## Abstract

Let *X* be an arbitrary separable symmetric function space on [0, 1]. By using a combination of the frame approach and the notion of the multiplicator space \(\mathscr {M}(X)\) of *X* with respect to the tensor product, we investigate the problem when the sequence of dyadic dilations and translations of a function \(f\in X\) is a representing system in the space *X*. The main result reads that this holds whenever \(\int _0^1 f(t)\,dt\ne 0\) and \(f\in \mathscr {M}(X).\) Moreover, the condition \(f\in \mathscr {M}(X)\) turns out to be sharp in a certain sense. In particular, we prove that a decreasing nonnegative function *f*, \(f\ne 0,\) from a Lorentz space \(\varLambda _{\varphi }\) generates an absolutely representing system of dyadic dilations and translations in \(\varLambda _{\varphi }\) if and only if \(f\in \mathscr {M}(\varLambda _{\varphi }).\)

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## Acknowledgements

The work of the Sergey V. Astashkin was supported by the Ministry of Education and Science of the Russian Federation, Project 1.470.2016/1.4 and by the RFBR Grant 18-01-00414. The work of the Pavel A. Terekhln was supported by the RFBR Grant 18-01-00414. The authors are very grateful to the referee for detailed and constructive criticism that helped us improve the presentation of the paper.

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Communicated by Krzysztof Stempak.

## Appendix

### Appendix

A central role in the proof of the Filippov–Oswald theorem [8] is played by condition (2) that has allowed to Filippov and Oswald to prove the following characterization: let \(I_{k,i}=\left[ \frac{i}{2^k},\frac{i}{2^{k+1}}\right] \) for \(k=0,1,\ldots \) and \(i=0,1,\ldots ,2^k-1.\) For \(n = 2^k + i\) set \(I_n := I_{k,i}\) and \(f_n:=f_{k,i}.\) If \(f\in L_p([0,1])\) (with \(1\le p<\infty \)) satisfies condition (1), then a subsystem \(\{f_{n_l}\}\subset \{f_n\}\) is a representing system in \(L_p([0,1])\) if and only if for every \(L\in \mathbb {N}\) the set \(\cup _{l=L}^\infty I_{n_l}\) has full Lebesgue measure in the interval [0, 1].

However, we show here that condition (2) is not satisfied by Lorentz spaces on [0, 1] different from \(L_1.\) This is an immediate consequence of the following connection of (2) with the smoothness of a separable symmetric function space on [0, 1] at the function, identically equal to 1. Recall that a Banach space *E* is *smooth* at an element \(x_0\in E,\)\(\Vert x_0\Vert _E=1,\) whenever there exists a unique \(x^*\in E^*\) with \(\Vert x^*\Vert _{E^*}=x^*(x_0)=1.\)

### Proposition 4

Let *X* be a separable symmetric function space on [0, 1]. Then, condition (2) is fulfilled for each \(f\in X\) such that \(\int _0^1 f(t)\,dt\ne 0\) if and only if *X* is smooth at 1.

### Proof

Firstly, let condition (2) be fulfilled for each \(f\in X\) such that \(\int _0^1 f(t)\,dt\ne 0.\) Assuming that *X* is not smooth at 1, we find two functions \(y_1\) and \(y_2,\)\(y_1\ne y_2,\) from the dual space \(X^*=X'\) such that

Let \(f\in X\) be an arbitrary function such that \(a:=\langle f,y_1\rangle >0\) and \(b:=-\langle f,y_2\rangle >0.\) Obviously, we can assume that \(\int _0^1 f(t)\,dt\ne 0.\) Then, we have

if \(\lambda \le 0,\) and similarly

if \(\lambda >0.\) This contradicts the condition.

