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All Journal Journal of the Indonesian Mathematical Society Jambura Journal of Mathematics Jurnal Ilmiah Matematika dan Pendidikan Matematika (JMP) Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi Pattimura Proceeding : Conference of Science and Technology Hilbert Journal of Mathematical Analysis
Hakim, Denny Ivanal
Institut Teknologi Bandung
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Rate of convergence of Kantorovich operator sequences near L1([0, 1]) Abdul Karim Munir Aszari; Denny Ivanal Hakim
Hilbert Journal of Mathematical Analysis Vol. 3 No. 1 (2024): Hilbert J. Math. Anal.
Publisher : KOMUNITAS Analisis Matematika INDONESIA

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.62918/hjma.v3i1.37

Abstract

The study of the rate of convergence of Kantorovich operator sequences has predominantly focused on the Lp spaces for 1< p<infty, yet the behaviour near L1([0,1]) remains less understood, particularly as p approaches 1. To bridge this gap, we investigate the rate of convergence within the framework of the grand Lebesgue spaces Lp ([0,1]), which encompass all Lp ([0,1]) spaces for 1<p<infty but remain a subset of L1([0,1]). Our approach leverages the intrinsic properties of Lp ([0,1]) to derive new results on the convergence rate of Kantorovich operator sequences. Specifically, our objective is to demonstrate that Kantorovich operators exhibit a significant rate of convergence within this broader context, thereby providing insights applicable to the boundary behavior as p to 1. We will then apply these findings to alpha-Holder continuous functions to further understand the convergence rate of Kantorovich operator sequences in these settings. This combined approach suggests that functions with derivatives in Lp ([0,1]) exhibit specific convergence rates under Kantorovich operators.
Inclusion between Bourgain-Morrey spaces and their weak types Pratama, Aria; Ivanal Hakim, Denny
Hilbert Journal of Mathematical Analysis Vol. 3 No. 2 (2025): Hilbert J. Math. Anal.
Publisher : KOMUNITAS Analisis Matematika INDONESIA

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.62918/hjma.v3i2.35

Abstract

This paper aims to explore the embedding relation between Bourgain-Morrey spaces and weak Bourgain-Morrey spaces. Bourgain-Morrey Spaces are important function spaces in harmonic analysis. In this article, we introduce weak Bourgain-Morrey spaces as certain generalization of weak Morrey spaces. We investigate the necessary and sufficient conditions for functions in weak Bourgain-Morrey spaces to also be elements in Bourgain-Morrey spaces, and vice versa. Our result are inclusion between weak Bourgain-Morrey spaces and inclusion of weak Bourgain- Morrey spaces into a certain Bourgain-Morrey spaces. These result generelize the inclusion result in Morrey spaces, weak Morrey spaces, and Bourgain- Morrey spaces.
Revisiting Kantorovich Operators in Lebesgue Spaces Obie, Maximillian Ventura; Taebenu, Erick Angga; Gunadi, Reinhart; Hakim, Denny Ivanal
Journal of the Indonesian Mathematical Society VOLUME 29 NUMBER 3 (NOVEMBER 2023)
Publisher : IndoMS

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jims.29.3.1596.289-298

Abstract

According to the Weierstrass Approximation Theorem, any continuous function on the closed and bounded interval can be approximated by polynomials. A constructive proof of this theorem uses the so-called Bernstein polynomials. For the approximation of integrable functions, we may consider Kantorovich operators as certain modifications for Bernstein polynomials. In this paper, we investigate the behaviour of Kantorovich operators in Lebesgue spaces. We first give an alternative proof of the uniform boundedness of Kantorovich operators in Lebesgue spaces by using the Riesz-Thorin Interpolation Theorem. In addition, we examine the convergence of Kantorovich operators in the space of essentially bounded functions. We also give an example related to the rate of convergence of Kantorovich operators in a subspace of Lebesgue spaces.
A note on some Endpoint Estimates of Commutators of Fractional Integral Operators Wijaya, Verrel Rievaldo; Hakim, Denny Ivanal; Setya Budhi, Marcus Wono
Journal of the Indonesian Mathematical Society VOLUME 29 NUMBER 3 (NOVEMBER 2023)
Publisher : IndoMS

