Tauberian class estimates for wavelet and non-wavelet ...cage.ugent.be/~jvindas/Talks_files/VindasTMSF2011.pdfTo present severalprecise characterizationsof the spaces of distributions

Statement of the problem and motivationThe non-wavelet case

The wavelet case

Tauberian class estimates for wavelet andnon-wavelet transforms of vector-valued

distributions

Jasson [email protected]

Department of MathematicsGhent University

Transform Methods and Special Functions – TMSF’ 20116th International Conference, Sofia, Bulgaria, October 21, 2011

J. Vindas Tauberian class estimates


The wavelet case

In this talk we study vector-valued distributions in terms ofintegral transforms

Mϕf(x , y) = (f ∗ ϕy )(x), (x , y) ∈ Rn × R+, (1)

where ϕy (t) = y−nϕ(t/y). We call such transforms regularizingtransforms.

Two important cases can be distinguished:1 The wavelet case:

∫Rn ϕ(t)dt = 0.

2 The non-wavelet case:∫

Rn ϕ(t)dt 6= 0.

Our aim is:To present several precise characterizations of the spacesof distributions with values in Banach spaces in terms ofnorm size estimates for (1).



The wavelet case


Mϕf(x , y) = (f ∗ ϕy )(x), (x , y) ∈ Rn × R+, (1)



∫Rn ϕ(t)dt = 0.


Rn ϕ(t)dt 6= 0.




The wavelet case


Mϕf(x , y) = (f ∗ ϕy )(x), (x , y) ∈ Rn × R+, (1)



∫Rn ϕ(t)dt = 0.


Rn ϕ(t)dt 6= 0.




The wavelet case

Tauberian class estimatesNon-degenerate test functions

General Notation

E always denotes a fixed Banach space with norm ‖ · ‖E .X stands for a (Hausdorff) locally convex topological vectorspace.S ′(Rn,X ) = Lb(S(Rn),X ), the space of X -valued tempereddistributions.Hn+1 = Rn × R+, the upper half-space.ϕ denotes the Fourier transform.



The wavelet case


Statement of the problem

Suppose that f takes a priori values in the “broad” space X , i.e.,f ∈ S ′(Rn,X ).

Suppose that the “narrower” spaceE is continuously embedded in X .

If we know that f takes values in E , f ∈ S ′(Rn,E), then (forsome k , l ,C):

‖Mϕf(x , y)‖E ≤ C(1 + y)k (1 + |x |)l

yk , (x , y) ∈ Hn+1. (2)

We call (2) a (Tauberian) class estimate.

Converse problem: Up to what extend does the class estimate(2) allow one to conclude that f actually takes values in E?The problem has a Tauberian character.



The wavelet case






‖Mϕf(x , y)‖E ≤ C(1 + y)k (1 + |x |)l

yk , (x , y) ∈ Hn+1. (2)





The wavelet case






‖Mϕf(x , y)‖E ≤ C(1 + y)k (1 + |x |)l

yk , (x , y) ∈ Hn+1. (2)





The wavelet case






‖Mϕf(x , y)‖E ≤ C(1 + y)k (1 + |x |)l

yk , (x , y) ∈ Hn+1. (2)





The wavelet case


Motivation

The stated problem was first raised and studied by Drozhzhinovand Zav’yalov. It gives a general setting to attack problemssuch as:

1 Classical Hardy-Littlewood-Karamata type Tauberiantheorems for various integral transforms (e.g., the Laplacetransform).

2 Stabilization in time for certain Cauchy problems (e.g., forthe heat equation).

3 Norm estimates for solutions to certain PDE (e.g., theSchrödinger equation)

4 Wavelet characterizations of important Banach spaces offunctions and distributions (e.g., Besov type spaces).

5 Pointwise and (micro-)local analysis.



The wavelet case


Motivation









The wavelet case


Motivation









The wavelet case


Local class estimates

We said that Mϕf satisfies a local class estimate if:1 Mϕf(x , y) takes values in E for almost all

(x , y) ∈ Rn × (0,1] and is measurable as an E-valuedfunction on Rn × (0,1], and,

2 (the local class estimate):

‖Mϕf(x , y)‖E ≤ C(1 + |x |)l

yk , for almost all (x , y) ∈ Rn×(0,1].

for some k , l ∈ N and C > 0.

Furthermore, we assume from now on that:The Banach space E is continuously embedded in thelocally convex space X .



The wavelet case


Non-degenerate test functions

Naturally, not all kernels ϕ will be well-suited to our problem.The good ones are:

DefinitionLet ϕ ∈ S(Rn). It is said to be degenerate if there is a raythrough the origin along which ϕ identically vanishes. Incontrary case, the test function it is said to be non-degenerate.

Our Tauberian kernels are the non-degenerate test functions.In Wiener Tauberian theory the Tauberian kernels arethose ϕ such that ϕ do not vanish at any point.In our theory the Tauberian kernels will be those ϕ suchthat ϕ do not identically vanish on any ray through theorigin.



The wavelet case








The wavelet case








The wavelet case

The non-wavelet case

For the non-wavelet case, we always obtain a fullcharacterization of S ′(Rn,E).

TheoremLet f ∈ S ′(Rn,X ) and let ϕ ∈ S(Rn) be such that

∫Rn ϕ(t)dt 6= 0.

Then,

f ∈ S ′(Rn,E) if and only if Mϕf satisfies a local class estimate



The wavelet case

The non-wavelet case

For the non-wavelet case, we always obtain a fullcharacterization of S ′(Rn,E).

