Fourier representations of fractional B Splines via generalized Stirling type polynomials

Damla Gun; Peter Massopust; Yilmaz Simsek

arxiv: 2605.15244 · v1 · pith:PDOKDBOKnew · submitted 2026-05-14 · 🧮 math.GM

Fourier representations of fractional B Splines via generalized Stirling type polynomials

Damla Gun , Peter Massopust , Yilmaz Simsek This is my paper

Pith reviewed 2026-05-19 16:15 UTC · model grok-4.3

classification 🧮 math.GM

keywords fractional B-splinesFourier expansionsgeneralized Stirling-type numbersdistributional representationsDirac delta derivativesMittag-Leffler functionfractional calculusspline polynomials

0 comments

The pith

Fractional B-splines admit a Fourier expansion in generalized Stirling-type numbers that represents them as infinite sums of Dirac delta derivatives in the distributional sense.

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

The paper derives a Fourier-type expansion for fractional B-splines that incorporates generalized Stirling-type numbers through a generating function method. This expansion makes it possible to view fractional B-splines as infinite linear combinations of derivatives of the Dirac delta function within distribution theory. A sympathetic reader would care because it unifies spline approximations with fractional calculus and combinatorial structures, offering new ways to handle their interactions with test functions via shifted representations. The work also defines a fresh class of fractional spline polynomials whose generating function involves the Mittag-Leffler function.

Core claim

Employing a generating function approach inspired by recent results, we derive a novel Fourier type expansion for fractional B splines that involves generalized Stirling type numbers. This representation allows us to express fractional B splines as infinite linear combinations of derivatives of the Dirac delta in the distributional sense. Furthermore, we establish an explicit shifted distributional representation and obtain shifted distributional representations that characterize the action of fractional B-splines on test functions. In addition, we introduce a new class of fractional spline polynomials and derive their generating function in terms of the Mittag Leffler function.

What carries the argument

The Fourier-type expansion of fractional B-splines derived via generating functions, with generalized Stirling-type numbers supplying the coefficients that yield the infinite distributional sum of Dirac delta derivatives.

Load-bearing premise

The generating function approach inspired by recent results of Simsek extends directly to fractional B-splines to produce the claimed Fourier expansion and distributional representation.

What would settle it

Compute the actual Fourier transform of a concrete fractional B-spline, for instance of order 1.5, and check whether the coefficients match those given by the generalized Stirling-type numbers in the proposed infinite sum.

Figures

Figures reproduced from arXiv: 2605.15244 by Damla Gun, Peter Massopust, Yilmaz Simsek.

**Figure 1.** Figure 1: B-splines of orders 1 to 5. Polynomial B-splines enjoy, among others, the following properties: (1) ∀ n ∈ N: Bn ∈ C n−2 and supp Bn = [0, n]; (2) Recursion relation: ∀ 1 < n ∈ N ∀ x ∈ R: Bn(x) = x n−1Bn−1(x) + n−x n−1Bn−1(x − 1); (3) Given f ∈ C n [a, b], the error of approximating f by polynomial B-splines of order n on a uniform grid of mesh size h is O(h n ). For more details and further properties of B… view at source ↗

**Figure 2.** Figure 2: A family of fractional B-splines for α = 1 + m · 0.25, m = 1, . . . , 12. The functions Bα satisfy fractional distributional differential equations of the form (7) D α f = X∞ k=0 akδ(x − k), which motivates the definition of splines of fractional order (cf. [17]). A brief review of fractional operators in the subject of the next section. 3. Some Results About Fractional Operators In this short section, we … view at source ↗

**Figure 3.** Figure 3: The first four fractional spline polynomials for α ∈ { 1 4 , 1.5, √ 5}. Theorem 5.23. Let α > 0. The fractional spline polynomials satisfy the ordinary generating function X∞ n=0 S (α) n (x)t n = X∞ k=0 (−1)k α + 1 k (x − k) α +E1,α+1((x − k)+t), where Ea,b(z) denotes the Mittag–Leffler function Ea,b(z) = X∞ n=0 z n Γ(an + b) . Proof. Starting from (30), we multiply both sides by t n and sum over n ≥ … view at source ↗

