Constructive Approximation in Mixed norm Spaces

Dhiraj Patel; Priyanka Majethiya; Shivam Bajpeyi

REVIEW 3 major objections 4 minor 42 references

Constructive Approximation in Mixed norm Spaces

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Kantorovich-type averaging sampling operators converge in the norm of mixed-norm Lebesgue and Orlicz spaces.

desk verdict The mixed-norm Lebesgue results are a solid extension, but the Orlicz half rests on a false density lemma, so Theorem 5.5 fails as stated and needs a Delta2 restriction. read the letter →

arxiv 2506.04787 v2 pith:SQLVNZCF submitted 2025-06-05 math.FA

Priyanka Majethiya , Shivam Bajpeyi , Dhiraj Patel This is my paper

classification math.FA MSC 94A2026D1546B0946E3047B34

keywords approximationoffunctionsKantorovich-typesamplingseriesmixednormLebesguespaceOrliczmodularconvergencekernelsdensitycompactlysupported

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that one family of averaging sampling operators, the Kantorovich-type series, can reconstruct arbitrary functions in mixed-norm Lebesgue and mixed-norm Orlicz spaces, not just continuous or bandlimited signals. If correct, it supplies a unified approximation scheme for function spaces in which different variables may have different integrability behavior. The central results are norm-convergence theorems: for mixed-norm Lebesgue spaces, $\lim_{w\to\infty}\|K_w f-f\|_{\vec P}=0$, and for mixed-norm Orlicz spaces, $\lim_{w\to\infty}\|K_w f-f\|_{\vec\Phi}=0$, under a non-decreasing exponent assumption and standard kernel conditions. The Orlicz half rests on a newly claimed density result asserting that compactly supported continuous functions are dense in mixed-norm Orlicz spaces.

What carries the argument

The central object is the multivariate Kantorovich-type sampling operator, a named family of operators that averages $f$ over small cubes before sampling. The argument is carried by three interacting devices: the kernel partition-of-unity condition $\sum_{k\in\mathbb Z^n}\chi(u-k)=1$ together with the finite absolute moment $m_0(\chi)$, which turn the series into a well-controlled convolution-like summand; Jensen's inequality iterated variable by variable, which works because the exponent tuple is assumed non-decreasing, $p_1\le p_2\le\cdots\le p_n$; and a modular inequality for the mixed-norm Orlicz modular $I^{\vec\Phi}$ that gives the boundedness estimate $I^{\vec\Phi}(\lambda K_w f)\le \|\chi\|_1(m_0(\chi))^{-n}\,I^{\vec\Phi}(\lambda(m_0(\chi))^n f)$. Convergence is then upgraded from compactly supported continuous functions to the whole space by density: $C_c(\mathbb R^n)$ dense in $L^{\vec P}(\mathbb R^n)$ for Lebesgue spaces, and the paper's Lemma 5.4 for Orlicz spaces, with a mixed-norm Vitali convergence theorem supplying the final limit interchange.

What would settle it

A concrete check is to test the density claim for the mixed-norm Orlicz space built from the exponential Orlicz functions $\phi_i(u)=e^u-1$: compute $I^{\vec\Phi}(\gamma(\chi_{V_m\setminus U_m}))$ for shrinking sets and see whether the limit is zero without relying on $1\in L^{\vec\Phi}$. If density of $C_c$ fails there, Theorem 5.5 lacks its approximation argument; if it succeeds, the gap lies only in the proof, not in the statement.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Kantorovich-type sampling operator $K_w f(x)=\sum_{k\in\mathbb Z^n}\chi(wx-k)\,w^n\int_{I_{k,w}} f(t)\,dt$, which replaces point samples by averages over cubes $I_{k,w}=\prod_{i=1}^n[k_i/w,(k_i+1)/w]$, is a bounded and convergent approximation scheme on mixed-norm Lebesgue spaces $L^{\vec P}(\mathbb R^n)$ and mixed-norm Orlicz spaces $L^{\vec\Phi}(\mathbb R^n)$. Theorem 4.5 states that $\|K_w f-f\|_{\vec P}\to 0$ for every $f\in L^{\vec P}(\mathbb R^n)$, and Theorem 5.5 states the same conclusion with the Orlicz norm $\|\cdot\|_{\vec\Phi}$. The proof proceeds through boundedness estimates for generalized and Kantorovich-type sampling operators, uniform convergence on compactly supported continuous functions, and an approximation argument that requires density of $C_c(\mathbb R^n)$ in the target space. The genuinely new load-bearing result for the Orlicz half is Lemma 5.4, which claims that $C_c(\mathbb R^n)$ is dense in $L^{\vec\Phi}(\mathbb R^n)$.

