On approximation properties by exponential type of Bernstein-Stancu Operators

Abstract

In the paper, we introduced a generalization of Bernstein-Stancu-Kantorovich operators that reproduces exponential functions. For appropriate function spaces, both the uniform and L^p convergence have been established. We proved that the new operators satisfy the Korovkin tests with the exponential functions and calculated the operators’ analytical expressions evaluated on various powers of e ^μxwhich is necessary to get the uniform convergence conclusion using the well-known Korovkin Theorem. Consequently, the convergence theorem for the new operators, which transfer the weighted space L_μ^p ([0,1]) to itself, has been established. Additionally, using the usual modulus of continuity of the estimated function in the continuous case, we provide quantitative estimates for the approximation error.

Keywords

• Bernstein-Kantorovich operators
• Exponential polynomials
• Modulus of continuity.

Üstel tip Bernstein-Stancu Operatörlerinin yaklaşım özellikleri üzerine

Abstract

Bu çalışmada üstel fonksiyonları yeniden üreten Bernstein-Stancu-Kantorovich operatörlerinin bir genellemesi sunulmuştur. Uygun fonksiyon uzayları için hem düzgün hem de L^p yakınsaması kurulmuştur. Yeni operatörlerin üstel fonksiyonu sağladığını kanıtladık ve iyi bilinen Korovkin Teoremini kullanarak düzgün yakınsaklık sonucunu elde etmek için gerekli olan e ^μxin çeşitli kuvvetlerine göre değerlendirilen operatörlerin analitik ifadelerini hesapladık. Sonuç olarak L_μ^p ([0,1]) ağırlıklı uzayını kendisine aktaran yeni operatörler için yakınsama teoremi kurulmuştur. Ek olarak, sürekli durumda tahmin edilen fonksiyonun olağan süreklilik modülünü kullanarak, yaklaşık hatası için niceliksel tahminler verilmiştir.

Keywords

• Bernstein-Kantorovich operatörleri
• Üstel polinomlar
• Süreklilik modülü

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