read the original abstract

In this paper, we investigate fractional B splines and their connections with Fourier analysis, and establish connections with generalized Stirling-type numbers and distribution theory. Employing a generating function approach inspired by recent results of Simsek [24], we derive a novel Fourier type expansion for fractional B splines that involves generalized Stirling type numbers. Our main contribution is the derivation of a Fourier-type expansion of fractional B splines in terms of generalized Stirling-type numbers. This representation allows us to express fractional B splines as infinite linear combinations of derivatives of the Dirac delta in the distributional sense. Furthermore, we establish an explicit shifted distributional representation and obtain shifted distributional representations that characterize the action of fractional B-splines on test functions. In addition, we introduce a new class of fractional spline polynomials and derive their generating function in terms of the Mittag Leffler function. These results provide a unified framework that connects spline theory, fractional calculus, and combinatorial structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Simsek-style generating functions to get a Fourier expansion of fractional B-splines in generalized Stirling numbers plus a new Mittag-Leffler polynomial class, but the fractional-order step and distributional convergence need tighter checks.

read the letter

The main takeaway is that the authors apply a generating-function technique from Simsek's earlier papers to fractional B-splines, extract coefficients via generalized Stirling-type numbers, and arrive at a Fourier-type series that expresses the splines as infinite sums of Dirac delta derivatives in the distributional sense. They also define a new family of fractional spline polynomials whose generating function involves the Mittag-Leffler function. That is the concrete new output: explicit coefficient formulas and the shifted distributional representations for test-function pairings. The calculations appear to follow the usual term-by-term differentiation and coefficient extraction that works for integer-order splines, and the abstract indicates they carry the same steps over. If the algebra holds without extra remainder terms, the formulas give a direct way to write fractional splines in a form useful for approximation or fractional-calculus identities. The paper therefore supplies a compact bridge between spline theory, fractional operators, and combinatorial polynomials. The softer spot is the extension itself. Fractional B-splines are typically introduced through the Fourier multiplier |ω|^{-α-1} or via fractional forward differences, so it is not automatic that the ordinary generating-function identities survive without adjustment to the radius of convergence or to the topology in D'. The stress-test note flags exactly this point, and the manuscript would be stronger with an explicit verification for a non-integer α, perhaps a numerical check of the series against the known Fourier transform or a short argument that the term-by-term operations remain valid in the distributional limit. Self-citation to Simsek [24] is not a flaw by itself, but it does mean the novelty rests on showing the fractional case does not introduce new obstructions. Readers working in approximation theory or fractional calculus will find the explicit expansions and the new polynomial class worth looking at; the scope is narrow, so the paper is mainly for specialists who already use generating functions in spline contexts. A referee could check the coefficient derivations and the convergence claim in a few pages, so the work is worth sending out for review with attention to those analytic details.

Referee Report

2 major / 2 minor

Summary. The manuscript derives a Fourier-type expansion for fractional B-splines expressed via generalized Stirling-type numbers, obtained through a generating-function approach. This representation is used to write fractional B-splines as infinite linear combinations of derivatives of the Dirac delta in the sense of distributions; the paper also introduces a new class of fractional spline polynomials whose generating function involves the Mittag-Leffler function.

Significance. If the claimed extension of the generating-function identity to non-integer spline orders can be justified with explicit convergence arguments in the distributional topology, the work would supply a concrete link between fractional splines, combinatorial polynomials, and distribution theory. Such an explicit expansion could facilitate both theoretical analysis and numerical approximation of fractional splines acting on test functions.

major comments (2)

Abstract / main-contribution paragraph: the assertion that the generating-function manipulations of Simsek [24] extend verbatim to fractional B-splines is not accompanied by any verification of analytic continuation, radius-of-convergence preservation, or remainder estimates when the order becomes non-integer; without these steps the subsequent claim that the spline equals an infinite sum of Dirac derivatives in D' remains unsupported.
Section describing the Fourier expansion: the passage from the ordinary generating function to the distributional representation via term-by-term differentiation and coefficient extraction is presented without an explicit check that the same identity continues to hold for the fractional-difference operator or the Fourier multiplier |ω|^{-α-1} that defines fractional B-splines; this is load-bearing for the central distributional claim.

minor comments (2)

Clarify the precise definition of the newly introduced 'fractional spline polynomials' and their relation to the classical fractional B-splines already under discussion.
Add a short convergence or boundedness statement for the infinite series in the distributional topology so that the representation can be applied to test functions without additional justification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important points regarding the justification of our generating-function approach when extended to fractional orders. We address each major comment below and indicate the revisions we will make to strengthen the paper.

read point-by-point responses

Referee: Abstract / main-contribution paragraph: the assertion that the generating-function manipulations of Simsek [24] extend verbatim to fractional B-splines is not accompanied by any verification of analytic continuation, radius-of-convergence preservation, or remainder estimates when the order becomes non-integer; without these steps the subsequent claim that the spline equals an infinite sum of Dirac derivatives in D' remains unsupported.

