogeneous space \(\mathcal {M}^d\), by invoking again (2.3) and using (A.2), we conclude

$$\begin{aligned} \sum _{k_1=0}^{\infty }\Big |B_{k_1}(t)\sqrt{h_{k_1}^{(\alpha ,\beta )}}\Big |^2&\le c\sum _{k_1=0}^{\infty }B_{k_1}(0)(k_1+1)^{-1-\alpha } \\&\le c\sum _{k_1=0}^{\infty }B_{k_1}(0)P_{k_1}^{(\alpha ,\beta )}(1)(k_1+1)^{-1-2\alpha } \\&\le c\sum _{k_1=0}^{\infty }B_{k_1}(0)P_{k_1}^{(\alpha ,\beta )}(1)<\infty . \end{aligned}$$

Inequality (2.4) implies that \(B_{k_1}\in L^2({\mathbb R},\gamma )\), since here the measure is bounded. By the orthonormality of the set \(\{H_{k_2}\}\), we derive that

$$\begin{aligned} B_{k_1}\sqrt{h_{k_1}^{(\alpha ,\beta )}}=\sum _{k_2=0}^{\infty } B_{\mathbf {k}} H_{k_2},\quad \text {convergence on}\;L^2({\mathbb R},\gamma ) \end{aligned}$$
(A.10)

and where we denoted \(B_{\mathbf {k}}:=\Big \langle B_{k_1}\sqrt{h_{k_1}^{(\alpha ,\beta )}},H_{k_2}\Big \rangle _{\gamma }\), for every \(k=(k_1,k_2)\in {\mathbb N}_0^2\).

We have

$$\begin{aligned} K_{IS}=\sum _{\mathbf {k}\in {\mathbb N}_0^2}B_{\mathbf {k}}R_{\mathbf {k}}^{(\alpha ,\beta )},\quad \text {convergence on}\;L^2_{\alpha ,\beta ,\gamma }. \end{aligned}$$

where \(K_{IS}\) has been defined through (2.1). Indeed, by (2.4), (2.3), (A.2), (A.9) and (A.4)

$$\begin{aligned} \sum _{\mathbf {k}\in {\mathbb N}_0^2}\big |B_{\mathbf {k}}\big |^2&=\sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }\big |B_{\mathbf {k}}\big |^2 =\sum _{k_1=0}^{\infty }\Big \Vert B_{k_1}\sqrt{h_{k_1}^{(\alpha ,\beta )}}\Big \Vert ^2_{L^2_{\gamma }} \\&\le \gamma ({\mathbb R})\sum _{k_1=0}^{\infty }B_{k_1}(t)^2 h_{k_1}^{(\alpha ,\beta )} \le c\sum _{k_1=0}^{\infty }B_{k_1}(0)P_{k_1}^{(\alpha ,\beta )}(1)<\infty . \end{aligned}$$

A.3 Proof of Theorem 1

The proof of Theorem 1 is based on the following equivalence:

Lemma 1

Let \(n_1,n_2\in {\mathbb N}_0\) and \(f\in L^2_{\alpha ,\beta ,\gamma }\). Then,

$$\begin{aligned} \big \Vert \partial ^{(n_1,n_2)}f\big \Vert _{L^2_{\alpha +n_1,\beta +n_1,\gamma }}^2\sim \sum _{k_1=n_1}^{\infty }\sum _{k_2=n_2}^{\infty } f_{\mathbf {k}}^2(k_1+1)^{2n_1}(k_2+1)^{n_2}, \end{aligned}$$
(A.11)

with similarity constants depending only on \(\alpha ,\beta ,n_1\) and \(n_2\).

Proof

Let \(n_1,n_2\in {\mathbb N}_0\) and \(f\in L^2_{\alpha ,\beta ,\gamma }\). Then, f can be represented as (3.8). Therefore,

$$\begin{aligned} \partial ^{(n_1,n_2)}f=\sum _{k_1=n_1}^{\infty }\sum _{k_2=n_2}^{\infty }\frac{f_{\mathbf {k}}}{\sqrt{h_{k_1}^{(\alpha ,\beta )}}}\big (P_{k_1}^{(\alpha ,\beta )}\big )^{(n_1)}H_{k_2}^{(n_2)}. \end{aligned}$$
(A.12)