Load-bearing premise

The load-bearing premise is that compactly supported continuous functions are dense in every mixed-norm Orlicz space, and the proof of that density swaps a limit past an integral using the constant function 1 as a dominating function; when an Orlicz component grows exponentially, the constant function 1 need not lie in the space.

Editorial extensions

If this is right

For every $f$ in a mixed-norm Lebesgue space with $1\le p_1\le\cdots\le p_n$, the averaged sampling reconstruction converges in the mixed norm without any continuity or bandlimitedness assumption.
The same convergence holds in mixed-norm Orlicz spaces, covering exponential and logarithmic integrability in individual variables, provided the claimed density of compactly supported continuous functions is valid.
The boundedness constants are explicit in terms of kernel data: in the Lebesgue case the operator norm is controlled by $(m_0(\chi))^{1-1/p_n}\|\chi\|_1^{1/p_n}$, and in the Orlicz case $\|K_w f\|_{\vec\Phi}\le (m_0(\chi))^n\|f\|_{\vec\Phi}$.
Concrete kernels that factor as products of Fejér, Jackson, B-spline, or Bochner–Riesz kernels satisfy the stated hypotheses, so the convergence theorem applies to these reconstruction schemes.
Generalized sampling operators without averaging are shown to be bounded on the subspace $\Delta^{\vec P}$ of functions whose samples over relatively separated sets have finite mixed $\ell^{\vec P}$ weight, extending the admissible-sequence framework to mixed norms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if the density lemma is repaired under a $\Delta_2$-condition on the Orlicz components, Theorem 5.5 would extend cleanly to exponential and logarithmic Orlicz spaces; without such a repair, the Orlicz convergence result should be read as conditional on density of $C_c$.
Beyond the paper: the same iterated-Jensen argument could be re-run for higher-order Kantorovich-type sampling operators, yielding mixed-norm analogues of the single-norm convergence theorems cited in the introduction.
Beyond the paper: the explicit role of the kernel's absolute moment suggests a quantitative program—tracking the tail decay of $\chi$ to derive convergence rates for $\|K_w f-f\|$ in terms of a mixed-norm modulus of smoothness, a direction the paper does not pursue.
Beyond the paper: the mixed-norm framework is natural for anisotropic smoothness, so the same operator family could be tested in mixed-norm Besov or Triebel–Lizorkin spaces, where iterated norms are already standard.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take after reading it carefully.

The genuinely new and correct part is the mixed-norm Lebesgue half. The boundedness estimates for generalized and Kantorovich-type sampling operators and the convergence theorem for L^P with nondecreasing exponents look right and are a legitimate extension of the known L^p/Orlicz results. The proofs are standard—convexity, Jensen, tail estimates—but they are written carefully. Theorem 5.1, the modular bound, is also fine as far as it goes.

The problem is the Orlicz section. Lemma 5.4, the density lemma that carries Theorem 5.5, is false for the class of Orlicz functions defined in the paper. The stress-test note is correct. Take n=1 and phi(u)=u for 0<=u<=1, phi(u)=infinity for u>1. That is an admissible Orlicz function under the paper's own definition, and L^phi is L^1 intersect L^infinity with the max norm. The indicator of [0,1] lies in L^phi, but no compactly supported continuous function can approximate it in sup norm to within 1/2, since the jump can't be smoothed without changing the sup norm. So C_c is not dense. The proof's dominating function 1 is not in L^phi (infinite modular), and the cited dominated convergence theorem from [6] covers mixed-norm Lebesgue, not Orlicz modulars. Since Kwf is continuous for the Fejer kernel, Theorem 5.5's convergence claim fails outright for that f.

