Continuous wavelet transform of Schwartz tempered distributions

Abstract: The continuous wavelet transform of Schwartz tempered distributions is investigated and derive the corresponding wavelet inversion formula (valid modulo a constant-tempered distribution) interpreting convergence in S0ðRÞ. But uniqueness theorem for the present wavelet inversion formula is valid for the space SFðRÞ obtained by filtering (deleting) (i) all non-zero constant distributions from the space S0ðRÞ, (ii) all non-zero constants that appear with a distribution as a union. As an example, in considering the distribution x 2 1þx2 1⁄4 1 1 1þx2 we would omit 1 and retain only 1 1þx2 . The wavelet kernel under consideration for determining the wavelet transform are those wavelets whose all the moments are non-zero. As an example, ð1þ kx 2x2Þe x2 is such a wavelet. k is an arbitrary constant. There exist many other classes of such wavelets. In our analysis, we do not use a wavelet kernel having any of its moments zero.


Background results
The Schwartz testing function space SðRÞ of rapid descent consists of infinitely differentiable functions ϕ defined on R such that ABOUT THE AUTHOR The Continuous wavelet transform on Schwartz tempered distributions was firstly introduced by Holschneider in 1995. Pathak(2004) studied the various properties of contiuous wavelet transform on Schwartz tempered distributions. Pandey and Upadhyay(2015) introduced the continuous wavelet transform using the concept of window functions. Motivated from the above results, the authors extend the continuous wavelet transform to Schwartz tempered distributions and investigate the corresponding wavelet inversion formula (valid modulo a constant tempered distribution) interpreting convergence in the weak distributional sense. This theory is true, when the wavelet kernel under consideration for determining the wavelet transform are those wavelets, whose all the moments are non-zero. The example of such types of wavelet kernel is also given in this work.

PUBLIC INTEREST STATEMENT
In the present work, the authors studied the continuous wavelet transform to Schwartz tempered distributions and found its inversion formula. This theory is useful for many researchers who are doing research work in image processing and signal processing. Researchers will be advantageous, who are using different types of integral transforms. The aforesaid theory is applicable, where many differential equations can be solved by exploiting the theory of Fourier transform. This work can be played an important role to study different types of integral equations. So the approach of this research paper is multidisciplinary in nature, which are applicable in mathematics, physics, and engineerings. sup tεR t m ϕ ðnÞ ðtÞ < 1 for each m; n ¼ 0; 1; 2; ::: .
The space SðRÞ is obviously metricize by the metric β defined by The fact that β is a metric is proved by using the fact that the function f ðxÞ ¼ x 1þx ; x ! 0, is an increasing function of x. It is well known that the topology generated by the metric β on SðRÞ is the same as that generated by the sequence of seminorms (1.2). Since the locally convex topological vector space SðRÞ is complete and metrizable, it is a Frechet space.
It is proved in (Chui, 1992) that this window function f also belongs to L 1 ðRÞ. A more general result in n-dimensions is proved in (Pandey & Upadhyay, 2015).
We denote the constant defined in (1.4) by C f 2π . Here,f ðλÞ is the Fourier transform of f which is given aŝ f ðλÞ ¼ l:i:m: (1:5) f ðxÞe Àixλ dx in L 2 ð RÞ norm; from ½p:751: (1:6) From Definition 2 it follows that the function ψ ¼ xe Àx 2 is a basic wavelet belonging to SðRÞ. This is e Àλ 2 is the Fourier transform of ψ 2 SðRÞ: Therefore, bounded. The theorem stated below helps us in constructing wavelets in various testing function spaces very simply.
Theorem 1.1. A window function f 2 L 2 ðRÞ is a basic wavelet if and only if general theorem in n-dimensions, n ! 1 is proved in (Pandey & Upadhyay, 2015). Since ϕ 2 D & L 2 ðRÞ is a window function, an element ϕ 2 D is a basic wavelet if and only if