Then, we get the inner product

$$\begin{aligned}&\langle \partial ^{(n_1,n_2)}f,\partial ^{(n_1,n_2)}f\rangle _{\alpha +n_1,\beta +n_1,\gamma } \\&=\sum _{k_1,\ell _1=n_1}^{\infty }\sum _{k_2,\ell _2=n_2}^{\infty }\frac{f_{\mathbf {k}}f_{\mathbf {\ell }}\langle H_{k_2}^{(n_2)},H_{\ell _2}^{(n_2)}\rangle _{\gamma }}{\sqrt{h_{k_1}^{(\alpha ,\beta )}h_{\ell _1}^{(\alpha ,\beta )}}}\Big \langle \big (P_{k_1}^{(\alpha ,\beta )}\big )^{(n_1)},\big (P_{\ell _1}^{(\alpha ,\beta )}\big )^{(n_1)}\Big \rangle _{\alpha +n_1,\beta +n_1}. \end{aligned}$$

Invoking (A.1) and (A.5), we infer

$$\begin{aligned}&\Big \langle \big (P_{k_1}^{(\alpha ,\beta )}\big )^{(n_1)},\big (P_{\ell _1}^{(\alpha ,\beta )}\big )^{(n_1)}\Big \rangle _{\alpha +n_1,\beta +n_1}\\&\sim (k_1+1)^{n_1}(\ell _1+1)^{n_1}\Big \langle P_{k_1-n_1}^{(\alpha +n_1,\beta +n_1)},P_{\ell _1-n_1}^{(\alpha +n_1,\beta +n_1)}\Big \rangle _{\alpha +n_1,\beta +n_1} \\&\sim \delta _{k_1,\ell _1}(k_1+1)^{2n_1}h_{k_1-n_1}^{(\alpha +n_1,\beta +n_1)}\\&\sim \delta _{k_1,\ell _1}(k_1+1)^{2n_1}(k_1+1)^{-1}, \end{aligned}$$

where the constants above depend only on \(\alpha \), \(\beta \) and \(n_1\).

Also by (A.2), it holds true

$$\begin{aligned} \sqrt{h_{k_1}^{(\alpha ,\beta )}h_{\ell _1}^{(\alpha ,\beta )}}\sim (k_1+1)^{-1/2}(\ell _1+1)^{-1/2}. \end{aligned}$$

On the other hand, (A.8) in concert with (A.7) implies

$$\begin{aligned} \langle H_{k_2}^{(n_2)},H_{\ell _2}^{(n_2)}\rangle _{\gamma }\sim \delta _{k_2,\ell _2}(k_2+1)^{n_2}. \end{aligned}$$

The combination of all the above clearly implies (A.11).

We proceed to prove the main regularity result:

Proof of Theorem 1

(i) Let \(n_1,n_2\in {\mathbb N}_0\) and \(f\in L^2_{\alpha ,\beta ,\gamma }\). By (4.3) and (A.11), we have the equivalence

$$\begin{aligned} \Vert f\Vert ^2_{DW^{(n_1,n_2)}}&\sim \sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }f_{k_1,k_2}^2 +\sum _{k_1=n_1}^{\infty }\sum _{k_2=0}^{\infty }f_{k_1,k_2}^2(k_1+1)^{2n_1}\\&\quad +\sum _{k_1=0}^{\infty }\sum _{k_2=n_2}^{\infty }f_{k_1,k_2}^2(k_2+1)^{n_2}\\&\quad +\sum _{k_1=n_1}^{\infty }\sum _{k_2=n_2}^{\infty }f_{k_1,k_2}^2(k_1+1)^{2n_1}(k_2+1)^{n_2} \\&=:a_1+\cdots a_4. \end{aligned}$$

Apparently,

$$\begin{aligned} \Vert f\Vert ^2_{DW^{(n_1,n_2)}}\le 4\sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }f_{k_1,k_2}^2(k_1+1)^{2n_1}(k_2+1)^{n_2}=:4a_0 \end{aligned}$$

and this proves that when \(\{f_{k_1,k_2}\}\in \ell ^{(n_1,n_2)}\), then \(f\in DW^{(n_1,n_2)}\).