This is a load-bearing flaw, not a typo. The fix is standard: assume the Orlicz tuple satisfies Delta2, or restrict to the order-continuous subspace, and then C_c density is true. With that restriction, the rest of the modular argument should go through. The Bochner-Riesz kernel formula also has a sign error in the exponent (should be -(N/2+gamma)), which is minor and easily fixed.

So: the Lebesgue-space half is publishable as a modest extension; the Orlicz half needs a serious correction before it can be trusted. People working on sampling series in mixed-norm spaces will find the Lebesgue half useful once the Orlicz claims are corrected. I would send it to a competent referee rather than desk-reject, because the flaw is sharp, identifiable, and repairable, and the correct Lebesgue results deserve an outlet. But I would not accept anything close to the current version.

Referee Report

3 major / 4 minor

Summary. The paper studies generalized sampling operators and Kantorovich-type sampling operators acting on mixed norm Lebesgue spaces L^{\vec P}(R^n) and mixed norm Orlicz spaces L^{\vec \Phi}(R^n). It proves boundedness of the generalized operators on a subspace \Delta_{\vec P}, boundedness of the Kantorovich operators on L^{\vec P}, convergence of K_w f to f in L^{\vec P} (Theorems 4.4 and 4.5), and then claims the analogous convergence in L^{\vec \Phi} (Theorem 5.5). The Orlicz-space argument is based on a density lemma, Lemma 5.4, asserting that C_c(R^n) is dense in L^{\vec \Phi}(R^n). Section 6 lists admissible kernels including Fej\'er, Jackson, B-spline, and Bochner\u2013Riesz kernels.

Significance. If correct, the Orlicz-space convergence theorem would be a useful extension of known sampling-operator results to a genuinely broader class of mixed norm spaces. The Lebesgue-space estimates and the modular inequality (5.1) appear to be plausible extensions of existing work and may be of independent interest. However, the central advertised contribution, convergence in mixed norm Orlicz spaces, rests on a density statement that is false under the paper's own definition of Orlicz function. The failure is explicit and demonstrable, so the main theorem of Section 5 is not established in the stated generality.

major comments (3)

[Section 5, Theorem 5.5] Lemma 5.4 is false as stated. The proof uses a dominated convergence theorem for the mixed norm modular I^{\vec \Phi}, citing [6, Section 2], but [6] treats mixed norm Lebesgue spaces, not Orlicz modulars, and the asserted domination by the constant function 1 is invalid because 1 need not belong to L^{\vec \Phi}. A concrete admissible Orlicz function on R is \phi(u)=u for 0\le u\le 1 and \phi(u)=\infty for u>1; it is convex, lower semicontinuous, \phi(0)=0, and \phi(u)>0 for u>0, so it satisfies the definition in Section 2.3. For this \phi, the Orlicz norm is \|h\|_\phi=\max(\|h\|_\infty,\|h\|_1), and f=\chi_{[0,1]} lies in L^\phi. Any g\in C_c(R) satisfies \|f-g\|_\infty\ge 1/2 by continuity at the jump, hence \|f-g\|_\phi\ge 1/2. Thus C_c(R) is not dense in L^\phi, and Lemma 5.4 fails in dimension one. The proof also omits the finite-measure truncation needed to approximate a general measurable set S by sets U_m\subset S\subset V_m with finite \mu(V_m\setminus U_m). A \Delta_2 assumption or a restriction to the order-continuous subspace E^{\vec \Phi} would be needed to repair the statement.
[Section 5, Theorem 5.3] Theorem 5.5 is false in the stated generality. Using the same Orlicz function as in the previous comment, take f=\chi_{[0,1]} in L^\phi(R). For the Fej\'er kernel, which is continuous and satisfies the kernel assumptions, each K_w f is a finite sum of continuous terms and hence continuous. Since \|K_w f-f\|_\phi\ge \|K_w f-f\|_\infty\ge 1/2 for every w, the claimed convergence \lim_w \|K_w f-f\|_\phi=0 fails outright. The issue is not a removable gap in a proof: the density lemma on which the theorem is based is false, and the displayed conclusion is contradicted by an elementary example within the paper's framework.
[Section 5, Theorem 5.3] The proof of Theorem 5.3 does not establish the stated modular convergence. The argument invokes the Vitali convergence theorem of [25] after proving uniform convergence, smallness of I^{\vec \Phi}(\lambda K_w f) outside a large cube, and equi-absolute continuity of the integrals \int_{B_2}\phi_2(\int_{B_1}\phi_1(\lambda|K_w f|)). Even if these three properties are granted, they concern K_w f, not K_w f-f, and the Vitali theorem in [25] is a statement for mixed norm L^p spaces, not for Orlicz modulars. The conclusion lim_w I^{\vec \Phi}(\lambda(K_w f-f))=0 therefore does not follow from the displayed estimates; additional arguments would be needed to control the modular of the difference.