Introduction
Let SðRÞ be the Schwartz testing function space of rapid descent and let sðRÞ be a subspace of SðRÞ so that every element ϕ 2 sðRÞ satisfies ð 1 À1 ϕðxÞdx ¼ 0, i.e., every element of sðRÞ is a basic wavelet. The subspace sðRÞ of SðRÞ is equipped with the topology induced by SðRÞ on sðRÞ. One can verify that the restriction of f 2 S 0 ðRÞ to sðRÞ is in s 0 ðRÞ and, therefore, in the following discussion the wavelet inversion formula that is valid for f 2 S 0 ðRÞ restricted to SðRÞ modulo a constant distribution, is also valid for elements of S 0 ðRÞ restricted to sðRÞ. More clearly, we have We extend the continuous wavelet transform to the Schwartz tempered distribution space S 0 ðRÞ, exploiting the structure formula (2:1) Here f 2 S 0 ðRÞ and ϕ 2 SðRÞ, and g is a function belonging to L 2 R ð Þ depending upon f and not on ϕ. The structure formula ð2:1Þ follows from the boundedness property of S 0 ðRÞ, i.e., for f 2 S 0 ðRÞ there exists a nonnegative integer m and a constant C>0 such that < f ; ϕ > j j Cρ m ðϕÞ; "ϕ 2 SðRÞ: (2:2) This is derived by virtue of the fact that ρ 0 ρ 1 ρ 2 . . . and the method of contradiction; using (2.2) we get for a non-negative integer m satisfying ðxÞ þ 1 þ x 2 À Á m m2xϕ ðmÞ ðxÞ 2 using Holder 0 sinequality: Now, using the Hahn Banach theorem (Yosida, 1995, p. 102, 105, 106), f can be extended to L 2 ðRÞ. Since S is dense in L 2 ðRÞ, this extension is unique.
By Riesz theorem, the dual of L 2 R ð Þ is homeomorphic to L 2 R ð Þ we get a function g 2 L 2 R ð Þ such that from (Akhiezer & Glazman, 1961, p. 33). This justifies the structure formula (2.1).
(2.3) Fact 1: If ψ is a wavelet belonging to SðRÞ, the continuous wavelet transform W f a; b ð Þ of f 2 S 0 ðRÞ in view of the relation ð2:1Þ can be proved to be when a ¼ 0: Using the classical wavelet inversion formula for L 2 ðRÞ functions as proved in (Boggess & Narcowich, 2001;Chui, 1992;Daubechies, 1990;Lebedeva & Postinikov, 2014), we will prove in the next section that Since two tempered distributions having the same continuous wavelet transform may differ by a constant distribution, our inversion formula will be valid modulo a constant distribution.The structure formula for f 2 S 0 ðRÞ reduces a functional analytic problem to a classical problem of analysis, i.e. a L 2 ðRÞ function theory.
Pathak (Pathak, 2004) extended the wavelet transform to Schwartz tempered distributions in the year 2004 using the method of adjoints, i.e.
From (2.3) it follows that the wavelet transform of a constant distribution is zero as the wavelet ψ belonging to the space SðR n Þ satisfies the condition Thus, two wavelets having the same wavelet transform may differ by a constant. Pathak (2004) was motivated to give the definition (2.3) for the wavelet transform of tempered distribution by the Parseval's type of relation for the wavelet transform He strengthened his result (definition) 2.2 (Pathak, 2004) further by proving some continuity results and boundedness property; but he did not derive the corresponding wavelet inversion formula. Since the wavelet transform of a constant distribution is zero the uniqueness theorem for the wavelet inversion formula will not be true; it will be valid modulo a constant distribution. In order that the uniqueness theorem may be valid we have to delete all non-zero constants distribution from the space S 0 ðR n Þ. In addition, we have to delete a non-zero constant distribution from a tempered distribution which is contained in it as a sum or difference. For example, in considering the distribution We delete the constant 1 and retain the tempered distribution À1 1þjxj 2 only.
The space S 0 ðR n Þ filtered this way is represented by the symbol S 0 F ðR n Þ, then the uniqueness theorem for the wavelet inversion formula will be valid for this space S 0 F ðR n Þ.
During the last five years several good results on the continuous wavelet transform of functions appeared. Notable amongst them is the work of Postnikov et al. (2016), Lebedeva & Postinikov (2014), who proved the wavelet inversion formula for functions in the year 2016 without a requirement of the admissibility condition.
Weisz (2013) proved the norm and a.e. convergence of inversion formula in L p and Wiener amalgam spaces. In 2014 he proved the inverse wavelet transform to summability means of Fourier transforms and obtained norm and almost everywhere convergence of the inversion formula for functions from the L p and Wiener amalgam spaces (Weisz, 2014). In 2015, Weisz (2015) also proved, using the summability methods of Fourier transform, norm convergence and convergence at Lebesgue points of the inverse wavelet transform for functions from the L p and Wiener amalgam spaces.
Our objective is to extend the continuous wavelet transform to Schwartz space S 0 ðRÞ and prove an inversion formula modulo a constant distribution and then extend the uniqueness theorem for the continuous wavelet transform of distributions to the space S 0 F ðRÞ; the space S F 0 ðRÞ is a subspace of the space S 0 ðRÞ.
Our spaces S 0 ðRÞ and S F 0 ðRÞ are big spaces and they contain the spaces L 1 ðRÞ, L p ðRÞ as considered by these authors.
The wavelets that we use as a kernel of the wavelet transform will not be any element of sðRÞ will be those elements of sðRÞ whose moments of any order will be non-zero. An example of one such wavelet is ð1 þ x À 2x 2 Þe Àx 2 . Many more such wavelets can be constructed by assigning arbitrary values to the constant k in the expression ð1 þ kx À 2x 2 Þe Àx 2 . Another set of such wavelet kernels can be constructed by assigning appropriate values to the constants k and b in the expression ð1 þ kx À bx 2 Þðe Àx 2 À e Àx 4 Þ. We first select b such that ð 1 À1 ð1 þ kx À bx 2 Þðe Àx 2 À e Àx 4 Þdx ¼ 0. The number b will be independent of k and therefore k can be assigned arbitarary real values, thereby proving the existence of wavelet kernels in sðRÞ whose any moment will be non-zero. Many more such wavelets (unaccountably many of them) can be constructed.
Our reason to avoid wavelets whose every moment is zero is that wavelet transform of every polynomial function will be zero, and our inversion formula will break. This situation is already dealt with by Holschneider (1995). He quotients out the space of tempered distributions by the space of all polylnomials.