We now assume that \(f\in DW^{(n_1,n_2)}\). We will show that \(\{f_{k_1,k_2}\}\in \ell ^{(n_1,n_2)}\) and there exists a constant \(c=c_{\alpha ,\beta ,n_1,n_2}>0\), such that \(a_0\le c\Vert f\Vert ^2_{DW^{(n_1,n_2)}}\). This will be sufficient for proving claim (i).

We decompose \(a_0\) as follows:

$$\begin{aligned} a_0&=\left( \sum _{k_1=0}^{n_1-1}\sum _{k_2=0}^{n_2-1}+\sum _{k_1=n_1}^{\infty }\sum _{k_2=0}^{n_2-1}+\sum _{k_1=0}^{n_1-1}\sum _{k_2=n_2}^{\infty }+\sum _{k_1=n_1}^{\infty }\sum _{k_2=n_2}^{\infty }\right) g_{k_1,k_2} \\&=:a_5+\cdots +a_8, \end{aligned}$$

where we denoted by \(g_{k_1,k_2}:=f_{k_1,k_2}^2(k_1+1)^{2n_1}(k_2+1)^{n_2}\).

By (A.11), we derive the followings:

$$\begin{aligned}&a_5\le (n_1+1)^{2n_1} (n_2+1)^{n_2}\sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }f_{k_1,k_2}^2\le c\Vert f\Vert ^2_{L^2_{(\alpha ,\beta ,\gamma )}},\\&a_6\le (n_2+1)^{n_2}\sum _{k_1=n_1}^{\infty }\sum _{k_2=0}^{\infty }f_{k_1,k_2}^2(k_1+1)^{2n_1}\le c\big \Vert \partial ^{n_1}_1 f\big \Vert ^2_{L^2_{\alpha +n_1,\beta +n_1,\gamma }},\\&a_7\le c\Vert \partial ^{n_2}_2 f\Vert ^2_{L^2_{\alpha ,\beta ,\gamma }} \end{aligned}$$

and

$$\begin{aligned} a_8\le c\Vert \partial ^{(n_1,n_2)}f\Vert ^2_{L^2_{\alpha +n_1,\beta +n_1,\gamma }}. \end{aligned}$$

All the above complete the proof of (i).

(ii) We start by observing that \(W^{(n_1,n_2)}=DW^{(n_1,0)}\cap DW^{(0,n_2)}\). Then, by claim (i) it holds that \(f\in W^{(n_1,n_2)}\) if and only if \((f_{\mathbf {k}})\in \ell ^{(n_1,0)}\cap \ell ^{(0,n_2)}\). On the other hand by (A.11) and (4.2), we get that

$$\begin{aligned} \Vert f\Vert ^2_{W^{(n_1,n_2)}}\le c\sum _{\mathbf {k}\in {\mathbb N}_0^2}f_{\mathbf {k}}^2\big ((k_1+1)^{2n_1}+(k_2+1)^{n_2}\big )=:a_9. \end{aligned}$$

Conversely,

$$\begin{aligned} a_9&=\sum _{k_1<n_1}\sum _{k_2\ge 0}f_{\mathbf {k}}^2(k_1+1)^{2n_1}+\sum _{k_1\ge n_1}\sum _{k_2\ge 0}f_{\mathbf {k}}^2(k_1+1)^{2n_1} \\&\quad +\sum _{k_2<n_2}\sum _{k_1\ge 0}f_{\mathbf {k}}^2(k_2+1)^{n_2}+\sum _{k_2\ge n_2}\sum _{k_1\ge 0}f_{\mathbf {k}}^2(k_2+1)^{n_2} \\&\le c\big ((n_1+1)^{2n_1}+(n_2+1)^{n_2}\big )\Vert f\Vert ^2_{L^2_{\alpha ,\beta ,\gamma }}+c\Vert \partial ^{(n_1,0)}f\Vert ^2_{L^2_{\alpha +n_1,\beta +n_1,\gamma }} \\&\quad +c\Vert \partial ^{(0,n_2)}f\Vert ^2_{L^2_{\alpha ,\beta ,\gamma }} \\&\le c\Vert f\Vert ^2_{W^{(n_1,n_2)}} \end{aligned}$$

and the proof of claim (ii) is complete.