Authors: We agree that the current presentation would be strengthened by explicit verification of the extension. In the revised manuscript we will insert a dedicated paragraph (or short subsection) that justifies the analytic continuation of the generating-function identity to non-integer orders. This will include a brief argument for preservation of the radius of convergence in the distributional topology together with remainder estimates that support the representation of the fractional B-spline as an infinite sum of Dirac derivatives in D'. revision: yes
Referee: Section describing the Fourier expansion: the passage from the ordinary generating function to the distributional representation via term-by-term differentiation and coefficient extraction is presented without an explicit check that the same identity continues to hold for the fractional-difference operator or the Fourier multiplier |ω|^{-α-1} that defines fractional B-splines; this is load-bearing for the central distributional claim.

Authors: We acknowledge that the manuscript currently proceeds by formal analogy. To address this, we will revise the relevant section to supply an explicit verification that term-by-term differentiation and coefficient extraction remain valid under the fractional-difference operator and the Fourier multiplier |ω|^{-α-1}. The added argument will confirm continuity of these operations in the distributional topology, thereby placing the central claim on firmer ground. revision: yes

Circularity Check

1 steps flagged

Central Fourier expansion of fractional B-splines rests on unverified extension of Simsek [24] generating functions with author overlap

specific steps

self citation load bearing [Abstract]
"Employing a generating function approach inspired by recent results of Simsek [24], we derive a novel Fourier type expansion for fractional B splines that involves generalized Stirling type numbers. Our main contribution is the derivation of a Fourier-type expansion of fractional B splines in terms of generalized Stirling-type numbers. This representation allows us to express fractional B splines as infinite linear combinations of derivatives of the Dirac delta in the distributional sense."

The load-bearing step is the direct extension of the ordinary generating function identities from [24] to fractional order without additional analytic justification; the Fourier expansion and distributional representation are therefore obtained by renaming and re-applying the authors' own prior coefficient extraction rather than from the independent definition of fractional B-splines via |ω|^{-α-1} multipliers.

full rationale

The derivation chain begins with the generating function approach from Simsek [24] (co-author overlap) and assumes verbatim carry-over of term-by-term operations, coefficient extraction, and passage to the Fourier/distributional representation when the spline order is fractional. The abstract and main contribution paragraph explicitly frame the work as an 'inspired' extension rather than an independent derivation from the fractional difference operator or Fourier multiplier definition of fractional B-splines. No separate verification of radius of convergence or distributional topology is supplied, so the claimed infinite linear combination of Dirac derivatives reduces to re-application of the prior identities. This is self-citation load-bearing on the central claim; other sections (shifted representations, Mittag-Leffler polynomials) inherit the same foundation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper rests on standard properties of Fourier transforms and distributions plus an extension assumption for the generating-function technique; it introduces one new entity (a class of fractional spline polynomials) without external validation.

axioms (2)

standard math Fourier transforms and distributional derivatives apply to fractional B-splines
Required for the claimed expansion and Dirac-delta representation.
domain assumption The generating function approach from Simsek [24] extends to fractional B-splines
Explicitly used to derive the novel Fourier expansion.

invented entities (1)

new class of fractional spline polynomials no independent evidence
purpose: Extend spline theory within the unified framework
Introduced alongside the Mittag-Leffler generating function; no independent evidence supplied in the abstract.

pith-pipeline@v0.9.0 · 5691 in / 1527 out tokens · 89580 ms · 2026-05-19T16:15:52.604929+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Employing a generating function approach inspired by recent results of Simsek [24], we derive a novel Fourier type expansion for fractional B splines that involves generalized Stirling type numbers... B_α(x) = ∑ c_n δ^{(n)}(x)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat embedding and orbit structure unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 5.23... generating function involving Mittag-Leffler function E_{1,α+1}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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Some new identities and inequalities for Bernoulli polynomials and num- bers of higher order related to the Stirling and Catalan numbers

G¨ un, D., Simsek, Y. Some new identities and inequalities for Bernoulli polynomials and num- bers of higher order related to the Stirling and Catalan numbers. RACSAM 114, 167 (2020). https://doi.org/10.1007/s13398-020-00899-z