Comparison of our results with that of Pathak
(1) Pathak (2004) followed the method of adjoints, whereas we have followed the method of embedding to define the wavelet transform of tempered distributions; but he did not prove the inversion formula.
(2) Pathak took a>0 whereas we took a 2 R, aÞ0, a more general result in this sense.
(3) We have proven the inversion formula for the wavelet transform of distributions giving the situation where our inversion formula has unique results and where it does not.
(4) Calculation of the wavelet transform is far easier by our method whereas calculation of the wavelet transform by Pathak's method is not quite as easy.
(5) We have proven the uniqueness theorem for the inversion formula for the wavelet transform for the space S 0 F ðRÞ, n ¼ 1 and the result can be extended for n>1.
Our objective is to prove the wavelet inversion formula for tempered distributions in the weak distributional sense and this will be accomplished in Chapter 3.
3. An integral wavelet transforms of schwartz tempered distributions in R and its inversion

Integral wavelet transform
In this section we will use three symbols F b , C ψ and F; the symbols F b and F stand for the Fourier transform of functions of b and t respectively, and the symbol C ψ stands for the admissibility constant which is defined in (1.4).
We require that C ψ be finite as in the derivation of the wavelet inversion formula. The expression C ψ appears in the denominator of the related expression, that is useful in our derivation of the inversion formula using the Fourier transform technique. But there exists an alternative reconstruction formula for the continuous wavelet transform, which is applicable even if the admissibility condition is violated see (Holschneider, 1995;Postnikov et al., 2016). jλj dλ k f k 2 2 þ2 1 k xf k 2 2 <1 by taking n ¼ 1 in (Pandey & Upadhyay, 2015, Theorem 3.1.).
Theorem 3.2. Let f 2 L 2 ðRÞ and ψ 2 sðRÞ & SðRÞ where sðRÞ and SðRÞ are spaces of functions as defined in Theorem 3.1. Then Proof. In (Pandey & Upadhyay, 2015, Theorem 3.1) the proof of above theorem is given.
Proof. It is a special case of (Pandey & Upadhyay, 2015, Theorem 3.1) see also (Daubechies, 1990). The functionψðaλÞ 2 SðRÞ. Hence the expression in the above integral which is in curly bracket is bounded. Therefore, the coefficient of e iλb in the integrand in the above integral belongs to L 2 ðRÞ as a function of λ; which implies that the above integral as a function of b belongs to L 2 b ðRÞ. Similarly, we can show that as a function of b and is infinitely differentiable with respect to b and each of its derivatives also belongs to L 2 b ðRÞ.
Therefore, W f ða; bÞ 2 L 2 ðRÞ as a function of b, aÞ0. The form of the structure formula we have chosen is valid for m ! 0. In fact, the structure formula for f when m ¼ 0 can also be derived from (2.1) by setting m ¼ 0.
(iii) W f ða; bÞ, @W @a and @W @b are uniformly bounded in a compact neighbourhood of the point ða; bÞ, aÞ0.
(iv) W f ða; bÞ is a continuous function of ða; bÞ everywhere on R 2 except possibly at a ¼ 0 (the b-axis).
(3.2) Proof. Using the structure formula (2.1) for f we have ; g 2 L 2 ðRÞ: (3:2) Therefore, using a standard result in analysis we have (3:3) A similar result for differentiation with respect to b can be proved. Therefore, we obtain ðiÞ @ k @a k W f ða; bÞ ¼ f ðtÞ; ) .
Using the boundedness property of f we get, where P is a polynomial of degree 2m. The non-negative integer m is the least possible value conforming to the boundedness property of f . These polynomials will be uniformly bounded in a compact neighborhood of ða; bÞ. Since aÞ0, 1 jaj mþ 1 2 will be also finite. Therefore, W f ða; bÞ is bounded in a compact neighborhood of ða; bÞ, aÞ0. Similar bounds can be established for the first partial derivatives of W f ða; bÞ with respect to a and b.
(iv) To prove (iv), we assume (i) and (ii) for k ¼ 1, and (iii). Now This is valid if W f is real-valued. If it is complex-valued, then we apply the mean value theorem of differential calculus separately for the real and imaginary part of W f .
