INVERSE CONTINUOUS WAVELET TRANSFORM IN WEIGHTED VARIABLE EXPONENT AMALGAM SPACES

The wavelet transform is an useful mathematical tool. It is a mapping of a time signal to the time-scale joint representation. The wavelet transform is generated from a wavelet function by dilation and translation. This wavelet function satisfies an admissible condition so that the original signal can be reconstructed by the inverse wavelet transform. In this study, we firstly give some basic properties of the weighted variable exponent amalgam spaces. Then we investigate the convergence of the θ−means of f in these spaces under some conditions. Finally, using these results the convergence of the inverse continuous wavelet transform is considered in these spaces.


Introduction
Recently, the variable exponent Lebesgue L p(:) (R d ) spaces and a class of nonlinear problems with variable exponential growth have been new and interesting topics. The space has several applications, such as electrorheological ‡uids (see [31]), elastic mechanics (see [43]) and image processing model. Moreover, the spaces L p(:) (R d ) and L p (R d ) have many common properties, such as Banach space, re ‡exivity, separability, uniform convexity, Hölder inequalities and embeddings. One of the most important di¤erences between these spaces is that the space L p(:) (R d ) is not translation invariant [27]. It is also well known that the maximal operator is bounded in L p(:) (R d ). For more comprehensive information (see [10], [12], [13] and [14]).
The amalgam of L p and l q on the real line is the space (L p ; l q ), which is also larger than the space L p , consisting of functions which are locally in L p and have l q 2020 Mathematics Subject Classi…cation. 65T60, 42C40, 42B08, 43A15, 46B15, 46E30. Keywords and phrases. Weighted variable exponent amalgam spaces, Inverse continuous wavelet transform, -summability.
oznur.kulak@amasya.edu.tr-Corresponding author; iaydin@sinop.edu.tr 0000-0003-1433-3159; 0000-0001-8371-3158. c 2 0 2 0 A n ka ra U n ive rsity C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rsity o f A n ka ra -S e rie s A 1 M a th e m a tic s a n d S ta tistic s 1172 Ö. KULAK, · I. AYDIN behavior at in…nity. Many di¤erent forms of amalgam spaces have been studied by some authors (see [25], [33], [24], [15] and [18]). Moreover, this space play important roles in recent developments in time frequency analysis and sampling theory, which are modern branches of harmonic analysis. Signal analysis and wireless communication issues are quite popular in amalgam spaces (see [20]). Variable exponent amalgam spaces L p(:) ; l q and some basic properties, such as Banach function space, Hölder type inequalities, interpolation, bilinear multipliers and the boundedness of maximal operator, have been investigated recently. Some interesting articles have been published on this subject, but not many. So there are many open problems in this function spaces [5], [21], [26], [30], [22], [28], [3], [7], [2], [6].
The so called -summation method is investigated by some authors, such as [36], [32], [38], [39], [40], [34], [8]. The -summation is de…ned by for an integrable function on R. This summability is a generalized form of the wellknown summability methods, like Fejér, Riesz, Weierstrass, Abel, etc. by a suitable chosen of . Feichtinger and Weisz ( [16], [17], [42]) showed that the -means T f converges to f almost everywhere and in norm as T ! 1 for f 2 L p (R d ); (L p ; l q ). Also we characterize the points of the set of a.e. convergence as the Lebesgue points. Moreover, Uribe and Fiorenza [10], Szarvas and Weisz [34] obtained similar results for the space L p(:) R d : In this study we will discuss the convergence of the inverse continuous wavelet transform in weighted variable exponent amalgam spaces. Also, we investigate the convergence of the -means of f almost everywhere and in norm in these spaces under which conditions. Hence we obtain more general results with respect to [34].