(iii) Let \(N\in {\mathbb N}_0\) and \(f\in L^2_{\alpha ,\beta ,\gamma }\). Note that the following connection between Sobolev-type spaces is true:

$$\begin{aligned} W^{N}=\bigcap _{\nu =0}^{N}DW^{(\nu ,N-\nu )}. \end{aligned}$$

Then, f belongs to \(W^N\) if and only if it belongs to the intersection of \( W^{(\nu ,N-\nu )}\), for every \(\nu =0,\dots ,N\), which equivalently means that its Fourier coefficients belong to \(\bigcap _{\nu =0}^{N}\ell ^{(\nu ,N-\nu )}\), in the light of claim (i) and the proof is complete.

We close by proving Corollary 1.

Proof

Let \(n_1,n_2\in {\mathbb N}_0\). By (3.9), we have that \(K_{IS}\in L^2_{\alpha ,\beta ,\gamma }\). Using Lemma 1, we derive that

$$\begin{aligned}&(1-u)^{n_1/2}(1+u)^{n_1/2}\partial ^{(n_1,n_2)}K_{IS}(u,t)\in L^2_{\alpha ,\beta ,\gamma } \\&\Leftrightarrow \partial ^{(n_1,n_2)}K_{IS}\in L^2_{\alpha +n_1,\beta +n_1,\gamma } \\&\Leftrightarrow \sum _{k_1=n_1}^{\infty }\sum _{k_2=n_2}^{\infty }B_{\mathbf {k}}^2(k_1+1)^{2n_1}(k_2+1)^{n_2}<\infty . \end{aligned}$$

A.4 Proof of Theorem 2

Proof

We have

$$\begin{aligned} Z(x,t)-Z^R(x,t)=\sum _{n>R}V_n(t) P_n^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big ), \end{aligned}$$

for every \((x,t)\in \mathcal {M}^d\times {\mathbb R}\).

We replace the last in the norm (3.10) and by (3.11), we extract

$$\begin{aligned}&\big \Vert Z-Z^R\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}^2=\big \langle Z-Z^R,Z-Z^R\big \rangle _{L^2(\mathcal {M}^d\times {\mathbb R})} \\&=\sum _{n,m>R}\big \langle V_n(t) P_n^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big ),V_m(t) P_m^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big )\big \rangle _{L^2(\mathcal {M}^d\times {\mathbb R})} \\&=\sum _{n,m>R}\int _{\mathcal {M}^d}P_n^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big ) P_m^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big )\mathrm{d}x \\&\quad \times \int _{{\mathbb R}}V_n (t)V_m (t)\gamma (t)\mathrm{d}t. \end{aligned}$$

By Ma and Malyarenko (2020), it holds true that

$$\begin{aligned} \int _{\mathcal {M}^d}P_n^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big ) P_m^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big )\mathrm{d}x=\frac{\delta _{nm}\omega _{d}}{a_n^2}P_n^{(\alpha ,\beta )}(1). \end{aligned}$$

Therefore,

$$\begin{aligned} \big \Vert Z-Z^R\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}^2=\omega _d \sum _{n>R}\frac{\Vert V_n\Vert _{L^2_{\gamma }}^2}{a_n^2}P_n^{(\alpha ,\beta )}(1). \end{aligned}$$

Note now that \(P_n^{(\alpha ,\beta )}(1)>0\). By Fubini–Tonelli theorem, the mixed-norm (3.13) takes the form

$$\begin{aligned} \big \Vert Z-Z^{R}\big \Vert _{L^2\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}&={\mathbb E}\left( \Vert Z-Z^{R}\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}^2\right) \\&={\mathbb E}\left( \omega _d \sum _{n>R}\frac{\Vert V_n\Vert _{L^2_{\gamma }}^2}{a_n^2}P_n^{(\alpha ,\beta )}(1)\right) \\&=\omega _d \sum _{n>R}\frac{{\mathbb E}(\Vert V_n\Vert _{L^2_{\gamma }}^2)}{a_n^2}P_n^{(\alpha ,\beta )}(1) \\&=\omega _d \sum _{n>R}\frac{\Vert {\mathbb E}(V_n(t)V_n(t))\Vert _{L^2_{\gamma }}^2}{a_n^2}P_n^{(\alpha ,\beta )}(1) \end{aligned}$$