TheoremLet f ∈ S ′(Rn,X ) and let ϕ ∈ S(Rn) be such that

∫Rn ϕ(t)dt 6= 0.

Then,

f ∈ S ′(Rn,E) if and only if Mϕf satisfies a local class estimate



The wavelet case

The wavelet case

The analysis of the wavelet case is more complicated.

We only obtain characterizations of S′(Rn,E) up to acorrection term that is totally controlled by the wavelet.From now on, we assume that ϕ ∈ S(Rn) is anon-degenerate wavelet, namely,∫

Rn ϕ(t)dt = 0 and ϕ is non-degenerate.

Definition

Let ϕ ∈ S(Rn) be non-degenerate. Given ω ∈ Sn−1, weconsider ϕω(r) := ϕ(rω) as a function of one variable r . Wedefine its index of non-degenerateness as

τ = inf{

r ∈ R+ : supp ϕω ∩ [0, r ] 6= ∅,∀ω ∈ Sn−1}.



The wavelet case

The wavelet case

The analysis of the wavelet case is more complicated.

We only obtain characterizations of S′(Rn,E) up to acorrection term that is totally controlled by the wavelet.From now on, we assume that ϕ ∈ S(Rn) is anon-degenerate wavelet, namely,∫

Rn ϕ(t)dt = 0 and ϕ is non-degenerate.

Definition

Let ϕ ∈ S(Rn) be non-degenerate. Given ω ∈ Sn−1, weconsider ϕω(r) := ϕ(rω) as a function of one variable r . Wedefine its index of non-degenerateness as

τ = inf{

r ∈ R+ : supp ϕω ∩ [0, r ] 6= ∅,∀ω ∈ Sn−1}.



The wavelet case

Wavelet caseLocal class estimates

TheoremLet f ∈ S ′(Rn,X ) and let ϕ ∈ S(Rn) be a non-degeneratewavelet with index τ .

Assume that Mϕf satisfies a local class estimate.

Then: for every r > τ , there is an X-valued entire function Gsuch that

f−G ∈ S ′(Rn,E),

where supp G ⊂ {t ∈ Rn : |t | < r}.



The wavelet case

Strongly non-degenerate wavelets

It is still possible to strengthen the previous result, but oneshould use the following kind of wavelets:

Definition

Let ϕ ∈ S(Rn) be a wavelet. We call ϕ strongly non-degenerateif there exist constants N ∈ N, r > 0, and C > 0 such that

C |u|N ≤ |ϕ(u)| , for all |u| ≤ r .



The wavelet case

Strongly non-degenerate wavelets

It is still possible to strengthen the previous result, but oneshould use the following kind of wavelets:

Definition

Let ϕ ∈ S(Rn) be a wavelet. We call ϕ strongly non-degenerateif there exist constants N ∈ N, r > 0, and C > 0 such that

C |u|N ≤ |ϕ(u)| , for all |u| ≤ r .



The wavelet case

Local class estimatesStrongly non-degenerate wavelets

TheoremLet f ∈ S ′(Rn,X ) and let ϕ ∈ S(Rn) be a stronglynon-degenerate wavelet. Then, the following two conditions areequivalent:

Mϕf satisfies a local class estimate.There is an X-valued entire function G such that

f−G ∈ S ′(Rn,E) and supp G ⊆ {0} .

CorollaryIf X is a normed space, the function G = P is indeed apolynomial with coefficients in X.



The wavelet case

Local class estimatesStrongly non-degenerate wavelets

TheoremLet f ∈ S ′(Rn,X ) and let ϕ ∈ S(Rn) be a stronglynon-degenerate wavelet. Then, the following two conditions areequivalent:

Mϕf satisfies a local class estimate.There is an X-valued entire function G such that

f−G ∈ S ′(Rn,E) and supp G ⊆ {0} .

CorollaryIf X is a normed space, the function G = P is indeed apolynomial with coefficients in X.



The wavelet case

Comments on the (Tauberian) theorems

The discussed theorems improve several earlier results ofDrozhzhinov and Zav’ylov.Main improvements:

Enlargement of the Tauberian kernels. Actually, our classof non-degenerate kernels is the optimal one.Our results are valid for general locally convex spaces X(Drozhzhinov and Zav’ylov considered normed spaces).



The wavelet case

References

For further results see our preprint (joint work with S. Pilipovic):Multidimensional Tauberian theorems for wavelets andnon-wavelet transforms, preprint (arXiv:1012.5090v2 ).

See also:Y. N. Drozhzhinov, B. I. Zav’yalov, Tauberian theorems forgeneralized functions with values in Banach spaces, Izv.Math. 66 (2002), 701–769.Y. N. Drozhzhinov, B. I. Zav’yalov, MultidimensionalTauberian theorems for Banach-space valued generalizedfunctions, Sb. Math. 194 (2003), 1599–1646.Y. N. Drozhzhinov, B. I. Zav’yalov, Applications ofTauberian theorems in some problems in mathematicalphysics, Teoret. Mat. Fiz. 157 (2008), 373–390.J. Vindas, S. Pilipovic, D. Rakic, Tauberian theorems for thewavelet transform, J. Fourier Anal. Appl. 17 (2011), 65–95.


Tauberian class estimates for wavelet and non-wavelet ...cage.ugent.be/~jvindas/Talks_files/VindasTMSF2011.pdfTo present severalprecise characterizationsof the spaces of distributions

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