minor comments (4)

[Section 6, Example (IV)] The Bochner\u2013Riesz kernel formula contains a sign error: the exponent should be -N/2-\gamma, not -(N/2-\gamma). As written, the kernel may fail to be in L^1(R^n) for the stated parameters.
[Section 6, Example (IV)] The dimension is written as N in the kernel formula but as n in the surrounding text and in the condition x\in R^n; the notation should be unified.
[Throughout] There are several typographical errors, including \"discueed\" in the introduction, \"denisty\" in the proof of Theorem 4.5, \"Lemmma\" in the proof of Theorem 5.3, and \"paritcularly\" in Section 2.4.
[Section 4, Theorem 4.4] In Step 3 of the proof of Theorem 4.4, the expression \|\chi\|_{L(1,1)(B_1\times B_2)} is used without defining this mixed L^{(1,1)} norm on a product of measurable sets; the notation should be introduced or replaced.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the approximation theorems are assembled from external kernel, density, and Vitali-convergence inputs, without fitting parameters or definitional self-reference.

full rationale

The paper's derivation chain is linear and does not reduce to its own inputs. Section 4 bounds K_w on mixed-norm Lebesgue spaces using convexity and Jensen's inequality (Theorem 4.2), proves convergence on C_c via the uniform result [21, Theorem 4.1] and the Vitali convergence theorem for mixed norms [25] (Theorem 4.4), and then extends to L^P by the classical density of C_c in mixed-norm Lebesgue spaces cited to [2] (Theorem 4.5). Section 5 repeats the same architecture for Orlicz modulars: Theorems 5.1 and 5.3 use convexity, Jensen, the same external uniform convergence result, and the Vitali theorem; Theorem 5.5 invokes Lemma 5.4, a density statement proved from simple functions and an external dominated convergence theorem. No parameter is fitted, no norm or space is defined in terms of the Kantorovich operator or its convergence, and no load-bearing step is justified by the authors' own prior work. The self-references [3] and [32] appear only in background remarks on Orlicz spaces and on the monotonicity convention, so they are not load-bearing. The known gap in Lemma 5.4, namely that the cited mixed-norm dominated convergence theorem [6, Section 2] is for Lebesgue mixed norms and that the constant function 1 need not have finite Orlicz modular, is a correctness or generality defect that could invalidate Theorem 5.5; it is not a circular reduction, because the lemma is not equivalent to the convergence claim being proved. Accordingly the circularity score is 1, indicating no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard kernel assumptions, the non-decreasing exponent restriction, and the density of C_c in mixed norm Orlicz spaces. The density proof itself depends on an unproved dominated convergence theorem for mixed norm Orlicz modulars, which is the main fragility. No free parameters are fitted and no new entities are postulated.

assumptions (5)

domain assumption The kernel chi satisfies partition of unity (3.1) and finite absolute moment (3.2).
Standing assumption defining admissible kernels; used in every theorem. Reasonable and standard for sampling series.
domain assumption Exponents in mixed norm Lebesgue space are non-decreasing: p1 <= p2 <= ... <= pn.
Stated in Section 2.2; used in Jensen steps in Theorems 3.4 and 4.2. Restricts the class of mixed norm spaces.
domain assumption C_c(R^n) is dense in mixed norm Orlicz space L^Phi(R^n).
Lemma 5.4; proof relies on a dominated convergence theorem for mixed norm modulars that is cited but not proved and not covered by the cited reference [6].
ad hoc to paper Dominated convergence for the mixed norm modular I^Phi.
Invoked in Lemma 5.4 proof without proof; cited to [6, Section 2] which treats only Lebesgue mixed norm spaces.
domain assumption Uniform convergence of K_w f to f for f in C_c(R^n), from [21, Theorem 4.1].
External theorem used in Step 1 of Theorems 4.4 and 5.3; not proved in the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Constructive Approximation in Mixed norm Spaces." pith.science (2026). https://pith.science/paper/SQLVNZCF

