<p>In a Hilbert space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{L}^{2}_{\alpha }(\mathbb {R}^{+})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mtext>L</mtext> <mi>α</mi> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha \ge -\frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, we study the translation operator associated with the linear canonical Fourier-Bessel transform. Based on this operator, we introduce the notion of a generalized modulus of smoothness in the space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{L}^{2}_{\alpha }(\mathbb {R}^{+})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mtext>L</mtext> <mi>α</mi> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The main result of this paper is the establishment of a new estimate theorem for the <i>K</i>-functional, yielding improved bounds in this framework. Furthermore, we prove analogues of Jackson’s and Bernstein’s inequalities adapted to the setting of the linear canonical Fourier-Bessel transform.</p>

Estimates of the K-functional and Jackson-type theorems for the linear canonical fourier-bessel transform

期刊论文
发表日期：

2026年5月8日

查看全文

M. El Hamma,
M. El Bouazizi,
H. Benlaajine

摘要

In a Hilbert space $\textrm{L}^{2}_{\alpha }(\mathbb {R}^{+})$ $L_{α}^{2} (R^{+})$ , with $\alpha \ge -\frac{1}{2}$ $α \geq - \frac{1}{2}$ , we study the translation operator associated with the linear canonical Fourier-Bessel transform. Based on this operator, we introduce the notion of a generalized modulus of smoothness in the space $\textrm{L}^{2}_{\alpha }(\mathbb {R}^{+})$ $L_{α}^{2} (R^{+})$ . The main result of this paper is the establishment of a new estimate theorem for the K-functional, yielding improved bounds in this framework. Furthermore, we prove analogues of Jackson’s and Bernstein’s inequalities adapted to the setting of the linear canonical Fourier-Bessel transform.