The first partial derivatives of W f ða; bÞ are bounded in a compact neighborhood of ða; bÞ where aÞ0 and ða þ Δa; b þ ΔbÞ lies in the neighborhood of ða; bÞ during the limiting process. Therefore, from (3.6) W f ða þ Δa; b þ ΔbÞ À W f ða; bÞ ! 0 as Δa; Δb ! 0: Note that differentiality results proved by us apply to the wavelet transform of tempered distributions, whereas Pathak's differentiality results apply to the wavelet transform of functions. Our results are a lot more general (Pathak, 2004).  When we use a structure formula for f , the distributional problem is converted into the classical one and so all lower limits and upper limits of the integral will be À 1 and 1, respectively.
(3.9) Proof. From (2.1), (3.7) can be written as Our aim to find the inversion formula interpreting convergence in the weak topology of S 0 ðRÞ, i.e., as in (3.8) In the following integrations we delete the region jaj ε and make changes in the order of integration and after that let ! 0. Therefore, in view of (3.6) and Theorems 3.4 and 3.5, the integral in (3.11) is meaningful (it exists) and now when operated against ϕ 2 SðRÞ, (3.11) becomes: [The above angular brackets represent integration with respect to x in R].
Therefore, we have ( 3:13) using integration by parts, note that the integral of the terms enclosed in the curly brackets with respect to t as a function of b is infinitely differentiable and belongs to L 2 b ðRÞ for each m ¼ 0; 1; 2; 3; . . . . So to evaluate the integral term we use integration by parts with the limit terms being zero. (3:14) In Equations (3.11)-(3.13) we could have taken finite limits of integration À M, N and then after integration by parts let M; N ! 1; the same results as shown above would have been obtained. Thus, there is no error involved in setting the lower and upper limits of the foregoing integrals as À 1 and 1. We also make use of the fact that Finally by distributional differentiation, (3.13) and (3.14) become ( 3:15) If we express the expressions in (3.15) as a fourfold iterated integral by removing the angular brackets, the two expressions will be fourfold iterated integrals in the order dtdb dadx. We wish to express them in the order dxdb dadt by switching the order of integrations. We cannot apply the Fubini-Tonelli theorem (Yosida, 1995), p.18 at this stage as none of the above iterated integrals is absolutely convergent. We therefore proceed as follows to apply Fubini's theorem to switch the order of integration. We number the above integrands in (3.15) jaja 2 : (3:17) Let K be a compact set of the " XBAT-space" given by ½ðx; b; a; tÞ : jxj X 1 ; jbj B 1 ; ε jaj A 1 ; and jtj T 1 : Then the fourfold iterated integrals of the integrands (3.16) and (3.17) by (Yosida, 1995), p.18, with respect to the measure dxdbda dt are absolutely convergent over compact set K and so are integrable. Therefore, switches in the order of integration over K can be done in 4! ways and all these 4! iterated integrals of integrands (3.16) and (3.17) are equal in view of Fubini's theorem. Our concern for the time being is the equality of the fourfold iterated integrals dx dbda dt and dtdb dadx of the above mentioned integrands over the compact set K, which is valid in view of Fubini's theorem. We now let X 1 ; B 1 ; A 1 and T 1 all tend to 1 and then we let ! 0; the fact that the fourfold iterated infinite integrals dxdb dadt of integrands (3.16) and (3.17) are now convergent is proved by using the Plancherel theorem with respect to the Fourier transform F b (Boggess & Narcowich, 2001, p. 107;Pandey & Upadhyay, 2015). Hence, (3:18) Therefore, Now using the wavelet inversion formula the triple integrals The results (i), (ii) and (iii) are all proved in (Pandey, Jha, & Singh, 2016), but the proofs of (i) and (ii) can also be found in (Boggess & Narcowich, 2001, p.258). This implies (in view of (iii)) that This explains the ambiguity in our inversion formula. For the validity of the uniqueness in our inversion formula f must belong to S 0 F ðRÞ.

Conclusion
In this present paper authors introduced the continuous wavelet transform on Schwartz tempered distributions and proved the corresponding wavelet inversion formula (valid modulo a constant distribution) interpreting convergence in the weak topology of S 0 ðRÞ.
We have observed that our aforesaid investigations are true when the wavelet kernel under cosideration for determining the wavelet transform are those wavelets whose all the moments are non-zero. Our entire results and facts are stated and proved as Lemmas and Theorems.