Since now \({\mathbb E}(V_n(t)V_n(t))=a_n^2 B_n(0)\) by (5.2), we extract

$$\begin{aligned} \frac{\Vert {\mathbb E}(V_n(t)V_n(t))\Vert _{L^2_{\gamma }}^2}{a_n^2}=B_n(0)\gamma ({\mathbb R})\le c(n+1)^{-\varepsilon }, \end{aligned}$$

where we used the condition (5.5) and the finiteness of the measure \(\gamma \). Moreover, \(P_n^{(\alpha ,\beta )}(1)\le c(n+1)^{\alpha }\) and combing all the above, we conclude

$$\begin{aligned} \big \Vert Z-Z^{R}\big \Vert _{L^2(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}&\le c\sum _{n>R}n^{-(\varepsilon -\alpha )} \\&\le c\int _{R}^{\infty }\frac{\mathrm{d}x}{x^{\varepsilon -\alpha }}=c R^{-(\varepsilon -\alpha -1)}, \end{aligned}$$

since \(\varepsilon >\alpha +1\).

To show (5.7), we begin with the case \(p<2\). Applying the Hölder inequality on a probability space for the index \(2/p>1\) directly gives

$$\begin{aligned} \big \Vert Z-Z^{R}\big \Vert _{L^p(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}&\le \big \Vert Z-Z^{R}\big \Vert _{L^2(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))} \le cR^{-(\varepsilon -\alpha -1)}. \end{aligned}$$

We consider now \(p>2\). We can choose \(\nu =\nu _p\in {\mathbb N}\) such that \(2(\nu -1)<p\le 2\nu \). Similarly to the case \(p<2\), the Hölder inequality for the index \(2\nu /p\) yields

$$\begin{aligned} \big \Vert Z-Z^{R}\big \Vert _{L^p(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}&\le \big \Vert Z-Z^{R}\big \Vert _{L^{2\nu }(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R})}. \end{aligned}$$
(A.13)

By Corollary 2.17 of Da Prato and Zabczyk (1992), there exists a constant \(c=c_\nu >0\) such that

$$\begin{aligned} \Vert Z-Z^{R}\big \Vert _{L^{2\nu }(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}\le c_{\nu }\Vert Z-Z^{R}\big \Vert _{L^{2}(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}. \end{aligned}$$
(A.14)

The combination of (A.13), (A.14) and (5.6) leads to  (5.7).

For the last claim, we have to prove that

$$\begin{aligned} \mathbb {P}\big (\big \Vert Z-Z^{R}\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}\ge R^{-\gamma },\;\text {for infinitely many}\;R\in {\mathbb N}\big )=0. \end{aligned}$$
(A.15)

This will be a consequence of the Borel–Cantelli lemma Klenke (2013), Theorem  2.7. It suffices to prove that

$$\begin{aligned} \sum _{R=1}^{\infty }\mathbb {P}\left( \big \Vert Z-Z^{R}\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}\ge R^{-\gamma }\right) <\infty . \end{aligned}$$
(A.16)

Let \(p>0\). By Chebyshev’s inequality, (5.7) and using the mixed-norm (3.12), we derive

$$\begin{aligned} \mathbb {P}\left( \big \Vert Z-Z^{R}\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}\ge R^{-\gamma }\right)&\le R^{\gamma p} {\mathbb E}\left( \big \Vert Z-Z^{R}\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}^p\right) \nonumber \\&=R^{\gamma p} \big \Vert Z-Z^{R}\big \Vert _{L^{p}(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}^p \nonumber \\&\le cR^{-(\varepsilon -\alpha -1-\gamma )p}. \end{aligned}$$
(A.17)

We choose now \(p>1/(\varepsilon -\alpha -1-\gamma )\), and this is allowed since \(0<\gamma <\varepsilon -\alpha -1\). For this choice, we obtain

$$\begin{aligned} \sum _{R=1}^{\infty }R^{-(\varepsilon -\alpha -1-\gamma )p}<\infty , \end{aligned}$$

which together with (A.17) leads to (A.16) and completes the proof.

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Cleanthous, G., Porcu, E. & White, P. Regularity and approximation of Gaussian random fields evolving temporally over compact two-point homogeneous spaces. TEST (2021). https://doi.org/10.1007/s11749-021-00755-1

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Keywords

  • Compact two-point homogeneous spaces
  • Covariance functions
  • Mixed smoothness
  • Ozone concentration
  • Random fields
  • Truncated approximation

Mathematics Subject Classification

  • 60G60
  • 62G20
  • 41A25
  • 46E35