@misc{pith2026250604787,
  author       = {Pith},
  title        = {Pith review of: Constructive Approximation in Mixed norm Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQLVNZCF}},
  note         = {Machine review of arXiv:2506.04787}
}

read the original abstract

The concept of mixed norm spaces has emerged as a significant interest in fields such as harmonic analysis. In addition, the problem of function approximation through sampling series has been particularly noteworthy in the realm of approximation theory. This paper aims to address both these aspects. Here we deal with the problem of function approximation in diverse mixed norm function spaces. We utilise the family of Kantorovich type sampling operators as approximator for the functions in mixed norm Lebesgue space, and mixed norm Orlicz space. The Orlicz spaces are well-known as a generalized family that encompasses many significant function spaces. We establish the boundedness of the family of generalized as well as Kantorovich type sampling operators within the framework of these mixed norm spaces.Further, we study the approximation properties of Kantorovich-type sampling operators in both mixed norm Lebesgue and Orlicz spaces. At the end, we discuss a few examples of suitable kernel involved in the discussed approximation procedure.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

[6]

Benedek, R

A. Benedek, R. Panzone, The space Lp with mixed norm, Duke Math. J. 28 (1961) 301–324

work page 1961
[25]

Galmarino, R

A.R. Galmarino, R. Panzone, Lp-spaces with mixed norm, for P a sequence, J. Math. Anal. Appl. 10 (3) (1965) 494–518

work page 1965
[1]

T. Acar, D. Costarelli, G. Vinti, Linear prediction and simultaneous approximation by m-th order Kantorovich-type sampling series, Banach J. Math. Anal. 14 (4) (2020) 1481–1508

work page 2020
[2]

Antoni´ c, I

N. Antoni´ c, I. Ivec, On the H¨ ormander–Mihlin theorem for mixed norm Lebesgue spaces, J. Math. Anal. Appl. 433 (1) (2016) 176–199

work page 2016
[3]

Bajpeyi, D

S. Bajpeyi, D. Patel, S. Sivananthan, Relevant sampling in a reproducing kernel subspace of Orlicz space, Anal. Appl. 22 (2024) 1075–1094

work page 2024
[4]

Bardaro, P.L

C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals, J. Math. Anal. Appl. 316 (1) (2006) 269–306

work page 2006
[5]

Bardaro, G

C. Bardaro, G. Vinti, P.L. Butzer, R.L. Stens, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process. 6 (1) (2007) 29–52

work page 2007
[7]

Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J

P.L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition 3 (1) (1983) 185–212. 25

work page 1983

Show all 42 references

[8]

Butzer, The sampling theorem and linear prediction in signal analysis, Jahresber

P.L. Butzer, The sampling theorem and linear prediction in signal analysis, Jahresber. Dtsch. Math.-Ver. 90 (1988) 1–70

1988
[9]

Butzer, A

P.L. Butzer, A. Fischer, R.L. Stens, Generalized sampling approximation of multivariate signals; theory and some applications, Note Mat. 10 (1990) 173–191

1990
[10]

Butzer, M

P.L. Butzer, M. Hauss, R.L. Stens, The sampling theorem and its unique role in various branches of mathematics, Mitt. Math. Ges. Hamburg 12 (3) (1991) 523–547

1991
[11]

Butzer, R.J

P.L. Butzer, R.J. Nessel, Fourier Analysis and Approximation, Vol. 1, Reviews in Group Representation Theory, Part A, Pure and Applied Mathematics Series, Vol. 7, (1971)

1971
[12]

Butzer, S

P.L. Butzer, S. Ries, R.L. Stens, Approximation of continuous and discontinuous func- tions by generalized sampling series, J. Approx. Theory 50 (1) (1987) 25–39

1987
[13]

Butzer, R.L

P.L. Butzer, R.L. Stens, Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Rev. 34 (1) (1992) 40–53