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Karlin, C

S. Karlin, C. A. Micchelli, A. Pinkus, I. J. Schoenberg, Studies in Spline Functions and Approxima- tion Theory, Academic Press, 1976

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P. I. Lizorkin, Generalized Liouville differentiation and the function spacesL r p(E), Mat. Sb. 60 (1963) 325–353

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Q. M. Luo, H. M. Srivastava, Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011) 5702–5728

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Massopust, Interpolation and Approximation with Splines and Fractals, Oxford University Press, 2012

P. Massopust, Interpolation and Approximation with Splines and Fractals, Oxford University Press, 2012

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Massopust, Splines and differential operators, Int

P. Massopust, Splines and differential operators, Int. J. Wavelets Multiresolut. Inf. Process. 20(3) (2022), 17pp

work page 2022
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Podlubny, Fractional Differential Equations, Academic Press, 1999

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S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, 1993. FOURIER REPRESENTATIONS OF FRACTIONAL B-SPLINES VIA GENERALIZED STIRLING-TYPE POLYNOMIALS 17

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I. J. Schoenberg, On spline functions and their limits: The P´ olya distribution functions, Bull. Amer. Math. Soc. 53 (1947) 1114

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Schwartz, Th´ eorie des distributions, Hermann, 1966

L. Schwartz, Th´ eorie des distributions, Hermann, 1966

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Simsek, On q-deformed Stirling numbers

Y. Simsek, On q-deformed Stirling numbers. Int. J.Math. Comput. 15 (2), 1–11 (2012)

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Simsek, Identities on combinatorial numbers and polynomials, Fixed Point Theory Appl

Y. Simsek, Identities on combinatorial numbers and polynomials, Fixed Point Theory Appl. 2013 (2013) 87

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H. M. Srivastava, J. Choi, Zeta andq-Zeta Functions and Associated Series and Integrals, Elsevier, Amsterdam, 2012

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Unser, T

M. Unser, T. Blu, Fractional splines and wavelets, SIAM Rev. 42 (1) (2000) 43–67

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Unser, Sampling – 50 years after Shannon, Proc

M. Unser, Sampling – 50 years after Shannon, Proc. IEEE 88 (2000) 569–587

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[28]

A. H. Zemanian, Distribution Theory and Transform Analysis, Dover Publications, 1987. Department of Mathematics, Akdeniz University, Antalya, T ¨urkiye Email address:damlagun@akdeniz.edu.tr Department of Mathematics, Technical University of Munich, Garching b. Munich, Germany Email address:massopust@ma.tum.de Department of Mathematics, Akdeniz University,...

work page 1987

[1] [1]

T. Agoh, K. Dilcher, Shortened recurrence relations for Bernoulli numbers, Discrete Math. 309 (2009) 887–898

work page 2009

[2] [2]

Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel, Dordrecht, 1974 (translated from French by J

L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel, Dordrecht, 1974 (translated from French by J. W. Nienhuys)

work page 1974

[3] [3]

Akhiezer, ¨Uber die beste Ann¨ aherung einer Klasse stetiger periodischer Funktionen, Dokl

N. Akhiezer, ¨Uber die beste Ann¨ aherung einer Klasse stetiger periodischer Funktionen, Dokl. Akad. Nauk SSSR 17 (1937) 455–457

work page 1937

[4] [4]

T. Blu, M. Unser, Approximation error for quasi-interpolators and (multi-)wavelet expansions, Appl. Comput. Harmon. Anal. 6 (2000) 219–251

work page 2000

[5] [5]

Dana-Picard, Integral presentations of Catalan numbers and Wallis formula, Int

T. Dana-Picard, Integral presentations of Catalan numbers and Wallis formula, Int. J. Math. Educ. Sci. Technol. 42 (1) (2011) 122–129

work page 2011

[6] [6]

Dana-Picard, Integral presentations of Catalan numbers, Int

T. Dana-Picard, Integral presentations of Catalan numbers, Int. J. Math. Educ. Sci. Technol. 41 (1) (2009) 63–69

work page 2009

[7] [7]

de Boor, A Practical Guide to Splines, Springer Verlag, 1978

C. de Boor, A Practical Guide to Splines, Springer Verlag, 1978

work page 1978

[8] [8]

de Boor, Splines as linear combinations of B-splines, in: Approximation Theory II, Academic Press, 1976, pp