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jims.29.3.1599.322-327

Abstract

It is known that fractional integral operators are not bounded from Lebesgue integrable functions to Lebesgue space for some particular related exponent. Based on some recent results by Schikorra, Spector, and Van Schaftingen, we investigate commutators of fractional integral operators on Lebesgue integrable functions. We establish a weak type estimates for these commutators generated by essentially bounded functions. Under the same assumption, we also prove that the norm of these commutators are dominated by the norm of the Riesz transform.
Pointwise Multipliers of Orlicz-Morrey Spaces Ifronika, Ifronika; Hakim, Denny Ivanal; Budhi, Wono Setya
Journal of the Indonesian Mathematical Society Vol. 31 No. 4 (2025): DECEMBER
Publisher : IndoMS

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jims.v31i4.1965

Abstract

We investigate the space of pointwise multipliers of Orlicz-Morrey spaces. Using the H\"older inequality in Orlicz-Morrey spaces, we prove that the space of pointwise multipliers of Orlicz-Morrey spaces contains an Orlicz-Morrey space. We also prove a partial reverse inclusion of this result. In addition, we des\-cribe the space of pointwise multipliers of Orlicz-Morrey spaces by adding some growth conditions on Young functions. Our results can be viewed as an extension of the results on pointwise multipliers of Morrey spaces.
Validasi Estimasi Batas Atas Luas Sofa dan Perhitungan Luas Sofa Gerver serta Analisis Mekanisme Pergerakan Sofa Hammersley Akbar, Muhammad Imam; Gumilang, Aji; Hakim, Denny Ivanal
Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi Volume 13 Issue 3 December 2025
Publisher : Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.37905/euler.v13i3.33457

Abstract

The Moving Sofa Problem concerns the planar shape of maximum area that can be moved around a right-angled corner in a two-dimensional hallway of unit width. The objectives of this study are (1) to validate the upper bound estimate of the sofa area obtained from the intersection of the straight corridor and the right-angled corridor models, (2) to analyze the movement mechanism of the Hammersley sofa through rotation paths and contact paths, and (3) to provide details of the Gerver sofa area calculation for numerical validation. The methods used include analysis of the function A(u, θ), which is the upper bound value function of the sofa area obtained from the area of the intersection of the straight corridor and the right-angled corridor, the application of the concepts of rotation paths, contact points, and contact paths to prove that the Hammersley sofa construction can pass through the right-angled corridor, and the calculation of the area using Green’s Theorem to validate the Gerver sofa area. The main results show that (1) the minimum upper bound of the function A(u, θ) reaches two times the square root of two under certain conditions, (2) the rotation path used proves that the Hammersley sofa satisfies the definition of a shape that can pass through a right-angled corridor, and (3) calculations using Green’s Theorem yield an area of approximately 2.2195 area units. The findings of this study clarify the geometric construction elements of the Hammersley and Gerver sofas, and provide validation details that have rarely been fully described before.
Co-Authors Abdul Karim Munir Aszari Akbar, Muhammad Imam Ali, Amrizal Marwan Amalina, Nur Dina Andri Setiawan Daniel Salim Gumilang, Aji Gunadi, Reinhart Hendra Gunawan Hendra Gunawan Ifronika, Ifronika Moch Taufik Hakiki Moch Taufik Hakiki Mu’afa Purwa Arsana Obie, Maximillian Ventura Oki Neswan Pratama, Aria Setya Budhi, Marcus Wono Taebenu, Erick Angga Wijaya, Verrel Rievaldo Wono Setya Budhi Wono Setya Budhi Yudatama, Rizma