1992
[14]

Butzer, R.L

P.L. Butzer, R.L. Stens, Linear prediction by samples from the past, Adv. Topics Shan- non Sampling Interpolation Theory, Springer Texts Electrical Engrg., Springer-Verlag, New York, (1993), 157–183

1993
[15]

Butzer, R.L

P.L. Butzer, R.L. Stens, Reconstruction of signals inLp(R)-space by generalized sampling series based on linear combinations of B-splines, Integral Transforms Spec. Funct. 19 (1–2) (2008) 35–58

2008
[16]

Cagini, D

C. Cagini, D. Costarelli, R. Gujar, M. Lupidi, G.A. Lutty, M. Seracini, G. Vinti, Improve- ment of retinal OCT angiograms by sampling Kantorovich algorithm in the assessment of retinal and choroidal perfusion, Appl. Math. Comput. 427 (2022) Paper No. 127152, 15 pp

2022
[17]

Cantarini, D

M. Cantarini, D. Costarelli, G. Vinti, Approximation of differentiable and non- differentiable signals by the first derivative of sampling Kantorovich operators, J. Math. Anal. Appl. 509 (1) (2022) Paper No. 125913, 20 pp

2022
[18]

Cleanthous, A.G

G. Cleanthous, A.G. Georgiadis, M. Nielsen, Molecular decomposition of anisotropic homogeneous mixed norm spaces with applications to the boundedness of operators, Appl. Comput. Harmon. Anal. 47 (2) (2019) 447–480

2019
[19]

Cluni, D

F. Cluni, D. Costarelli, A.M. Minotti, G. Vinti, Multivariate sampling Kantorovich op- erators: approximation and applications to civil engineering, EURASIP Proc. SampTA (2013) 400–403

2013
[20]

Costarelli, A.M

D. Costarelli, A.M. Minotti, G. Vinti, Approximation of discontinuous signals by sam- pling Kantorovich series, J. Math. Anal. Appl. 450 (2) (2017) 1083–1103

2017
[21]

Costarelli, G

D. Costarelli, G. Vinti, Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces, Boll. Unione Mat. Ital. (9) 4 (3) (2011) 445–468

2011
[22]

Edmunds, M

D.E. Edmunds, M. Krbec, Two limiting cases of Sobolev imbeddings, Houston J. Math. 21 (1) (1995) 119–128

1995
[23]

Evseev, A

N. Evseev, A. Menovschikov, Bounded operators on mixed norm Lebesgue spaces, Com- plex Anal. Oper. Theory 13 (2019) 2239–2258

2019
[24]

Finol, L

C.E. Finol, L. Maligranda, On a decomposition of some functions, Ann. Soc. Math. Pol., Ser. 1: Comment. Math. Prace Mat. 30 (2) (1991) 285–291

1991
[26]

Gr¨ ochenig, J.L

K. Gr¨ ochenig, J.L. Romero, J. St¨ ockler, Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions, Invent. Math. 211 (2018) 1119–1148

2018
[27]

Jerri, The Shannon sampling theorem—Its various extensions and applications: A tutorial review, Proc

A.J. Jerri, The Shannon sampling theorem—Its various extensions and applications: A tutorial review, Proc. IEEE 65 (11) (1977) 1565–1596

1977
[28]

Jiang, W

Y. Jiang, W. Sun, Adaptive sampling of time-space signals in a reproducing kernel subspace of mixed Lebesgue space, Banach J. Math. Anal. 14 (2020) 821–841. 26

2020
[29]

Kim, Elliptic and parabolic equations with measurable coefficients in Lp-spaces with mixed norms, Methods Appl

D. Kim, Elliptic and parabolic equations with measurable coefficients in Lp-spaces with mixed norms, Methods Appl. Anal. 15 (4) (2008) 437–467

2008
[30]

Kita, On Hardy–Littlewood maximal functions in Orlicz spaces, Math

H. Kita, On Hardy–Littlewood maximal functions in Orlicz spaces, Math. Nachr. 183 (1) (1997) 135–155

1997
[31]