Conversely, suppose that *X* is smooth at 1 but, however, there is a function \(f\in X,\)\(\int _0^1 f(t)\,dt\ne 0,\) such that

Then, clearly, the projection \(P(a\cdot 1+b\cdot f):= a\cdot 1,\)\(a,b\in \mathbb {R},\) defined on the subspace, spanned by 1 and *f*, has norm 1. Therefore, by Hahn–Banach Theorem, we have

Hence, there exists a sequence \(\{y_n\}\subset X^*=X'\) such that \(\Vert y_n\Vert _{X^*}\le 1,\)\(\langle f,y_n\rangle =0,\)\(n=1,2,\ldots ,\) and \(\langle 1,y_n\rangle \rightarrow 1\) as \(n\rightarrow \infty .\) Since the closed unit ball in \(X^*\) is weakly\(^*\) compact, we can find a subsequence \(\{y_{n_k}\}\subset \{y_n\}\) such that \(y_{n_k}\rightarrow \tilde{y}\) weakly\(^*\) for some \(\tilde{y}\in X^*,\)\(\Vert \tilde{y}\Vert _{X^*}\le 1.\) This implies that \(\langle f,\tilde{y}\rangle =0\) and \(\langle 1,\tilde{y}\rangle =1.\) On the other hand, since \(\Vert x\Vert _1\le \Vert x\Vert _X\) (see Sect. 2.1), we have

Therefore, taking into account that *X* is smooth at 1, from the preceding equations we deduce that \(\tilde{y}(t)\equiv 1\) and so \(\langle f,1\rangle =\int _0^1 f(t)\,dt=0,\) which contradicts the hypothesis.

\(\square \)

### Corollary 5

Let \(\varphi \) be an increasing convex function on [0, 1], \(\varphi (0)=0,\)\(\varphi (1)=1,\) and \(\lim _{t\rightarrow 0}\varphi (t)/t=\infty .\) Then there is a function \(f\in \varLambda _{\varphi }\) such that \(\int _0^1 f(t)\,dt\ne 0\) and for each \(\lambda \in \mathbb {R}\) we have

### Proof

Recall that isometrically \((\varLambda _{\varphi })^*=M_{\varphi },\) where \(M_{\varphi }\) is the Marcinkiewicz space with the norm

[11, Theorem II.5.2]. One can easily check that from properties of \(\varphi \) it follows that both functions \(y_1(t)\equiv 1\) and \(y_2(t)=\varphi '(t)\) belong to \(M_{\varphi },\)\(y_1\ne y_2,\) and

This means that the space \(\varLambda _{\varphi }\) is not smooth at 1. Therefore, applying Proposition 5, we get the desired result. \(\square \)

### Remark 2

A careful inspection of the proof of Lemma 2 from the paper [8] shows that, in fact, this proof is based on using the well-known Weak Greedy Algorithm. In the case of \(L_p,\)\(1\le p<\infty ,\) everything that is needed to apply it is condition (2). However, if we try to prove an analogue of the Filippov–Oswald theorem for a general separable symmetric function space *X* on [0, 1], the following much more restrictive conditions are required:

(a) \(f\in \mathscr {M}(X);\)

(b) \(dist_{\mathscr {M}(X)}(1,X_{0,f})<1;\)

(c) \(\sup _{\Vert x\Vert _{\mathscr {M}(X)}\le 1}\liminf _{k\rightarrow \infty }dist_{\mathscr {M}(X)}(x,X_{k,f})<1.\)

Here, as before, \(X_{k,f}=\text {span}((f_{\alpha })_{|\alpha |=k}]),\)\(k=0,1,2,\ldots ,\) and for every Banach space *Y*, \(L\subset Y\) and \(y_0\in Y\) we set

In contrast to that, according to Theorem 2, the only condition \(f\in \mathscr {M}(X)\) (together with (1)) assures that the sequence of dyadic dilations and translations of *f* is an absolutely representing system in the separable symmetric function space *X*. Thus, we see that the frame approach, used in this paper, works under less restrictive conditions and so has wider applicability than the above Weak Greedy Algorithm, used in [8] (cf. [18]).

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Astashkin, S.V., Terekhin, P.A. Representing Systems of Dilations and Translations in Symmetric Function Spaces.
*J Fourier Anal Appl* **26, **13 (2020). https://doi.org/10.1007/s00041-019-09715-8

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### Keywords

- Sequence of dilations and translations
- Symmetric function space
- Representing system
- Tensor product
- Frame
- Lorentz space

### Mathematics Subject Classification

- Primary 46E30
- Secondary 46B70
- 42C15
- 46B15