Weighted Variable Exponent Lebesgue and Amalgam spaces
In this section we give some required de…nitions and information about wavelet transform and weighted variable exponent amalgam spaces. De…nition 1. Let x 2 R d ; s 2 R and s 6 = 0. The continuous wavelet transform is de…ned by for f and g, where D s is the dilation operator, and T x is the translation operator, i.e., [19]. If is radial, non-increasing as a function on (0; 1), non-negative, bounded, jf j and 2 L 1 R d , then is a radial majorant of f: If in addition (:) ln (j:j + 2) 2 L 1 R d , then is a radial log-majorant of f: where B (a; ) = x 2 R d : kx ak < : Then the operators S f and S;T f are de…ned by where 0 < S < T < 1. Let de…ne the operator C 0 g; with Then C 0 g; is …nite [29], where g and both have radial log-majorants.
Let g and be radial, i.e., R R d (g ) (x) dx = 0: Assume that g and have a radial log-majorant. Then we get at every Lebesgue point for any f 2 L p R d (1 p < 1). The convergence is proved with respect to L p -norm for T = 1; [29]. Under some similar conditions, Weisz has proved similar results [41].
Banach space. If p(:) = p is a constant function, then the norm k:k p(:) coincides with the usual Lebesgue norm k:k p ; [27]. A measurable and locally integrable function ! : R d ! (0; 1) is called a weight function. The weighted modular is de…ned by [4].
De…nition 5. The maximal operator M is de…ned by for f 2 L 1 loc (R). Hästö and Diening [23] de…ned the class A p(:) consists of those weights ! such that where ß denotes the set of all balls in R d , If p (:) satis…es the following inequality yj   INVERSE CONTINUO US WAVELET TRANSFO RM   1175 for all x; y 2 R d , then p (:) provides the local log-Hölder continuity condition. Moreover, if the inequality holds for some p 1 > 1; C > 0 and all x 2 R d ; then we say that p (:) satis…es the local log-Hölder decay condition. We denote by P log (R d ) the class of variable exponents which are log-Hölder continuous, i.e. which satisfy the local log-Hölder continuity condition and local log-Hölder decay condition [4], [37].
Let p 2 P log (R d ) and ! ; l q coincides with L p(:) ; l q (see [7], [26]) : In 2014, Meskhi and Zaighum showed that the maximal operator is bounded in weighted variable exponent amalgam spaces under some conditions [30].

3.
Summability on the Weighted Variable Exponent Wiener Amalgam Spaces  Theorem 2. Let 1 p (:) ; q < 1 and 0 < c !. Assume that has radial majorant. Then; i) The limit is valid for any Lebesgue point of f 2 L p(:) ! ; l q . ii) If in addition 1 q p p (:) p + < 1; then the following limit equality is available for all f 2 L   ! ; l q by Theorem 2.1 in [34]. ii) Also, if we follow Theorem 3.8 in [34], Theorem 2.3 in [9], Theorem 5.11 in [10], and Theorem 8 in [1], then we have that iii) Let > 0 be given. Using Proposition 1, it is obtained that the following inequality kf gk L p(:) ! ;l q < Ö. KULAK, · I. AYDIN is valid for g 2 C c (R), whose compact support suppg is K: Using i) and Proposition 2 in [7], we have that ! ; l q the following relation holds; Proof. Let f 2 L 1 ! \ L p(:) ! ; l q and y 2 R. Then we have decomposition of S f (y) as y x s dxdtds = I II + III by from [29], [34] . Also it is well known that where by proof of Theorem 1.1 in [29]. On the other hand, Szarvas and Weisz proved that ' and have radial majorants by Theorem 5.1 in [34] in case g and have radial log-majorants. Since g; have radial log-majorants, by Lemma 2.5 in [29]. Therefore we get where (y) = ' (y) (y) If '; have radial majorants, then = ' have radial majorant, that is, is a non-negative and non-increasing function, and belongs to the space L 1 \ L 1 . So it is obtained that and 2 L 1 ; l 1 . Then using Hölder inequality and Lemma 1, we have Hence the function S is linear and bounded from L 1 ! \ L  Proof. i) Since p (:) 2 P log (R) and 1 < p p (:) p + < 1; then A 1 A p(:) [4]. By Theorem 2 and Theorem 4, we deduce that for all Lebesgue points of f 2 L p(:) ! ; l q . On the other hand, using Theorem 5.2 in [34] , we have that R R (y) dy = C 0 g; and lim S!0 + S f (x) = C 0 g; f (x) : ii) By Theorem 5.2 in [34] we can write the equality S;T f (x) = S f (x) T f (x) for x 2 R. Then using (i), Theorem 2 and Theorem 4, we obtain that Corollary 1. Assume that g; have radial log-majorants, R R (g ) (x) dx = 0. If ! 2 A 1 and 0 < c !, then the following statements are valid for any f 2 L p(:) ! ; l q ; i) lim S!0 + s f (x) = C 0 g; f (x) a:e: ii) If in addition 1 q p (:) p + < 1; then lim S!0 + ;T !1 s;T f (x) = C 0 g; f (x) a:e: Proof. Let f 2 L p(:) ! ; l q . Then by Lemma 1, we have f 2 L 1 ; l 1 . It is known that if f 2 L 1 ; l 1 , then real numbers almost everywhere is a Lebesgue point of f , [16], [17]. Hence by the Theorem 5, we complete the proof.