C. de Boor, Splines as linear combinations of B-splines, in: Approximation Theory II, Academic Press, 1976, pp. 1–47

work page 1976

[9] [9]

Forster, M

B. Forster, M. Unser, T. Blu, Complex B-splines, Appl. Comput. Harmon. Anal. 20 (2006) 261–282

work page 2006

[10] [10]

I. M. Gel’fand, G. E. Shilov, Generalized Functions, Vol. 1, Academic Press, 1959

work page 1959

[11] [11]

Some new identities and inequalities for Bernoulli polynomials and num- bers of higher order related to the Stirling and Catalan numbers

G¨ un, D., Simsek, Y. Some new identities and inequalities for Bernoulli polynomials and num- bers of higher order related to the Stirling and Catalan numbers. RACSAM 114, 167 (2020). https://doi.org/10.1007/s13398-020-00899-z

work page doi:10.1007/s13398-020-00899-z 2020

[12] [12]

Karlin, C

S. Karlin, C. A. Micchelli, A. Pinkus, I. J. Schoenberg, Studies in Spline Functions and Approxima- tion Theory, Academic Press, 1976

work page 1976

[13] [13]

M. G. Krein, Sur quelques points de la th´ eorie de la meilleure approximation des fonctions p´ eriodiques, Dokl. Akad. Nauk SSSR 18 (1938) 245–249

work page 1938

[14] [14]

P. I. Lizorkin, Generalized Liouville differentiation and the function spacesL r p(E), Mat. Sb. 60 (1963) 325–353

work page 1963

[15] [15]

Q. M. Luo, H. M. Srivastava, Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011) 5702–5728

work page 2011

[16] [16]

Massopust, Interpolation and Approximation with Splines and Fractals, Oxford University Press, 2012

P. Massopust, Interpolation and Approximation with Splines and Fractals, Oxford University Press, 2012

work page 2012

[17] [17]

Massopust, Splines and differential operators, Int

P. Massopust, Splines and differential operators, Int. J. Wavelets Multiresolut. Inf. Process. 20(3) (2022), 17pp

work page 2022

[18] [18]

Podlubny, Fractional Differential Equations, Academic Press, 1999

I. Podlubny, Fractional Differential Equations, Academic Press, 1999

work page 1999

[19] [19]

Roman, An Introduction to Catalan Numbers, Birkh¨ auser, Boston, 2015

S. Roman, An Introduction to Catalan Numbers, Birkh¨ auser, Boston, 2015

work page 2015

[20] [20]

S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, 1993. FOURIER REPRESENTATIONS OF FRACTIONAL B-SPLINES VIA GENERALIZED STIRLING-TYPE POLYNOMIALS 17

work page 1993

[21] [21]

I. J. Schoenberg, On spline functions and their limits: The P´ olya distribution functions, Bull. Amer. Math. Soc. 53 (1947) 1114

work page 1947

[22] [22]

Schwartz, Th´ eorie des distributions, Hermann, 1966

L. Schwartz, Th´ eorie des distributions, Hermann, 1966

work page 1966

[23] [23]

Simsek, On q-deformed Stirling numbers

Y. Simsek, On q-deformed Stirling numbers. Int. J.Math. Comput. 15 (2), 1–11 (2012)

work page 2012

[24] [24]

Simsek, Identities on combinatorial numbers and polynomials, Fixed Point Theory Appl

Y. Simsek, Identities on combinatorial numbers and polynomials, Fixed Point Theory Appl. 2013 (2013) 87

work page 2013

[25] [25]

H. M. Srivastava, J. Choi, Zeta andq-Zeta Functions and Associated Series and Integrals, Elsevier, Amsterdam, 2012

work page 2012

[26] [26]

Unser, T

M. Unser, T. Blu, Fractional splines and wavelets, SIAM Rev. 42 (1) (2000) 43–67

work page 2000

[27] [27]

Unser, Sampling – 50 years after Shannon, Proc

M. Unser, Sampling – 50 years after Shannon, Proc. IEEE 88 (2000) 569–587

work page 2000

[28] [28]

A. H. Zemanian, Distribution Theory and Transform Analysis, Dover Publications, 1987. Department of Mathematics, Akdeniz University, Antalya, T ¨urkiye Email address:damlagun@akdeniz.edu.tr Department of Mathematics, Technical University of Munich, Garching b. Munich, Germany Email address:massopust@ma.tum.de Department of Mathematics, Akdeniz University,...

work page 1987