Krasnoselski ˘ ı, Convex Functions and Orlicz Spaces, Vol

M.A. Krasnoselski ˘ ı, Convex Functions and Orlicz Spaces, Vol. 4311, US Atomic Energy Commission, (1960)

1960
[32]

Kumar, D

A. Kumar, D. Patel, S. Sivananthan. Sampling and reconstruction in reproducing kernel subspaces of mixed Lebesgue spaces. J. Pseudodiffer. Oper. Appl. 11 (2020) 843-868

2020
[33]

Maligranda, Orlicz Spaces and Interpolation, Sem

L. Maligranda, Orlicz Spaces and Interpolation, Sem. Mat. 5, Universidade Estadual de Campinas, Departamento de Matem´ atica, Campinas (1989) iii+206 pp

1989
[34]

Maligranda, Calder´ on–Lozanovskii construction for mixed norm spaces, Acta Math

L. Maligranda, Calder´ on–Lozanovskii construction for mixed norm spaces, Acta Math. Hungar. 103 (4) (2004) 279–302

2004
[35]

Milman, A note on L(p, q) spaces and Orlicz spaces with mixed norms, Proc

M. Milman, A note on L(p, q) spaces and Orlicz spaces with mixed norms, Proc. Amer. Math. Soc. 83 (4) (1981) 743–746

1981
[36]

Musielak, Orlicz Spaces and Modular Spaces, Vol

J. Musielak, Orlicz Spaces and Modular Spaces, Vol. 1034, Springer, 2006

2006
[37]

Orlicz, ¨Uber konjugierte Exponentenfolgen, Studia Math

W. Orlicz, ¨Uber konjugierte Exponentenfolgen, Studia Math. 3 (1) (1931) 200–211

1931
[38]

Orlicz, A note on modular spaces I, Bull

W. Orlicz, A note on modular spaces I, Bull. Acad. Polon. Sci. S´ er. Sci. Math. Astronom. Phys. 9 (1961) 157–162

1961
[39]

Orlova, G

O. Orlova, G. Tamberg, On approximation properties of generalized Kantorovich-type sampling operators, J. Approx. Theory 201 (2016) 73–86

2016
[40]

Rao, Z.D

M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces, Monogr. Textbooks Pure Appl. Math. 146, Marcel Dekker, Inc., New York, 1991, xii+449 pp

1991
[41]

Rao, Z.D

M.M. Rao, Z.D. Ren, Applications of Orlicz Spaces, Monogr. Textbooks Pure Appl. Math. 250, Marcel Dekker, Inc., New York, 2002, xii+464 pp

2002
[42]

Shannon, Communication in the presence of noise, Proc

C.E. Shannon, Communication in the presence of noise, Proc. I.R.E. 37 (1949) 10–21. 27

1949

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [6]

Benedek, R

A. Benedek, R. Panzone, The space Lp with mixed norm, Duke Math. J. 28 (1961) 301–324

work page 1961

[2] [25]

Galmarino, R

A.R. Galmarino, R. Panzone, Lp-spaces with mixed norm, for P a sequence, J. Math. Anal. Appl. 10 (3) (1965) 494–518

work page 1965

[3] [1]

T. Acar, D. Costarelli, G. Vinti, Linear prediction and simultaneous approximation by m-th order Kantorovich-type sampling series, Banach J. Math. Anal. 14 (4) (2020) 1481–1508

work page 2020

[4] [2]

Antoni´ c, I

N. Antoni´ c, I. Ivec, On the H¨ ormander–Mihlin theorem for mixed norm Lebesgue spaces, J. Math. Anal. Appl. 433 (1) (2016) 176–199

work page 2016

[5] [3]

Bajpeyi, D

S. Bajpeyi, D. Patel, S. Sivananthan, Relevant sampling in a reproducing kernel subspace of Orlicz space, Anal. Appl. 22 (2024) 1075–1094

work page 2024

[6] [4]

Bardaro, P.L

C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals, J. Math. Anal. Appl. 316 (1) (2006) 269–306

work page 2006

[7] [5]

Bardaro, G

C. Bardaro, G. Vinti, P.L. Butzer, R.L. Stens, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process. 6 (1) (2007) 29–52

work page 2007

[8] [7]

Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J

P.L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition 3 (1) (1983) 185–212. 25

work page 1983